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Improved estimation of population mean in simple random sampling using information on auxiliary attribute
Applied Mathematics and Computation 218 (2012) 7798–7812
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Improved estimation of population mean in simple random sampling using information on auxiliary attribute Housila P. Singh, Ramkrishna S. Solanki ⇑ School of Studies in Statistics, Vikram University, Ujjain 456010, India
a r t i c l e
i n f o
Keywords: Auxiliary variable Study variable Proportion Bias Mean square error
a b s t r a c t This paper addresses the problem of estimating the population mean with known population proportion of an auxiliary variable. A class of estimators is defined which includes the estimators recently proposed by Shabbir and Gupta (2007) [10] and Abd-Elfattah et al. (2010) [1]. The usual unbiased estimator and Naik and Gupta (1996) [15] estimator are also the member of the proposed class of the estimators. The bias and mean square error (MSE) expressions of the proposed class are obtained up to first order of approximation. Asymptotically optimum estimator (AOE) in the class of estimators is identified alongwith its mean square error formula. The correct MSE and minimum MSE expressions of Shabbir and Gupta (2007) [10] estimator are also given. It has been shown that the proposed class of estimators is more efficient than the usual unbiased estimator, usual linear regression estimator and estimators/classes of estimators due to Naik and Gupta (1996) [15], Jhajj et al. (2008) [9], Shabbir and Gupta (2007) [10] estimator, Singh et al. (2008) [13] and Abd-Elfattah et al. (2010) [1]. The double sampling version of the proposed class of estimators is proposed alongwith its properties. Numerical illustrations are given in support of the present study. Ó 2012 Elsevier Inc. All rights reserved.
1. Introduction It is well known that precision of the estimates of the population mean or total of the study variable y can be considerably improved by the use of known information on an auxiliary variable x which is highly correlated with the study variable y. However in many practical situations, instead of existence of auxiliary variables there exist some auxiliary attributes / (say), which are highly correlated with the study variable y, such as. (i) Sex (/) and height of the persons (y), (ii) Amount of milk produced (y) and a particular breed of the cow (/), (iii) Amount of yield of wheat crop and a particular variety of wheat (/) etc. (see Jhajj et al. [9]). Consider a sample of size n drawn by simple random sampling without replacement (SRSWOR) from the population U of size N. Let yi and /i denote the observations on the study variable y and the auxiliary attribute /, respectively for the ith unit (i = 1, 2, . . . , N) of the population U. We note that /i[=1], if ith unit of the population possesses attribute / and /i[=0], otherh P i P wise (i = 1, 2, . . . , N). Let A ¼ Ni¼1 /i and a ¼ ni¼1 /i denote the total number of units in the population U and sample
⇑ Corresponding author. E-mail addresses: [email protected] (H.P. Singh), [email protected] (R.S. Solanki). 0096-3003/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2012.01.047
H.P. Singh, R.S. Solanki / Applied Mathematics and Computation 218 (2012) 7798–7812
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b ða=nÞ denote proportion of units in the population U and sample respectively possessing attribute /. Let P[=(A/N)] and P½¼ respectively possessing attribute /. Taking into consideration the point biserial correlation coefficient between auxiliary attribute / and the study variable y, h i P Naik and Gupta [15] defined the ratio estimator for population mean Y ¼ N 1 Ni¼1 yi of the study variable y (when the prior information of population proportion of units, possessing the same attribute is available) as follows
b ðP= PÞ; t1 ¼ y
ð1:1Þ
P ¼ n1 ni¼1 yi is the sample mean of the study variable y. where y The mean square error (MSE) of the ratio estimator t1, up to first order of approximation is given by
h i MSEðt 1 Þ ¼ fð1 f Þ=ngY 2 C 2y þ C 2P ð1 2kP Þ ;
ð1:2Þ
where,
f ¼ ðn=NÞ;
C 2y ¼ S2y =Y 2 ;
S2y ¼ ðN 1Þ1
N X ðyi YÞ2 ;
C 2P ¼ S2/ =P2 ;
i¼1
S2/
¼ ðN 1Þ
1
N X ð/i PÞ2 ;
kP ¼ ½ðqPb C y Þ=C P ;
qPb ¼ ½Sy/ =ðSy S/ Þ;
i¼1
Sy/ ¼ ðN 1Þ1
N X ðyi YÞð/i PÞ: i¼1
(sample mean) is It is well known under SRSWOR that the variance of the usual unbiased estimator y
Þ ¼ fð1 f Þ=ngY 2 C 2y : Varðy
ð1:3Þ
as Jhajj et al. [9] have suggested the class of estimators for the population mean Y
; uÞ; t2 ¼ hðy b ; uÞ is the function of ðy ; uÞ such that [hðy ; 1Þ ¼ Y; 8Y] and the function hðy ; uÞ satisfies certain reguwhere u½¼ P=P and hðy larity conditions similar to those given in Jhaji et al. [[9], Sec. 3, p. 45]. The minimum MSE of the class of estimators t2, up to first order of approximation is given by
MSEmin ðt 2 Þfð1 f Þ=ngS2y 1 q2Pb ;
ð1:4Þ
which equals to the variance [or MSE] of the usual linear regression estimator defined as
b ^/ ðP PÞ; þb tReg ¼ ½y
ð1:5Þ
^/ ¼ sy/ =s2 is the sample regression coefficient [or sample estimate of population regression coefficient where b / P P b/ ¼ Sy/ =S2/ ], with s2/ ¼ ðn 1Þ1 ni¼1 ð/i PÞ2 and sy/ ¼ ðn 1Þ1 ni¼1 ðyi YÞð/i PÞ (i.e. MSEmin(t2) = MSE(tReg)). Motivated by Ray and Singh [14] and Roy [6], Shabbir and Gupta [10] suggested the ratio-type estimator for the population mean Y as
b b ½d1 þ d2 ðP PÞðP= t3 ¼ y PÞ;
ð1:6Þ
where (d1, d2) are suitably chosen constants whose sum need not be unity. In case where, the population proportion P of the auxiliary attribute / is unknown, Jhajj et al. [9] and Shabbir and Gupta [10] have suggested the two phase sampling version of their estimators t2 and t3 with properties. It is to be mentioned that the mean square error of the ratio-type estimatort3, to the first order of approximation obtained by Shabbir and Gupta [10] is not correct in single as well as in two phase sampling. Singh et al. [13] suggested the ratio-type estimators for estimating the population mean Y of the study variable y in simple random sampling using known parameters of the auxiliary attribute /, such as, coefficient of variation (Cp), coefficient of kurtosis (b2(/)) and point biserial correlation coefficient (qpb) as,
b b ^/ ðP PÞgðP= þb ts1 ¼ fy PÞ;
ð1:7Þ
ts2
b b þ b ð/Þg; ^/ ðP PÞg½fP þb ¼ fy þ b2 ð/Þg=f P 2
ð1:8Þ
ts3
b b þ C P Þg; ^/ ðP PÞgfðP þb ¼ fy þ C P Þ=ð P
ð1:9Þ
ts4
b b ^/ ðP PÞg½fb þb ¼ fy 2 ð/ÞP þ C P g=fb2 ð/Þ P þ C P g;
ð1:10Þ
ts5
b b ^/ ðP PÞg½fC þb ¼ fy P P þ b2 ð/Þg=fC P P þ b2 ð/Þg:
ð1:11Þ
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Motivated by Kadilar and Cingi [4], one may also define the following ratio-type estimators for the population mean Y as
b b ^/ ðP PÞgfðC þb ts6 ¼ fy P P þ qPb Þ=ðC P P þ qPb Þg;
ð1:12Þ
b ^/ ðP PÞgfð þb qPb P þ C P Þ=ðqPb Pb þ C P Þg; ts7 ¼ fy
ð1:13Þ
b b ^/ ðP PÞg½fb þb ts8 ¼ fy 2 ð/ÞP þ qPb g=fb2 ð/Þ P þ qPb g;
ð1:14Þ
b ^/ ðP PÞg½f þb ts9 ¼ fy qPb P þ b2 ð/Þg=fqPb Pb þ b2 ð/Þg;
ð1:15Þ
b b þ q Þg: ^/ ðP PÞgfðP þb ts10 ¼ fy þ qPb Þ=ð P Pb
ð1:16Þ
where CP and b2(/) are the population coefficient of variation and the population coefficient of kurtosis of auxiliary attribute /, respectively. To the first order of approximation, the MSEs of the ratio-type estimators tsi, are given by
h i MSEðt si Þ ¼ fð1 f Þ=ng R2i S2/ þ S2y 1 q2Pb ;
ði ¼ 1; 2; . . . ; 10Þ;
ð1:17Þ
where
R1 ¼ ðY=PÞ;
R2 ¼ ½Y=fP þ b2 ð/Þg;
R4 ¼ ½fYb2 ð/Þg=fb2 ð/ÞP þ C P g; R6 ¼ ½ðYC P Þ=ðC P P þ qPb Þ;
R3 ¼ ½Y=ðP þ C p Þ;
R5 ¼ ½ðYC P Þ=fC P P þ b2 ð/Þg;
R7 ¼ ½ðY qPb Þ=ðqPb C P P þ C P Þ;
R8 ¼ ½fYb2 ð/Þg=fb2 ð/ÞP þ qPb g;
R9 ¼ ½ðY qPb Þ=fqPb P þ b2 ð/Þg;
R10 ¼ ½Y=ðP þ qPb Þ: Keeping the form of the ratio-type estimators tsi, (i = 1, 2, . . . , 10); one may suggest the class of estimators for population mean Y as
b b þ gÞg; ^/ ðP PÞfðwP þb t 4 ¼ ½y þ gÞðw P
ð1:18Þ
where w and g are either real numbers or function of known parameters of the auxiliary attribute / such as CP, b2(/) and qPb. The sum of w and g is not necessarily equal to unity. Note that the ratio-type estimators tsi, (i = 1, 2, . . . , 10); are members of the class of estimators t4 for different values of w and g. To the first order of approximation, the MSE of the class of estimators t4 is given by
h i MSEðt 4 Þ ¼ fð1 f Þ=ng R S2/ þ S2y 1 q2Pb ;
ð1:19Þ
where R ¼ ½ðwYÞ=ðwP þ gÞ. The MSE of the class of estimators t4 is minimum, when R⁄[=0]. Thus the resulting minimum MSE of t4 is given by
MSEmin ðt4 Þ ¼ fð1 f Þ=ngS2y 1 q2Pb ;
ð1:20Þ
which is equal to the variance (or MSE) of the usual linear regression estimator tReg (i.e. MSEmin(t4) = MSE(tReg)). Motivated by Kadilar and Cingi [5], Abd-Elfattah et al. [1] have suggested the following classes of estimators for the population mean Yusing the information on coefficient of kurtosis (b2(/)) and coefficient of variation (CP), respectively as
( ) ( ) # b b ^/ ðP PÞ ^/ ðP PÞ þb þb y y fP þ b2 ð/Þg ; P þ m2 b b þ b ð/Þg P fP 2 " ( # ( ) ) b b ^/ ðP PÞ ^/ ðP PÞ þb þb y y ¼ m1 ðP þ C P Þ ; P þ m2 b b þ b ð/Þg P fP 2 ( ) ) " ( # b b ^/ ðP PÞ ^/ ðP PÞ þb þb y y fb2 ð/ÞP þ C P g ; P þ m2 ¼ m1 b þ CP g b P fb2 ð/Þ P " ( # ( ) ) b b ^/ ðP PÞ ^/ ðP PÞ þb þb y y ¼ m1 fC P P þ b2 ð/Þg ; P þ m2 b þ b ð/Þg b P fC P P "
ta1 ¼ m1
ð1:21Þ
t a2
ð1:22Þ
t a3 t a4
ð1:23Þ
ð1:24Þ
2
where (m1, m2) are weights that satisfy the condition [(m1 + m2) = 1]. To the first order of approximation, the common minimum MSE of the classes of estimators taj is given by
MSEmin ðtaj Þ ¼ fð1 f Þ=ngS2y 1 q2Pb ;
ðj ¼ 1; 2; 3; 4Þ ¼ MSEðt Reg Þ:
ð1:25Þ
Further following the procedure outlined in Singh and Agnihotri [[7], p. 77], Abd-Elfattah et al. [1] suggested the ratio-type class of estimators for population mean Y as
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b þ gÞg; fðwP þ gÞ=ðw P t a5 ¼ y
ð1:26Þ
where w and g are either real numbers or function of known parameters of the auxiliary attribute / such as CP, b2(/) and qPb. Note that the sum of w and g is not necessarily equal to unity. The following ratio-type estimators
b þ C P Þg; fðP þ C P Þ=ð P ta5ð1Þ ¼ y
ð1:27Þ
b þ b ð/Þg; ½fP þ b2 ð/Þg=f P ¼y 2
ð1:28Þ
b þ C P g; ½fb2 ð/ÞP þ C P g=fb2 ð/Þ P ta5ð3Þ ¼ y
ð1:29Þ
b þ b ð/Þg; ½fC P P þ b2 ð/Þg=fC P P ta5ð4Þ ¼ y 2
ð1:30Þ
b þ q Þg; fðP þ qPb Þ=ð P ta5ð5Þ ¼ y Pb
ð1:31Þ
ta5ð2Þ
are members of the ratio-type class of estimators t a5 for (w, g)[=(1, CP), {1, b2(/)}, {b 2(/), CP}, {CP, b2(/)}, (1, qPb)] respectively. Some more ratio-type estimators on the lines of Kadilar and Cingi [5] can be generated from the ratio-type class of estimators ta5 for (w, g)[=(CP, qPb), (qPb, CP), {b2(/), qPb}, {qPb, b2(/)}], respectively as
b þ q Þg; fðC P P þ qPb Þ=ðC P P ta5ð6Þ ¼ y Pb
ð1:32Þ
ta5ð7Þ
b þ C P Þg; fðqPb P þ C P Þ=ðqPb P ¼y
ð1:33Þ
ta5ð8Þ
b þ q g; ½fb2 ð/ÞP þ qPb g=fb2 ð/Þ P ¼y Pb
ð1:34Þ
ta5ð9Þ
b þ b ð/Þg: ½fqPb P þ b2 ð/Þg=fqPb P ¼y 2
ð1:35Þ
To the first order of approximation, the MSEs of the ratio-type estimators ta5ðjÞ are given by
h i MSEft a5ðjÞ g ¼ fð1 f Þ=ngY 2 C 2y þ C 2P mj mj 2kP ;
ðj ¼ 1; 2; . . . ; 9Þ;
ð1:36Þ
where
m1 ¼ ½P=ðP þ C p Þ; m2 ¼ ½P=fP þ b2 ð/Þg; m3 ¼ ½b2 ð/ÞP=fb2 ð/ÞP þ C P g; m4 ¼ ½C P P=fC P P þ b2 ð/Þg; m5 ¼ ½P=ðP þ qPb Þ; m6 ¼ ½ðC P PÞ=ðC P P þ qPb Þ; m7 ¼ ½ðqPb PÞ=ðqPb P þ C P Þ; m8 ¼ ½fb2 ð/ÞPg=fb2 ð/ÞP þ qPb g; m9 ¼ ½ðqPb PÞ=fqPb P þ b2 ð/Þg The relevant references in this context are Sisodia and Dwivedi [2], Singh and Tailor [8], Upadhyaya and Singh [11] and Singh and Agnihotri [7]. To the first order of approximation, the minimum MSE of the ratio-type class of estimators t a5 is given by
MSEmin ðt a5 Þ ¼ fð1 f Þ=ngS2y 1 q2Pb ¼ MSEðt Reg Þ:
ð1:37Þ
From (1.2), (1.3), (1.4), (1.17), (1.20), (1.25) and (1.37) we have
Þ MSEðt Reg Þ½¼ MSEmin ðt2 Þ ¼ MSEmin ðt4 Þg ¼ MSEmin ðt aj Þ ¼ fð1 f Þ=ngY 2 C 2y q2Pb P 0; Varðy
ð1:38Þ 2
2
MSEðt 1 Þ MSEðt Reg Þ½¼ MSEmin ðt 2 Þ ¼ MSEmin ðt 4 Þg ¼ MSEmin ðt aj Þ ¼ fð1 f Þ=ngY ðC P qPb C y Þ P 0;
ð1:39Þ
2
MSEðt si Þ MSEðtReg Þ½¼ MSEmin ðt 2 Þ ¼ MSEmin ðt 4 Þg ¼ MSEmin ðt aj Þ ¼ fð1 f Þ=ngðRi S/ Þ P 0; ði ¼ 1; 2; . . . ; 10; j ¼ 1; 2; . . . ; 5Þ:
ð1:40Þ
, ratio estimator t1 due to Naik and Gupta [15] It is observed from (1.38), (1.39) and (1.40) that the usual unbiased estimator y and ratio-type estimators tsi, (i = 1, 2, . . . , 10) due to Singh et al. [13] are inferior to the usual linear regression estimator tReg, class of estimators t2 (at their optimum condition) due to Jhajj et al. [9], class of estimators t4 (at their optimum condition) and classes of estimators t aj (at their optimum conditions), (j = 1, 2, . . . , 5) due to Abd-Elfattah et al. [1]. Thus the procedure adopted by the authors Naik and Gupta [15], Jhajj et al. [9], Singh et al. [13] and Abd-Elfattah et al. [1] are unable to give estimators better than the usual linear regression estimators tReg. It poses a question. Is there any procedure which can give estimator better than the usual linear regression estimator tReg? Indeed Shabbir and Gupta [10] have given the procedure to answer this question. But the derivation of the result concerning MSE expression of the ratio-type estimator t3 proposed by them is not correct. So the entire results of the paper by Shabbir and Gupta [10] are erroneous. In this paper we have made an effort to propose the class of estimators for the population mean Y which includes the ratio-type estimator t3 envisaged by Shabbir and Gupta [10] and Abd-Elfattah et al. [1] class of estimators t a5 . We have obtained the bias and MSE expressions of the proposed class of estimators, up to first order of approximation. Asymptotic optimum estimator (AOE) in the proposed class of estimators is identified alongwith its MSE formula. The correct MSE expression of Shabbir and Gupta [10] ratio-type estimator t3 is also given. Two phase sampling version of the proposed class of estimators is also given alongwith its properties. Numerical illustration is given in support of the present study.
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2. The suggested class of estimators We define the following class of estimators for population mean Y as
!a wP þ dg b ½d1 þ d2 ðP PÞ t¼y ; b þ dg wP
ð2:1Þ
where w and g are either real numbers or function of known parameters of the auxiliary attribute /, such as Cp, b2(/) qPb and kP. The scalar a takes values 1, (for product-type estimator) and +1 (for ratio-type estimator); d is an integer which takes b þ dgÞ and (wP + dg) are non-negative and (d1, d2) are suitably values +1 and 1 for designing the estimators such that ðw P chosen constants such that the sum of the constants (d1, d2) need not be unity (i.e. (d1 + d2) – 1). Motivation of proposing the class of estimators t is taking from Upadhyaya and Singh [11], Kadilar and Cingi [3–5] and Shabbir and Gupta [10]. The class of estimators t reduces to following set of known estimators/classes of estimators for different values of constants (d1, d2, w, g, d, a): (The usual unbiased estimator), (i) for ðd1 ; d2 ; aÞ ¼ ð1; 0; 0Þ; t ! y (ii) for (w, g, a, d) = (1, 0, 1, 1), t ? t3 [10], (iii) for ðd1 ; d2 ; a; dÞ ¼ ð1; 0; 1; 1Þ; t ! ta5 [1]. To obtain the bias and MSE of suggested class of estimators t, we write
¼ Yð1 þ e0 Þ; y b ¼ Pð1 þ e1 Þ; P such that
Eðe0 Þ ¼ Eðe1 Þ ¼ 0 and up to first order of approximation, we have
E e20 ¼ fð1 f Þ=ngC 2y ; E e21 ¼ fð1 f Þ=ngC 2P ; Eðe0 e1 Þ ¼ fð1 f Þ=ngqPb C P C y : Expressing the suggested class of estimators t in terms of e’s, we have
t ¼ Yð1 þ e0 Þðd1 d2 Pe1 Þð1 þ me1 Þa ;
ð2:2Þ
where m = [(wP)/(wP + dg)]. We assume that jme1j < 1, so that (1 + me1)a is expandable. Expanding the right hand side of (2.2), multiplying out and neglecting terms of e’s having power greater than two, we have
ða þ 1Þ t ffi Y d1 ð1 þ e0 Þ ðd1 am þ d2 PÞðe1 þ e0 e1 Þ þ am d1 m þ d2 P e21 2 or
ða þ 1Þ ðt YÞ ffi Y d1 ð1 þ e0 Þ ðd1 am þ d2 PÞðe1 þ e0 e1 Þ þ am d1 m þ d2 P e21 1 : 2
ð2:3Þ
Taking expectation on both sides of (2.3), we get the bias of the suggested class of estimators t to the first order of approximation as
ð1 f Þ 2 d1 ða þ 1Þ C P am BðtÞ ¼ Y ðd1 1Þ þ m þ d2 P ðd1 am þ d2 PÞkP : n 2
ð2:4Þ
Squaring both sides of (2.3) and neglecting terms of e’s having power greater than two, we have
h 2 ðt YÞ2 ffi Y 2 1 þ d1 ð1 þ e0 Þ2 þ ðamd1 þ d2 PÞ2 e21 2d1 ð1 þ e0 Þþ2ðd1 am þ d2 PÞðe1 þ e0 e1 Þ d1 ða þ 1Þ d ða þ 1Þ 2am m þ d2 P e21 2d1 ðamd1 þ d2 PÞðe1 þ 2e0 e1 Þ þ 2amd1 1 m þ d2 P e21 2 2 n o h 2 2 ¼ Y 2 1 þ d1 1 þ 2e0 2ame1 þ e20 þ að2a þ 1Þm2 e21 4ame0 e1 þd2 P 2 e21 2d1 d2 Pðe1 þ 2e0 e1 Þ 2amPe21 aða þ 1Þ 2 2 2d1 1 þ e0 amðe1 þ e0 e1 Þ þ m e1 2d2 P e1 þ e0 e1 ame21 : ð2:5Þ 2
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Taking expectation of both sides of (2.5), we get the MSE of the suggested class of estimators t to the first order of approximation as
h i 2 2 MSEðtÞ ¼ Y 2 1 þ d1 A þ d2 B þ 2d1 d2 C 2d1 D 2d2 E ;
ð2:6Þ
where
h n oi A ¼ 1 þ fð1 f Þ=ng C 2y þ amC 2P ðð2a þ 1Þm 4kP Þ ; B ¼ fð1 f Þ=ngP2 C 2P ; C ¼ f2ð1 f Þ=ngPC 2P ðam kP Þ; h i D ¼ 1 þ fð1 f Þ=ngamC 2P ffða þ 1Þ=2gm kP g ; E ¼ fð1 f Þ=ngPC 2P ðam kP Þ: The MSE of the suggested class of estimators t is minimized for
d1 ¼ ½ðBD CEÞ=ðAB C 2 Þ ¼ d1 ;
ðsayÞ;
d2 ;
ðsayÞ:
2
d2 ¼ ½ðAE CDÞ=ðAB C Þ ¼
Thus the resulting minimum MSE of class t is obtained as
" MSEmin ðtÞ ¼ Y 2 1
ðBD2 2CDE þ AE2 Þ ðAB C 2 Þ
# ð2:7Þ
:
Thus we established the following theorem. Theorem 2.1. To the first order of approximation,
"
MSEðtÞ P Y
2
1
ðBD2 2CDE þ AE2 Þ
#
ðAB C 2 Þ
with equality holding if
d1 ¼ d1
and d2 ¼ d2 :
Remark 2.1. For d1 = 1, the suggested class of estimators t reduces to the class of estimators as
b ½1 þ d2 ðP PÞ t ¼ y
wP þ dg b þ dg wP
!a
ð2:8Þ
:
Putting d1 = 1 in (2.5) and (2.7), we get the bias and MSE of the class of estimators t⁄, respectively as
ð1 f Þ 2 ða þ 1Þm YC P am kP þ Pd2 ðam kP Þ ; n 2 h i 2 2 MSEðt Þ ¼ Y 1 þ A 2D þ d2 B þ 2d2 ðC EÞ ;
Bðt Þ ¼
ð2:9Þ ð2:10Þ
where
h i ð1 þ A 2DÞ ¼ fð1 f Þ=ng C 2y þ amC 2P ðam 2kP Þ ; ðC EÞ ¼ fð1 f Þ=ngPC 2p ðam kP Þ: Minimization of MSE(t⁄) with respect to d2, yields the optimum values of d2 as
d2 ¼ fðC EÞ=Bg ¼ ½ðam kP Þ=P ¼ d ;
ðsayÞ:
⁄
Thus the resulting minimum MSE of class t is obtained as
MSEmin ðt Þ ¼ fð1 f Þ=ngS2y 1 q2Pb ¼ MSEðtReg Þ: Thus we state the following theorem. Theorem 2.2. To the first order of approximation,
MSEðt Þ P fð1 f Þ=ngS2y 1 q2Pb
ð2:11Þ
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with equality holding if
d2 ¼ d : Remark 2.2. For [(w, g, a, d) = (1, 0, 1, 1)] in (2.4) and (2.6), we get the bias and MSE expressions of the Shabbir and Gupta [10] ratio-type estimator t3 to the first order of approximation, respectively as
Bðt 3 Þ ¼ Y½ðd1 1Þ þ fð1 f Þ=ngC 2P ðd1 þ d2 PÞð1 kP Þ; h i 2 2 MSEðt 3 Þ ¼ Y 2 1 þ d1 A þ d2 B þ 2d1 d2 C 2d1 D 2d2 E ;
ð2:12Þ ð2:13Þ
where
h n oi A ¼ 1 þ fð1 f Þ=ng C 2y þ C 2P ð3 4kP Þ ; B ¼ B ¼ fð1 f Þ=ngP 2 C 2P ; C ¼ f2ð1 f Þ=ngPC 2P ð1 kP Þ; D ¼ ½1 þ fð1 f Þ=ngC 2P ð1 kP Þ; E ¼ fð1 f Þ=ngPC 2P ð1 kP Þ: The MSE of the ratio-type estimator t3 is minimized for
d1 ¼ ½ðB D C E Þ=ðA B C 2 Þ ¼ d10 ;
ðsayÞ;
d20 ;
ðsayÞ:
2
d2 ¼ ½ðA E C D Þ=ðA B C Þ ¼ Explicit expressions of
d10
and
d20
are respectively obtained as
h i ¼ 1 fð1 f Þ=ngC 2P ð1 kP Þð1 2kP Þ h n oi = 1 þ fð1 f Þ=ng C 2y C 2P ð1 2kP Þ2 ; h n oi d20 ¼ ðk 1Þ 1 fð1 f Þ=ng C 2y þ C 2P ð1 2kP Þ h n oi = 1 þ fð1 f Þ=ng C 2y C 2P ð1 2kP Þ2 :
d10
Thus the resulting minimum MSE of the ratio-type estimator t3 is given by
h i " # 2 Þ 2 ð1 f Þ C 2y þ C 2P 1 kP kP ð1f C P ð1 kP Þ2 C 2y C 2P n ðB D2 2C D E þ A E2 Þ 2 h n o i ¼Y 1 : MSEmin ðt3 Þ ¼ Þ ðA B C 2 Þ C 2y C 2P ð1 2kP Þ2 n 1 þ ð1f n
ð2:14Þ
We note that (2.13) and (2.14) is the correct MSE (minimum MSE) expression of Shabbir and Gupta [10] ratio-type estimator t3. The MSE (minimum MSE) expression of ratio-type estimatort3 obtained by Shabbir and Gupta [[10] p.3, Eq. (2.4) and (2.7)] is not correct. From (1.4), (1.20), (1.25), (1.37), (2.11) and (2.14) we have
MSEðt Reg Þ½¼ MSEmin ðt 2 Þ ¼ MSEmin ðt 4 Þg ¼ MSEmin ðt aj Þ ¼ MSEmin t Þ MSEmin ðt 3 Þ ¼
Y 2 ½B ðA D Þ þ C ðE C Þ2 B ðA B C 2 Þ
P 0;
ðj ¼ 1; 2; . . . ; 5Þ:
ð2:15Þ
From (2.15), it is observed that the ratio-type estimator t3 [10] is more efficient than the usual linear regression estimator tReg, Jhajj et al. [9] class of estimators t2, class of estimators t4, Abd-Elfattah et al. [1] classes of estimators taj ; ðj ¼ 1; 2; . . . ; 5Þ , , ratio estimator t1 due to Naik and Gupta [15] and ratio-type class of estimators t⁄ and hence the usual unbiased estimator y estimators tsi, (i = 1, 2, . . . , 10) due to Singh et al. [13] at their optimum conditions. In addition to many, the estimators cited in Tables 1 and 2 are members of the suggested class of estimators t. ðjÞ Similarly some product-type estimators t pi ; ði ¼ 1; 2; . . . ; 20; j ¼ 1; 2Þ can be generated from the suggested class of estimators t.
3. Empirical study In this section we compare the performance of different estimators considered here using two population data sets as earlier considered by Shabbir and Gupta [10] and Abd-Elfattah et al. [1]. The description of population data sets are as follows.
H.P. Singh, R.S. Solanki / Applied Mathematics and Computation 218 (2012) 7798–7812 Table 1 Some members (ratio-type) of the class t, when d[ = 1]. Ratio-type estimator (a = 1)
w
g
ð1Þ 2 b b þ S/ Þ ½d1 þ d2 ðP PÞ½ðNP tr1 ¼ y þ S/ Þ=ðNP P ð1Þ 2 b b þ f Þ ½d1 þ d2 ðP PÞ½ðNP tr2 ¼ y þ f Þ=ðNP P ð1Þ b b ½d1 þ d2 ðP PÞ½fb tr3 ¼ y 2 ð/ÞP þ kP g=fb2 ð/Þ P þ ð1Þ 2 b b þ kP Þ ½d1 þ d2 ðP PÞ½ðNP tr4 ¼ y þ kP Þ=ðNP P ð1Þ b b þ 1Þ ½d1 þ d2 ðP PÞ½ðNP tr5 ¼ y þ 1Þ=ðN P ð1Þ b b þ C P Þ ½d1 þ d2 ðP PÞ½ðNP þ C P Þ=ðN P tr6 ¼ y ð1Þ b bþ ½d1 þ d2 ðP PÞ½ðNP tr7 ¼ y þ Pb Þ=ðN P Pb Þ ð1Þ b b þ S/ Þ ½d1 þ d2 ðP PÞ½ðNP tr8 ¼ y þ S/ Þ=ðN P ð1Þ b b þ f Þ ½d1 þ d2 ðP PÞ½ðNP tr9 ¼ y þ f Þ=ðN P ð1Þ b b þ gÞ ½d1 þ d2 ðP PÞ½ðNP þ gÞ=ðN P tr10 ¼ y ð1Þ b b þ kP Þ ½d1 þ d2 ðP PÞ½ðNP þ kP Þ=ðN P tr11 ¼ y ð1Þ b bþ ½d1 þ d2 ðP PÞ½ðnP tr12 ¼ y þ Pb Þ=ðn P Pb Þ ð1Þ b b tr13 ¼ y½d1 þ d2 ðP PÞ½ðnP þ S/ Þ=ðn P þ S/ Þ
NP
S/
q
kP g
q
NP
f
b2(/ )
kP
NP
kP
N
1
N
CP
N
qPb
N
S/
N
f
N
g = (1 f)
N
kP
n
qPb
n
S/
b b þ f Þ ½d1 þ d2 ðP PÞ½ðnP tr14 ¼ y þ f Þ=ðn P ð1Þ b b þ gÞ ½d1 þ d2 ðP PÞ½ðnP ¼y þ gÞ=ðn P t
n
f
n
g = (1 f)
ð1Þ b b þ kP Þ ½d1 þ d2 ðP PÞ½ðnP þ kP Þ=ðn P tr16 ¼ y ð1Þ b b þ Pg t ¼ y½d1 þ d2 ðP PÞ½fb ð/ÞP þ Pg=fb ð/Þ P
n
kP
b2(/)
P
q
q
ð1Þ
r15
r17
2
2
ð1Þ 2 b b þ PÞ ½d1 þ d2 ðP PÞ½ðNP þ PÞ=ðNP P tr18 ¼ y ð1Þ b b þ PÞ ½d1 þ d2 ðP PÞ½ðNP ¼y þ PÞ=ðN P t
NP
P
N
P
ð1Þ b b þ PÞ ½d1 þ d2 ðP PÞ½ðnP þ PÞ=ðn P tr20 ¼ y
n
P
r19
Table 2 Some members (ratio-type) of the class t, when d [1]. Ratio-type estimator (a = 1)
w
g
ð2Þ 2 b b S/ Þ ½d1 þ d2 ðP PÞ½ðNP tr1 ¼ y S/ Þ=ðNP P ð2Þ 2 b b f Þ ½d1 þ d2 ðP PÞ½ðNP tr2 ¼ y f Þ=ðNP P ð2Þ b b ½d1 þ d2 ðP PÞ½fb tr3 ¼ y 2 ð/ÞP kP g=fb2 ð/Þ P kP g ð2Þ 2 b b kP Þ ½d1 þ d2 ðP PÞ½ðNP tr4 ¼ y kP Þ=ðNP P ð2Þ b b 1Þ ½d1 þ d2 ðP PÞ½ðNP tr5 ¼ y 1Þ=ðN P ð2Þ b b C P Þ ½d1 þ d2 ðP PÞ½ðNP C P Þ=ðN P tr6 ¼ y ð2Þ b b ½d1 þ d2 ðP PÞ½ðNP tr7 ¼ y Pb Þ=ðN P Pb Þ ð2Þ b b S/ Þ ½d1 þ d2 ðP PÞ½ðNP tr8 ¼ y S/ Þ=ðN P ð2Þ b b f Þ ½d1 þ d2 ðP PÞ½ðNP tr9 ¼ y f Þ=ðN P ð2Þ b b gÞ ½d1 þ d2 ðP PÞ½ðNP gÞ=ðN P tr10 ¼ y ð2Þ b b kP Þ ½d1 þ d2 ðP PÞ½ðNP kP Þ=ðN P tr11 ¼ y ð2Þ b b ½d1 þ d2 ðP PÞ½ðnP tr12 ¼ y Pb Þ=ðn P Pb Þ ð2Þ b b tr13 ¼ y½d1 þ d2 ðP PÞ½ðnP S/ Þ=ðn P S/ Þ ð2Þ b b f Þ ½d1 þ d2 ðP PÞ½ðnP tr14 ¼ y f Þ=ðn P ð2Þ b b gÞ ½d1 þ d2 ðP PÞ½ðnP gÞ=ðn P tr15 ¼ y ð2Þ b b kP Þ ½d1 þ d2 ðP PÞ½ðnP kP Þ=ðn P tr16 ¼ y ð2Þ b b Pg tr17 ¼ y½d1 þ d2 ðP PÞ½fb2 ð/ÞP Pg=fb2 ð/Þ P ð2Þ 2 b b PÞ ½d1 þ d2 ðP PÞ½ðNP tr18 ¼ y PÞ=ðNP P ð2Þ b b PÞ ½d1 þ d2 ðP PÞ½ðNP PÞ=ðN P tr19 ¼ y ð2Þ b b PÞ ½d1 þ d2 ðP PÞ½ðnP PÞ=ðn P tr20 ¼ y
NP
S/
q
q
q
q
NP
f
b2 (/)
kP
NP
kP
N
1
N
CP
N
qPb
N
S/
N
f
N
g = (1 f)
N
kP
n
qPb
n
S/
n
f
n
g = (1 f)
n
kP
b2(/)
P
NP
P
N
P
n
P
Population I [Source: Sukhatme and Sukhatme [12], p. 256]. y = Number of villages in the circles. / = A circle consisting more than five villages. N ¼ 89; n ¼ 23; Y ¼ 3:36; P ¼ 0:124; qPb ¼ 0:766; C y ¼ 0:601; C P ¼ 2:678; b2 ð/Þ ¼ 6:162. Population II [Source: Sukhatme and Sukhatme [12], p. 256].
7805
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H.P. Singh, R.S. Solanki / Applied Mathematics and Computation 218 (2012) 7798–7812
y = Number of villages in the circles. / = A circle consisting more than five villages. N ¼ 89; n ¼ 23; Y ¼ 1102; P ¼ 0:124; qPb ¼ 0:624; C y ¼ 0:65; C P ¼ 2:678; b2 ð/Þ ¼ 6:162: , Naik and Gupta [15] ratio We have computed the percent relative efficiencies (PREs) of the usual unbiased estimator y estimator t1, Jhajj et al. [9] class of estimators t2, Shabbir and Gupta [10] ratio-type estimatort3, Singh et al. [13] ratio-type estimators tsi(i = 1, 2, . . . , 10), class of estimators t4, Abd-Elfattah et al. [1] classes of estimators taj ðj ¼ 1; 2; . . . ; 5Þ, ratio-type ðjÞ estimators t a5ðjÞ ðj ¼ 1; 2; . . . ; 9Þ, class of estimators t⁄, usual linear regression estimator tReg and ratio-type estimators tri (which are members of suggested class of estimators (t) (i = 1, 2, . . . , 20; j = 1, 2) with respect to the usual unbiased estimator (at their optimum conditions) and displayed in Tables 3–5. y It is observed from Tables 3–5 that. The usual linear regression estimator tReg (or the class of estimators t2 due to Jhajj et al. [9], the class of estimators t4, the classes of estimators t aj ðj ¼ 1; 2; . . . ; 5Þ due to Abd-Elfattah et al. [1] and class of estimators t⁄ at their optimum conditions) , ratio estimator t1 [15], ratio-type estimators tsi [13], (i = 1, 2, . . . , 10) is more efficient than the usual unbiased estimator y and ta5ðjÞ ; ðj ¼ 1; 2; . . . ; 9Þ in both population data sets I and II. The performance of the estimators t1, ts1, ts4, ts6, ts8 and ta5ð8Þ in both the population data sets I and II. are even worse than the usual unbiased estimator y ðjÞ The ratio-type estimators tri (which are members of suggested class of estimators t), (i = 1, 2, . . . , 20; j = 1, 2) perform better than the usual linear regression estimators tReg and hence than the estimators/classes of estimators ; t1 ; t2 ; t4 ; taj ; ðj ¼ 1; 2; . . . ; 5Þ; t ; t si ; ði ¼ 1; 2; . . . ; 10Þ and t a5ðjÞ ; ðj ¼ 1; 2; . . . ; 9Þ. Further we note that the proposed estimay ð1Þ tors t ri ; ði ¼ 1; 2; . . . ; 20Þ are inferior to Shabbir and Gupta [10] ratio-type estimator t3 (with corrected version of MSE). ð1Þ This shows that the suggested class of estimators t with d = 1 (i.e. tri ; i ¼ 1; 2; . . . ; 20Þ play no significant role in improving ð2Þ the efficiency of the estimators. Table 4 clearly indicates that the estimators t ri ; ði ¼ 1; 2; . . . ; 20Þ are superior to all the
Table 3 . PREs of different estimators with respect to y Estimator
ð1Þ tr1 ð1Þ tr2 ð1Þ tr3 ð1Þ tr4 ð1Þ tr5 ð1Þ tr6 ð1Þ tr7 ð1Þ tr8 ð1Þ tr9 ð1Þ tr10
Þ PREð; y
Estimator
Population I
Population II
300.18
186.62
308.21
189.09
302.64
188.63
320.19
193.60
328.43
194.98
300.18
186.62
334.15
197.61
346.58
199.93
348.97
200.56
334.78
196.75
Þ PREð; y Population I
Population II
ð1Þ tr11 ð1Þ tr12 ð1Þ tr13 ð1Þ tr14 ð1Þ tr15 ð1Þ tr16 ð1Þ tr17 ð1Þ tr18 ð1Þ tr19 ð1Þ tr20
351.89
201.51
334.15
197.61
322.22
193.22
328.43
194.98
297.92
185.91
336.81
197.90
Estimator
Þ PREð; y
312.85
190.48
328.43
194.98
353.57
201.76
342.04
198.72
Table 4 . PREs of different estimators with respect to y Estimator
ð2Þ tr1 ð2Þ tr2 ð2Þ tr3 ð2Þ tr4 ð2Þ tr5 ð2Þ tr6 ð2Þ tr7 ð2Þ tr8 ð2Þ tr9 ð2Þ tr10 ⁄
Þ PREð; y Population I
Population II
730.92
262.76
541.80
239.15
648.68
242.48
443.18
218.93
410.57
215.24
730.92
262.76
395.04
209.90
372.03
206.39
368.71
205.57
393.56
211.46
Indicates the largest efficiency.
Population I ð2Þ tr11 ð2Þ tr12 ð2Þ tr13 ð2Þ tr14 ð2Þ tr15 ð2Þ tr16 ð2Þ tr17 ð2Þ tr18 ð2Þ tr19 ð2Þ tr20
Population II
364.99
204.44
928.50⁄
250.83
433.61
220.07
410.57
215.24
848.50
273.48⁄
389.09
209.41
491.59
230.96
410.57
215.24
363.01
204.16
379.15
208.10
7807
H.P. Singh, R.S. Solanki / Applied Mathematics and Computation 218 (2012) 7798–7812 Table 5 . PREs of different estimators with respect to y Þ PREð; y
Estimator
⁄
Estimator
Population I
Population II
y t1 t2 [or t4or t aj or t⁄ or tReg ]
100 7.13 241.99
100 7.79 163.77
t3
Þ PREð; y Population I
Population II
ts9 ts10 ta5ð1Þ
239.34 125.21 135.72
163.07 92.84 124.12
358.13
202.93
ta5ð2Þ
114.54
110.48
ts1
4.93
5.69
ta5ð3Þ
215.97
143.87
ts2
237.55
162.01
ta5ð4Þ
142.28
127.97
ts3
221.18
155.31
ta5ð5Þ
230.24
162.84
ts4
71.86
69.11
ta5ð6Þ
133.08
79.25
ts5
214.98
152.67
ta5ð7Þ
126.67
115.08
ts6 ts7
44.86 229.07
37.62 160.25
ta5ð8Þ ta5ð9Þ
39.33 110.99
30.18 106.51
ts8
18.64
17.38
(j = 1,2, . . . , 5).
estimators presented in the paper including Shabbir and Gupta [10] ratio-type estimator t3 for both the population data ð2Þ sets I and II. It follows that the proposed class of estimators t with d = 1 (i.e. tri ; i ¼ 1; 2; . . . ; 20Þ play significant role in improving the efficiency of the estimators. However this conclusion should not extrapolated, in general. ð2Þ The estimator tr12 (based on n, P and qPb) gives the largest efficiency over other competitors in population I, while the ð2Þ estimator t r15 (based on n, P and g) has the largest efficiency over other competitors in population II. Thus taking into consideration the above discussions, we have concluded that the proposal of the class of estimators t is justified. 4. Two phase sampling In Section 2 we have discussed the problem of estimating the population mean Y of the study variable y assuming that the information about the proportion P of population units possessing the auxiliary attribute /, highly correlated with study variable y, is known in advance. However in many practical situations it may happen that the proportion P is not known, before the start of the survey. In such situations, we generally apply the double (two phase) sampling design to furnish an efficient b 0 denote the proportion of units possessing attribute / in the first phase sample of estimator for the population mean Y. Let P b denote the proportion of units possessing attribute / in the second phase sample of size n[
b 0 = PÞ: b ð P t1d ¼ y
ð4:1Þ
The MSE of the ratio estimator t1d, up to first order approximation is given by
MSEðt 1d Þ ¼ Y 2
ð1 f Þ 2 1 1 2 Cy þ 0 C P ð1 2kP Þ : n n n
ð4:2Þ
The class of estimators of population mean Y in two phase sampling due to Jhajj et al. [9] is given by
; u0 Þ; t2d ¼ hðy
ð4:3Þ
b P b 0 Þ and hðy ; u0 Þ is the function of ðy ; u0 Þ such that ½hðy ; 1Þ ¼ Y; 8Y and the function hðy ; u0 Þ satisfies certain regwhere u ½¼ ð P= ularity conditions similar to those given in Jhajj et al. [[9] (Sec. 3, p. 45)]. The minimum MSE of the class of estimators t2d, up to terms of order n1, is given by 0
MSEmin ðt 2d Þ ¼
1 1 2 1 1 2 S ; 0 Sy 1 q2Pb þ n n n0 N y
ð4:4Þ
which equals to the approximate MSE of the two phase sampling usual linear regression estimator, defined as
b 0 PÞ; b ^/ ð P þb tRegd ¼ ½y i.e., MSEmin(t2d) = MSE(tRegd). Shabbir and Gupta [10] suggested the ratio-type estimator for population mean Y in two phase sampling as
ð4:5Þ
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H.P. Singh, R.S. Solanki / Applied Mathematics and Computation 218 (2012) 7798–7812
b 0 PÞð b P b 0 = PÞ; b ½w1 þ w2 ð P t3d ¼ y
ð4:6Þ
where (w1, w2) are suitably chosen constants whose sum is not necessarily equal to ‘unity’. Two phase sampling version of Singh et al. [13] ratio-type estimators tsi, (i = 1, 2, . . . , 10) are given by
b 0 PÞfð b b 0 Þ=ð PÞg; b ^/ ð P þb tsd1 ¼ ½y P b0
b0
b0
b0
ð4:7Þ
b b þ b ð/Þg; ^/ ð P PÞ½f þb tsd2 ¼ ½y P þ b2 ð/Þg=f P 2
ð4:8Þ
b b þ C P Þg; ^/ ð P PÞfð þb tsd3 ¼ ½y P þ C P Þ=ð P b0
ð4:9Þ
b b ^/ ð P PÞ½fb þb tsd4 ¼ ½y 2 ð/Þ P þ C P g=fb2 ð/Þ P þ C P g; b0
b0
ð4:10Þ
b0
b b ^/ ð P PÞ½fC þb tsd5 ¼ ½y P P þ b2 ð/Þg=fC P P þ b2 ð/Þg;
ð4:11Þ
b0
ð4:12Þ
b b ^/ ð P PÞfðC þb tsd6 ¼ ½y P P þ qPb Þ=ðC p P þ qPb Þg; b0
b0
b ^/ ð P PÞfð þb qPb P þ C P Þ=ðqPb Pb þ C P Þg; tsd7 ¼ ½y b0 b0
ð4:13Þ
b0
b b ^/ ð P PÞ½fb þb tsd8 ¼ ½y 2 ð/Þ P þ qPb g=fb2 ð/Þ P þ qPb g;
ð4:14Þ
b0
ð4:15Þ
b b q P þ b ð/Þg=fq P ^/ ð P PÞ½f þb tsd9 ¼ ½y 2 Pb Pb þ b2 ð/Þg; b0
b0
b0
b b þ q Þg: ^/ ð P PÞfð þb P þ qPb Þ=ð P tsd10 ¼ ½y Pb
ð4:16Þ
To the first order of approximation, the MSEs of the ratio-type estimator tsdi are given by
MSEðt sdi Þ ¼
o 1 1 1 1 n 2 S2 ; 0 Sy 1 q2Pb þ S2/ R2i þ n n n0 N y
ð4:17Þ
where Ri, (i = 1, 2, . . . , 10) are same as defined in Section 1. Keeping the form of the ratio-type estimators tsdi, (i = 1, 2, . . . , 10); one may suggest the class of estimators for population mean Y as
b 0 PÞfðw b b 0 þ gÞ=ðw P b þ gÞg; ^/ ð P þb t4d ¼ ½y P
ð4:18Þ
where (w, g) are same as defined for the class of estimators t4 in Section 1. Note that the ratio-type estimators tsdi , (i = 1, 2, . . . , 10); are members of the class of estimators t4d for different values of w and g. To the first order of approximation, the MSE of class of the estimators t4d is given by
MSEðt 4d Þ ¼
o 1 1 1 1 n 2 0 Sy 1 q2pb þ R2 S2/ þ S2 ; n n n0 N y
ð4:19Þ
where R⁄ is same as defined earlier. The MSE of the class of estimators t4d is minimized for R⁄ = 0. Thus the resulting minimum MSE of t4d is given by
MSEmin ðt4d Þ ¼
1 1 2 1 1 2 0 Sy 1 q2Pb þ S ; n n n0 N y
ð4:20Þ
which equals to the approximate MSE of the usual linear regression estimator tRegd (i.e. MSEmin(t4d) = MSE(tRegd)). Motivated by Abd-Elfattah et al. [1], one may define the classes of estimators in two phase sampling as
"
# b 0 PÞg b b 0 PÞg b ^/ ð P ^/ ð P þb þb fy b 0 þ b ð/Þg ; b 0 þ m2 fy fP P 2 b b þ b ð/Þg P fP 2 " # b 0 PÞg b b 0 PÞg b ^/ ð P ^/ ð P þb þb fy fy 0 0 b b ðP þ CP Þ ; P þ m2 ¼ m2 b b þ b ð/Þg P fP 2 " # b 0 PÞg b b 0 PÞg b ^/ ð P ^/ ð P þb þb fy b 0 þ m2 fy b0 þ CP g ; fb2 ð/Þ P P ¼ m1 b b þ b ð/Þg P fP 2 " # 0 b b b b ^ ^ fy þ b/ ð P PÞg b 0 fy þ b/ ð P 0 PÞg 0 b fC P P þ b2 ð/Þg ; P þ m2 ¼ m1 b b þ b ð/Þg P fP 2
tad1 ¼ m1
ð4:21Þ
tad2
ð4:22Þ
tad3 tad4
ð4:23Þ
ð4:24Þ
where (m1, m2) are weights that satisfies the condition (m1 + m2) = 1. The common minimum MSE of the classes of estimators tadj , to the first order of approximation is given by
MSEmin ðtadj Þ ¼
1 1 2 1 1 2 S ¼ MSEðt Regd Þ; 0 Sy 1 q2Pb þ n n n0 N y
The following two phase sampling estimators
ðj ¼ 1; 2; . . . 4Þ:
ð4:25Þ
H.P. Singh, R.S. Solanki / Applied Mathematics and Computation 218 (2012) 7798–7812
7809
b 0 þ C P Þ=ð P b þ C P Þg; fð P tad5ð1Þ ¼ y
ð4:26Þ
b 0 þ b ð/Þg=f P b þ b ð/Þg; ½f P tad5ð2Þ ¼ y 2 2
ð4:27Þ
b 0 þ C P g=fb ð/Þ P b þ C P g; ½fb2 ð/Þ P tad5ð3Þ ¼ y 2
ð4:28Þ
b0
b þ b ð/Þg; ½fC P P þ b2 ð/Þg=fC P P tad5ð4Þ ¼ y 2
ð4:29Þ
b 0 þ q Þ=ð P b þ q Þg; fð P tad5ð5Þ ¼ y Pb Pb
ð4:30Þ
b0
b þ q Þg; fðC P P þ qPb Þ=ðC P P tad5ð6Þ ¼ y Pb
ð4:31Þ
b 0 þ C P Þ=ðq P b fðqPb P tad5ð7Þ ¼ y Pb þ C P Þg;
ð4:32Þ
b 0 þ q g=fb ð/Þ P b þ q g; ½fb2 ð/Þ P ¼y 2 Pb Pb
ð4:33Þ
b 0 þ b ð/Þg=fq P b ½fqPb P tad5ð9Þ ¼ y 2 Pb þ b2 ð/Þg;
ð4:34Þ
tad5ð8Þ
etc. are members of the following ratio-type class of estimators
b 0 þ gÞ=ðw P b þ gÞg; fðw P tad5 ¼ y
ð4:35Þ
where (w, g) are same as defined for the class t4 in Section 1. To the first order of approximation, the MSEs of the ratio-type estimators tad5ðjÞ ; ðj ¼ 1; 2; . . . ; 9Þ are given by
MSEft ad5ðjÞ g ¼ Y 2
1f 2 1 1 2 Cy þ 0 C P mj mj 2kP ; n n n
ð4:36Þ
where, mj ; ðj ¼ 1; 2; . . . ; 9Þ are same as defined in Section 1. To the first order of approximation, the minimum MSE of class tad5 is given by
MSEmin ðt ad5 Þ ¼
1 1 2 1 1 2 0 Sy 1 q2Pb þ S ¼ MSEmin ðtRegd Þ: 0 n n n N y
ð4:37Þ
5. The Suggested class of estimators in two phase The double sampling version of the suggested class of estimators t Section 2 for the population mean Y is defined by
b 0 PÞ b ½w1 þ w1 ð P td ¼ y
!a b 0 þ dg wP ; b þ dg wP
ð5:1Þ
where (w, g) are either real numbers or function of known parameters of the auxiliary attribute / such as Cp, b2(/) qPb and kp. The scalar a takes values 1 (for product-type estimator) and +1 (for ratio-type estimator); d is an integer which takes values b 0 þ dgÞ and ðw P b þ dgÞ are non-negative and (w1, w2) are suitably chosen +1 and 1 for designing the estimators such that ðw P constants whose sum is not necessarily equal to unity. To the first order of approximation, the bias and MSE of the class of estimators td are respectively obtained as
1 1 2 w1 ða þ 1Þm Bðtd Þ ¼ Y ðw1 1Þ þ 0 C P am þ w2 P ðw1 am þ w2 PÞkP ; n n 2 MSEðt d Þ ¼ Y 2 1 þ w21 A1 þ w22 B1 þ 2w1 w2 C 1 2w1 D1 2w2 E1 ; where
ð1 f Þ 2 1 1 A1 ¼ 1 þ Cy þ 0 amC 2P fð2a þ 1Þm 4kP g ; n n n
1 1 2 2 P CP ; B1 ¼ n n0
1 1 C 1 ¼ 2P 0 ðam kP ÞC 2P ; n n
1 1 ða þ 1Þ D1 ¼ 1 þ 0 amC 2P m kP ; n n 2
1 1 E1 ¼ PC 2P ðam kP Þ: n n0 The MSE of the class of estimators td is minimized for
ð5:2Þ ð5:3Þ
7810
H.P. Singh, R.S. Solanki / Applied Mathematics and Computation 218 (2012) 7798–7812
h i w1 ¼ ðB1 D1 C 1 E1 Þ= A1 B1 C 21 ¼ w1 ; h i w2 ¼ ðA1 E1 C 1 D1 Þ= A1 B1 C 21 ¼ w2 ;
ðsayÞ; ðsayÞ:
Thus the resulting minimum MSE of the class td is obtained as
3 B1 D21 2C 1 D1 E1 þ A1 E21 5: MSEmin ðtd Þ ¼ Y 2 41 A1 B1 C 21 2
ð5:4Þ
Now we established the following theorem. Theorem 5.1. To the first order of approximation,
3 B1 D21 2C 1 D1 E1 þ A1 E21 5 MSEðt d Þ P Y 2 41 A1 B1 C 21 2
with equality holding if
w1 ¼ w1
and w2 ¼ w2 :
Remark 5.1. For w1 = 1, the class of estimators td reduces to the class of estimators
!a b0 b 0 PÞ b w P þ dg : ½1 þ w2 ð P td ¼ y b þ dg wP
ð5:5Þ
It can be easily shown to the first order of approximation that the minimum MSE of the class t d is
MSEmin t d ¼
1 1 2 1 1 2 0 Sy 1 q2Pb þ S ¼ MSEðtRegd Þ: n n n0 N y
ð5:6Þ
In addition to many, the estimators cited in Tables 6 and 7 are members of the proposed class of estimators td. ðjÞ Similarly some product-type estimators tdpi ; ði ¼ 1; 2; . . . ; 20; j ¼ 1; 2Þ can be generated from the suggested class of estimators.
6. Empirical study In this section we compare the performance of different estimators considered here in two phase sampling using the same data sets as we discussed before. The description of population data sets are as follows.
Table 6 Some members (ratio-type) of the class td, when d [1]. Ratio-type estimator (a = 1)
w
ð1Þ b 0 PÞ½fb b b0 b ½w1 þ w2 ð P tdr1 ¼ y 2 ð/Þ P þ kP g=fb2 ð/Þ P þ ð1Þ 0 0 b b b b ½w1 þ w2 ð P PÞ½ðN P þ 1Þ=ðN P þ 1Þ tdr2 ¼ y ð1Þ b 0 PÞ½ðN b b 0 þ C P Þ=ðN P b þ C P Þ ½w1 þ w2 ð P P tdr3 ¼ y ð1Þ b 0 PÞ½ðN b b0 þ bþ ½w1 þ w2 ð P tdr4 ¼ y P Þ=ðN P Pb Pb Þ ð1Þ b 0 PÞ½ðN b b 0 þ S/ Þ=ðN P b þ S/ Þ ½w1 þ w2 ð P tdr5 ¼ y P ð1Þ b 0 PÞ½ðN b b 0 þ f Þ=ðN P b þ f Þ ½w1 þ w2 ð P tdr6 ¼ y P ð1Þ 0 0 b b b b þ gÞ tdr7 ¼ y½w1 þ w2 ð P PÞ½ðN P þ gÞ=ðN P ð1Þ b 0 PÞ½ðN b b 0 þ kP Þ=ðN P b þ kP Þ ½w1 þ w2 ð P P tdr8 ¼ y ð1Þ b 0 PÞ½ðn b b0 þ bþ ½w1 þ w2 ð P tdr9 ¼ y P Þ=ðn P Pb Pb Þ ð1Þ b 0 PÞ½ðn b b 0 þ S/ Þ=ðn P b þ S/ Þ ½w1 þ w2 ð P tdr10 ¼ y P ð1Þ b 0 PÞ½ðn b b 0 þ f Þ=ðn P b þ f Þ ½w1 þ w2 ð P tdr11 ¼ y P
b2(/)
kP
N
1
N
CP
q
q
ð1Þ tdr12 ð1Þ tdr13
q
q
b 0 PÞ½ðn b b 0 þ gÞ=ðn P b þ gÞ ½w1 þ w2 ð P ¼y P b 0 PÞ½ðn b b 0 þ kP Þ=ðn P b þ kP Þ ½w1 þ w2 ð P ¼y P
kP g
g
N
qPb
N
S/
N
f
N
g = (1 f)
N
kP
n
qPb
n
S/
n
f
n
g = (1 f)
n
kP
7811
H.P. Singh, R.S. Solanki / Applied Mathematics and Computation 218 (2012) 7798–7812 Table 7 Some members (ratio-type) of the class td when, d[=1]. Ratio-type estimator (a = 1)
w
ð2Þ b 0 PÞ½fb b b0 b ½w1 þ w2 ð P tdr1 ¼ y 2 ð/Þ P kP g=fb2 ð/Þ P ð2Þ b 0 PÞ½ðN b b 0 1Þ=ðN P b 1Þ ½w1 þ w2 ð P tdr2 ¼ y P ð2Þ b 0 PÞ½ðN b b 0 C P Þ=ðN P b C P Þ ½w1 þ w2 ð P P tdr3 ¼ y ð2Þ b 0 PÞ½ðN b b0 b ½w1 þ w2 ð P tdr4 ¼ y P Pb Þ=ðN P Pb Þ ð2Þ b 0 PÞ½ðN b b 0 S/ Þ=ðN P b S/ Þ ½w1 þ w2 ð P tdr5 ¼ y P ð2Þ b 0 PÞ½ðN b b 0 f Þ=ðN P b f Þ ½w1 þ w2 ð P tdr6 ¼ y P ð2Þ b 0 PÞ½ðN b b 0 gÞ=ðN P b gÞ ½w1 þ w2 ð P P tdr7 ¼ y ð2Þ b 0 PÞ½ðN b b 0 kP Þ=ðN P b kP Þ ½w1 þ w2 ð P P tdr8 ¼ y ð2Þ b 0 PÞ½ðn b b0 b ½w1 þ w2 ð P tdr9 ¼ y P Pb Þ=ðn P Pb Þ ð2Þ b 0 PÞ½ðn b b 0 S/ Þ=ðn P b S/ Þ ½w1 þ w2 ð P tdr10 ¼ y P ð2Þ b 0 PÞ½ðn b b 0 f Þ=ðn P b f Þ ½w1 þ w2 ð P tdr11 ¼ y P
b2(/)
kp
N
1
N
CP
q
q
kP g
q
q
ð2Þ b 0 PÞ½ðn b b 0 gÞ=ðn P b gÞ ½w1 þ w2 ð P P tdr12 ¼ y ð2Þ b 0 PÞ½ðn b b 0 kP Þ=ðn P b kP Þ ½w1 þ w2 ð P P tdr13 ¼ y
g
N
qPb
N
S/
N
f
N
g = (1 f)
N
kP
n
qPb
n
S/
n
f
n
g = (1 f)
n
kP
Table 8 . PREs of different estimators with respect to y Estimator
ð1Þ tdr1 ð1Þ tdr2 ð1Þ tdr3 ð1Þ tdr4 ð1Þ tdr5 ð1Þ tdr6 ð1Þ tdr7 ð1Þ tdr8 ð1Þ tdr9 ð1Þ tdr10 ð1Þ tdr11 ð1Þ tdr12 ð1Þ tdr13 ⁄
Þ PREð; y Population I
Population II
177.79
144.39
181.92
146.20
177.36
143.79
182.76
146.92
184.48
147.53
184.80
147.70
182.85
146.69
185.19
147.94
176.77
144.10
180.98
145.71
181.92
146.20
176.96
143.57
183.14
146.99
Estimator
Þ PREð; y Population I
Population II
ð2Þ tdr1 ð2Þ tdr2 ð2Þ tdr3 ð2Þ tdr4 ð2Þ tdr5 ð2Þ tdr6 ð2Þ tdr7 ð2Þ tdr8 ð2Þ tdr9 ð2Þ tdr10 ð2Þ tdr11 ð2Þ tdr12 ð2Þ tdr13
207.12
156.66
191.83
151.23
210.21
159.96
190.25
150.01
187.68
149.16
187.29
148.96
190.09
150.37
186.84
148.68
215.49⁄
158.08
193.98
152.29
191.83
151.23
213.63
161.52⁄
189.61
149.89
Indicates the largest efficiency.
Population I (Source: Sukhatme and Sukhatme [12], p. 256). y = Number of villages in the circles. / = A circle consisting more than five villages. N ¼ 89; n ¼ 23; n0 ¼ 45; Y ¼ 3:36; P ¼ 0:124; qPb ¼ 0:766; C y ¼ 0:601; C P ¼ 2:678; b2 ð/Þ ¼ 6:162: Population II (Source: Sukhatme and Sukhatme [12], p. 256). y = Number of villages in the circles. / = A circle consisting more than five villages. N ¼ 89; n ¼ 23; n0 ¼ 45; Y ¼ 1102; P ¼ 0:124; qPb ¼ 0:624; C y ¼ 0:65; C P ¼ 2:678; b2 ð/Þ ¼ 6:162. , Naik and Gupta [15] ratio We have computed the percent relative efficiencies (PREs) of the usual unbiased estimator y estimator t1d, Jhajj et al. [9] class of estimators t2d, Shabbir and Gupta [10] ratio-type estimatort3d, Singh et al. [13] ratio-type estimators tsdi, (i = 1, 2, . . . , 10); class of estimators t4d, Abd-Elfattah et al. [1] classes of estimators t adj ; ðj ¼ 1; 2; . . . ; 5Þ; ratiotype estimators tad5ðjÞ ; ðj ¼ 1; 2; . . . ; 9Þ; class of estimators t d , usual linear regression estimator tRegd and ratio-type estimators ðjÞ t dri (which are members of suggested class of estimators td) (i = 1, 2, . . . , 13; j = 1, 2), with respect to the usual unbiased esti (at their optimum conditions) and displayed in Tables 8 and 9. mator y
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H.P. Singh, R.S. Solanki / Applied Mathematics and Computation 218 (2012) 7798–7812
Table 9 . PREs of different estimators with respect to y Estimator
Þ PREð; y
Estimator
Population I
Population II
y t1d t2d [or t4d or tadj or td or tRegd]
100 10.43 163.09
100 11.36 134.54
tsd9 tsd10 tad5ð1Þ
t3d
Þ PREð; y Population I
Population II
162.29 115.30 121.00
175.97 114.41 114.69
185.99
148.30
tad5ð2Þ
109.13
106.67
tsd1
7.30
8.38
tad5ð3Þ
154.80
125.16
tsd2
161.74
133.75
tad5ð4Þ
124.36
116.83
tsd3
156.54
130.68
tad5ð5Þ
159.47
134.12
tsd4
79.48
77.24
tad5ð6Þ
119.60
85.28
tsd5
154.46
129.44
tad5ð7Þ
116.12
109.45
tsd6
55.24
47.77
tad5ð8Þ
49.58
39.61
tsd7
159.10
132.96
tad5ð9Þ
106.99
104.20
tsd8
25.79
24.19
It is observed from Tables 8 and 9 that the trends of the suggested estimators as well as other estimators presented in Section 4 are same as observed for the estimators in single phase sampling. However the performance of ratio-type estimator ð2Þ ð2Þ tdr9 appears to be the best among all the estimators (in two phase) discussed here in population I, while the estimator tdr12 is proved to be the best among the entire estimators (of two phase) in population II. Thus the proposal of the class of estimators td is justified in two phase (double) sampling. 7. Conclusion We have developed the generalized version of Shabbir and Gupta [10] ratio-type estimator which includes the estimators suggested by Naik and Gupta [15], Singh et al. [13], Abd-Elfattah et al. [1] and usual unbiased estimator. Asymptotic optimum estimator (AOE) in the proposed class of estimators is identified alongwith its mean square error formula. It has been shown that the AOE of the proposed class of estimators is more efficient than the estimators/classes of estimators due to Naik and Gupta [15], Jhajj et al. [9], Shabbir and Gupta [10], Singh et al. [13], Abd-Elfattah et al. [1] and usual linear regression estimator. Theoretical results are also supported through two natural population data sets earlier used by Shabbir and Gupta [10] and Abd-Elfattah et al. [1]. Double sampling version of the suggested class is also discussed along with its properties. Acknowledgement Authors wish to thank the Editor-in-Chief and two anonymous referees for their helpful comments that aided in improving this article. References [1] A.M. Abd-Elfattah, E.A. El-Sherpieny, S.M. Mohamed, O.F. Abdou, Improvement in estimating the population mean in simple random sampling using information on auxiliary attribute, Applied Mathematics and Computation 215 (2010) 4198–4202. [2] B.V.S. Sisodia, V.K. Dwivedi, A modified ratio estimator using coefficient of an auxiliary variable, Journal of the Indian Society of Agricultural Statistics 33 (1981) 13–18. [3] C. Kadilar, H. Cingi, Ratio estimators for the population variance in simple and stratified random sampling, Applied Mathematics and Computation 173 (2) (2006) 1047–1059. [4] C. Kadilar, H. Cingi, An improvement in estimating the population mean by using the correlation coefficient, Hacettepe Journal of Mathematics and Statistics 35 (1) (2006) 103–109. [5] C. Kadilar, H. Cingi, Improvement in estimating the population mean in simple random sampling, Applied Mathematics Letters 19 (2006) 75–79. [6] D.C. Roy, A regression-type estimator in two phase sampling using two auxiliary variable, Pakistan Journal of Statistics 19 (3) (2003) 281–290. [7] H.P. Singh, N. Agnihotri, A general procedure of estimating population mean using auxiliary information in sample surveys, Statistics in Transition-new Series 8 (1) (2008) 71–87. [8] H.P. Singh, R. Tailor, Use of known correlation coefficient in estimating the finite population mean, Statistics in Transition 6 (2003) 555–560. [9] H.S. Jhajj, M.K. Sharma, L.K. Grover, A family of estimators of population mean using information on auxiliary attribute, Pakistan Journal of Statistics 22 (1) (2006) 43–50. [10] J. Shabbir, S. Gupta, On Estimating the finite population mean with known population proportion of an auxiliary variable, Pakistan Journal of Statistics 23 (1) (2007) 1–9. [11] L.N. Upadhyaya, H.P. Singh, Use of transformed auxiliary variable in estimating the finite population mean, Biometrical Journal 41 (5) (1999) 627–636. [12] P.V. Sukhatme, B.V. Sukhatme, Sampling Theory of Surveys with Applications, Asia Publishing House, New Delhi, India, 1970. [13] R. Singh, P. Chauhan, N. Sawan, F. Smarandache, Ratio estimators in simple random sampling using information on auxiliary attribute, Pakistan Journal of Statistics and Operation Research 4 (1) (2008) 47–53. [14] S.K. Ray, R.K. Singh, Difference-cum-ratio type estimators, Journal of the Indian Statistical Association 19 (1981) 147–151. [15] V.D. Naik, P.C. Gupta, A note on estimating of mean with known population of an auxiliary character, Journal of Indian Society of Agricultural Statistics 48 (2) (1996) 151–158.
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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Improved estimation of population mean in simple random sampling using information on auxiliary attribute Housila P. Singh, Ramkrishna S. Solanki ⇑ School of Studies in Statistics, Vikram University, Ujjain 456010, India
a r t i c l e
i n f o
Keywords: Auxiliary variable Study variable Proportion Bias Mean square error
a b s t r a c t This paper addresses the problem of estimating the population mean with known population proportion of an auxiliary variable. A class of estimators is defined which includes the estimators recently proposed by Shabbir and Gupta (2007) [10] and Abd-Elfattah et al. (2010) [1]. The usual unbiased estimator and Naik and Gupta (1996) [15] estimator are also the member of the proposed class of the estimators. The bias and mean square error (MSE) expressions of the proposed class are obtained up to first order of approximation. Asymptotically optimum estimator (AOE) in the class of estimators is identified alongwith its mean square error formula. The correct MSE and minimum MSE expressions of Shabbir and Gupta (2007) [10] estimator are also given. It has been shown that the proposed class of estimators is more efficient than the usual unbiased estimator, usual linear regression estimator and estimators/classes of estimators due to Naik and Gupta (1996) [15], Jhajj et al. (2008) [9], Shabbir and Gupta (2007) [10] estimator, Singh et al. (2008) [13] and Abd-Elfattah et al. (2010) [1]. The double sampling version of the proposed class of estimators is proposed alongwith its properties. Numerical illustrations are given in support of the present study. Ó 2012 Elsevier Inc. All rights reserved.
1. Introduction It is well known that precision of the estimates of the population mean or total of the study variable y can be considerably improved by the use of known information on an auxiliary variable x which is highly correlated with the study variable y. However in many practical situations, instead of existence of auxiliary variables there exist some auxiliary attributes / (say), which are highly correlated with the study variable y, such as. (i) Sex (/) and height of the persons (y), (ii) Amount of milk produced (y) and a particular breed of the cow (/), (iii) Amount of yield of wheat crop and a particular variety of wheat (/) etc. (see Jhajj et al. [9]). Consider a sample of size n drawn by simple random sampling without replacement (SRSWOR) from the population U of size N. Let yi and /i denote the observations on the study variable y and the auxiliary attribute /, respectively for the ith unit (i = 1, 2, . . . , N) of the population U. We note that /i[=1], if ith unit of the population possesses attribute / and /i[=0], otherh P i P wise (i = 1, 2, . . . , N). Let A ¼ Ni¼1 /i and a ¼ ni¼1 /i denote the total number of units in the population U and sample
⇑ Corresponding author. E-mail addresses: [email protected] (H.P. Singh), [email protected] (R.S. Solanki). 0096-3003/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2012.01.047
H.P. Singh, R.S. Solanki / Applied Mathematics and Computation 218 (2012) 7798–7812
7799
b ða=nÞ denote proportion of units in the population U and sample respectively possessing attribute /. Let P[=(A/N)] and P½¼ respectively possessing attribute /. Taking into consideration the point biserial correlation coefficient between auxiliary attribute / and the study variable y, h i P Naik and Gupta [15] defined the ratio estimator for population mean Y ¼ N 1 Ni¼1 yi of the study variable y (when the prior information of population proportion of units, possessing the same attribute is available) as follows
b ðP= PÞ; t1 ¼ y
ð1:1Þ
P ¼ n1 ni¼1 yi is the sample mean of the study variable y. where y The mean square error (MSE) of the ratio estimator t1, up to first order of approximation is given by
h i MSEðt 1 Þ ¼ fð1 f Þ=ngY 2 C 2y þ C 2P ð1 2kP Þ ;
ð1:2Þ
where,
f ¼ ðn=NÞ;
C 2y ¼ S2y =Y 2 ;
S2y ¼ ðN 1Þ1
N X ðyi YÞ2 ;
C 2P ¼ S2/ =P2 ;
i¼1
S2/
¼ ðN 1Þ
1
N X ð/i PÞ2 ;
kP ¼ ½ðqPb C y Þ=C P ;
qPb ¼ ½Sy/ =ðSy S/ Þ;
i¼1
Sy/ ¼ ðN 1Þ1
N X ðyi YÞð/i PÞ: i¼1
(sample mean) is It is well known under SRSWOR that the variance of the usual unbiased estimator y
Þ ¼ fð1 f Þ=ngY 2 C 2y : Varðy
ð1:3Þ
as Jhajj et al. [9] have suggested the class of estimators for the population mean Y
; uÞ; t2 ¼ hðy b ; uÞ is the function of ðy ; uÞ such that [hðy ; 1Þ ¼ Y; 8Y] and the function hðy ; uÞ satisfies certain reguwhere u½¼ P=P and hðy larity conditions similar to those given in Jhaji et al. [[9], Sec. 3, p. 45]. The minimum MSE of the class of estimators t2, up to first order of approximation is given by
MSEmin ðt 2 Þfð1 f Þ=ngS2y 1 q2Pb ;
ð1:4Þ
which equals to the variance [or MSE] of the usual linear regression estimator defined as
b ^/ ðP PÞ; þb tReg ¼ ½y
ð1:5Þ
^/ ¼ sy/ =s2 is the sample regression coefficient [or sample estimate of population regression coefficient where b / P P b/ ¼ Sy/ =S2/ ], with s2/ ¼ ðn 1Þ1 ni¼1 ð/i PÞ2 and sy/ ¼ ðn 1Þ1 ni¼1 ðyi YÞð/i PÞ (i.e. MSEmin(t2) = MSE(tReg)). Motivated by Ray and Singh [14] and Roy [6], Shabbir and Gupta [10] suggested the ratio-type estimator for the population mean Y as
b b ½d1 þ d2 ðP PÞðP= t3 ¼ y PÞ;
ð1:6Þ
where (d1, d2) are suitably chosen constants whose sum need not be unity. In case where, the population proportion P of the auxiliary attribute / is unknown, Jhajj et al. [9] and Shabbir and Gupta [10] have suggested the two phase sampling version of their estimators t2 and t3 with properties. It is to be mentioned that the mean square error of the ratio-type estimatort3, to the first order of approximation obtained by Shabbir and Gupta [10] is not correct in single as well as in two phase sampling. Singh et al. [13] suggested the ratio-type estimators for estimating the population mean Y of the study variable y in simple random sampling using known parameters of the auxiliary attribute /, such as, coefficient of variation (Cp), coefficient of kurtosis (b2(/)) and point biserial correlation coefficient (qpb) as,
b b ^/ ðP PÞgðP= þb ts1 ¼ fy PÞ;
ð1:7Þ
ts2
b b þ b ð/Þg; ^/ ðP PÞg½fP þb ¼ fy þ b2 ð/Þg=f P 2
ð1:8Þ
ts3
b b þ C P Þg; ^/ ðP PÞgfðP þb ¼ fy þ C P Þ=ð P
ð1:9Þ
ts4
b b ^/ ðP PÞg½fb þb ¼ fy 2 ð/ÞP þ C P g=fb2 ð/Þ P þ C P g;
ð1:10Þ
ts5
b b ^/ ðP PÞg½fC þb ¼ fy P P þ b2 ð/Þg=fC P P þ b2 ð/Þg:
ð1:11Þ
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H.P. Singh, R.S. Solanki / Applied Mathematics and Computation 218 (2012) 7798–7812
Motivated by Kadilar and Cingi [4], one may also define the following ratio-type estimators for the population mean Y as
b b ^/ ðP PÞgfðC þb ts6 ¼ fy P P þ qPb Þ=ðC P P þ qPb Þg;
ð1:12Þ
b ^/ ðP PÞgfð þb qPb P þ C P Þ=ðqPb Pb þ C P Þg; ts7 ¼ fy
ð1:13Þ
b b ^/ ðP PÞg½fb þb ts8 ¼ fy 2 ð/ÞP þ qPb g=fb2 ð/Þ P þ qPb g;
ð1:14Þ
b ^/ ðP PÞg½f þb ts9 ¼ fy qPb P þ b2 ð/Þg=fqPb Pb þ b2 ð/Þg;
ð1:15Þ
b b þ q Þg: ^/ ðP PÞgfðP þb ts10 ¼ fy þ qPb Þ=ð P Pb
ð1:16Þ
where CP and b2(/) are the population coefficient of variation and the population coefficient of kurtosis of auxiliary attribute /, respectively. To the first order of approximation, the MSEs of the ratio-type estimators tsi, are given by
h i MSEðt si Þ ¼ fð1 f Þ=ng R2i S2/ þ S2y 1 q2Pb ;
ði ¼ 1; 2; . . . ; 10Þ;
ð1:17Þ
where
R1 ¼ ðY=PÞ;
R2 ¼ ½Y=fP þ b2 ð/Þg;
R4 ¼ ½fYb2 ð/Þg=fb2 ð/ÞP þ C P g; R6 ¼ ½ðYC P Þ=ðC P P þ qPb Þ;
R3 ¼ ½Y=ðP þ C p Þ;
R5 ¼ ½ðYC P Þ=fC P P þ b2 ð/Þg;
R7 ¼ ½ðY qPb Þ=ðqPb C P P þ C P Þ;
R8 ¼ ½fYb2 ð/Þg=fb2 ð/ÞP þ qPb g;
R9 ¼ ½ðY qPb Þ=fqPb P þ b2 ð/Þg;
R10 ¼ ½Y=ðP þ qPb Þ: Keeping the form of the ratio-type estimators tsi, (i = 1, 2, . . . , 10); one may suggest the class of estimators for population mean Y as
b b þ gÞg; ^/ ðP PÞfðwP þb t 4 ¼ ½y þ gÞðw P
ð1:18Þ
where w and g are either real numbers or function of known parameters of the auxiliary attribute / such as CP, b2(/) and qPb. The sum of w and g is not necessarily equal to unity. Note that the ratio-type estimators tsi, (i = 1, 2, . . . , 10); are members of the class of estimators t4 for different values of w and g. To the first order of approximation, the MSE of the class of estimators t4 is given by
h i MSEðt 4 Þ ¼ fð1 f Þ=ng R S2/ þ S2y 1 q2Pb ;
ð1:19Þ
where R ¼ ½ðwYÞ=ðwP þ gÞ. The MSE of the class of estimators t4 is minimum, when R⁄[=0]. Thus the resulting minimum MSE of t4 is given by
MSEmin ðt4 Þ ¼ fð1 f Þ=ngS2y 1 q2Pb ;
ð1:20Þ
which is equal to the variance (or MSE) of the usual linear regression estimator tReg (i.e. MSEmin(t4) = MSE(tReg)). Motivated by Kadilar and Cingi [5], Abd-Elfattah et al. [1] have suggested the following classes of estimators for the population mean Yusing the information on coefficient of kurtosis (b2(/)) and coefficient of variation (CP), respectively as
( ) ( ) # b b ^/ ðP PÞ ^/ ðP PÞ þb þb y y fP þ b2 ð/Þg ; P þ m2 b b þ b ð/Þg P fP 2 " ( # ( ) ) b b ^/ ðP PÞ ^/ ðP PÞ þb þb y y ¼ m1 ðP þ C P Þ ; P þ m2 b b þ b ð/Þg P fP 2 ( ) ) " ( # b b ^/ ðP PÞ ^/ ðP PÞ þb þb y y fb2 ð/ÞP þ C P g ; P þ m2 ¼ m1 b þ CP g b P fb2 ð/Þ P " ( # ( ) ) b b ^/ ðP PÞ ^/ ðP PÞ þb þb y y ¼ m1 fC P P þ b2 ð/Þg ; P þ m2 b þ b ð/Þg b P fC P P "
ta1 ¼ m1
ð1:21Þ
t a2
ð1:22Þ
t a3 t a4
ð1:23Þ
ð1:24Þ
2
where (m1, m2) are weights that satisfy the condition [(m1 + m2) = 1]. To the first order of approximation, the common minimum MSE of the classes of estimators taj is given by
MSEmin ðtaj Þ ¼ fð1 f Þ=ngS2y 1 q2Pb ;
ðj ¼ 1; 2; 3; 4Þ ¼ MSEðt Reg Þ:
ð1:25Þ
Further following the procedure outlined in Singh and Agnihotri [[7], p. 77], Abd-Elfattah et al. [1] suggested the ratio-type class of estimators for population mean Y as
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H.P. Singh, R.S. Solanki / Applied Mathematics and Computation 218 (2012) 7798–7812
b þ gÞg; fðwP þ gÞ=ðw P t a5 ¼ y
ð1:26Þ
where w and g are either real numbers or function of known parameters of the auxiliary attribute / such as CP, b2(/) and qPb. Note that the sum of w and g is not necessarily equal to unity. The following ratio-type estimators
b þ C P Þg; fðP þ C P Þ=ð P ta5ð1Þ ¼ y
ð1:27Þ
b þ b ð/Þg; ½fP þ b2 ð/Þg=f P ¼y 2
ð1:28Þ
b þ C P g; ½fb2 ð/ÞP þ C P g=fb2 ð/Þ P ta5ð3Þ ¼ y
ð1:29Þ
b þ b ð/Þg; ½fC P P þ b2 ð/Þg=fC P P ta5ð4Þ ¼ y 2
ð1:30Þ
b þ q Þg; fðP þ qPb Þ=ð P ta5ð5Þ ¼ y Pb
ð1:31Þ
ta5ð2Þ
are members of the ratio-type class of estimators t a5 for (w, g)[=(1, CP), {1, b2(/)}, {b 2(/), CP}, {CP, b2(/)}, (1, qPb)] respectively. Some more ratio-type estimators on the lines of Kadilar and Cingi [5] can be generated from the ratio-type class of estimators ta5 for (w, g)[=(CP, qPb), (qPb, CP), {b2(/), qPb}, {qPb, b2(/)}], respectively as
b þ q Þg; fðC P P þ qPb Þ=ðC P P ta5ð6Þ ¼ y Pb
ð1:32Þ
ta5ð7Þ
b þ C P Þg; fðqPb P þ C P Þ=ðqPb P ¼y
ð1:33Þ
ta5ð8Þ
b þ q g; ½fb2 ð/ÞP þ qPb g=fb2 ð/Þ P ¼y Pb
ð1:34Þ
ta5ð9Þ
b þ b ð/Þg: ½fqPb P þ b2 ð/Þg=fqPb P ¼y 2
ð1:35Þ
To the first order of approximation, the MSEs of the ratio-type estimators ta5ðjÞ are given by
h i MSEft a5ðjÞ g ¼ fð1 f Þ=ngY 2 C 2y þ C 2P mj mj 2kP ;
ðj ¼ 1; 2; . . . ; 9Þ;
ð1:36Þ
where
m1 ¼ ½P=ðP þ C p Þ; m2 ¼ ½P=fP þ b2 ð/Þg; m3 ¼ ½b2 ð/ÞP=fb2 ð/ÞP þ C P g; m4 ¼ ½C P P=fC P P þ b2 ð/Þg; m5 ¼ ½P=ðP þ qPb Þ; m6 ¼ ½ðC P PÞ=ðC P P þ qPb Þ; m7 ¼ ½ðqPb PÞ=ðqPb P þ C P Þ; m8 ¼ ½fb2 ð/ÞPg=fb2 ð/ÞP þ qPb g; m9 ¼ ½ðqPb PÞ=fqPb P þ b2 ð/Þg The relevant references in this context are Sisodia and Dwivedi [2], Singh and Tailor [8], Upadhyaya and Singh [11] and Singh and Agnihotri [7]. To the first order of approximation, the minimum MSE of the ratio-type class of estimators t a5 is given by
MSEmin ðt a5 Þ ¼ fð1 f Þ=ngS2y 1 q2Pb ¼ MSEðt Reg Þ:
ð1:37Þ
From (1.2), (1.3), (1.4), (1.17), (1.20), (1.25) and (1.37) we have
Þ MSEðt Reg Þ½¼ MSEmin ðt2 Þ ¼ MSEmin ðt4 Þg ¼ MSEmin ðt aj Þ ¼ fð1 f Þ=ngY 2 C 2y q2Pb P 0; Varðy
ð1:38Þ 2
2
MSEðt 1 Þ MSEðt Reg Þ½¼ MSEmin ðt 2 Þ ¼ MSEmin ðt 4 Þg ¼ MSEmin ðt aj Þ ¼ fð1 f Þ=ngY ðC P qPb C y Þ P 0;
ð1:39Þ
2
MSEðt si Þ MSEðtReg Þ½¼ MSEmin ðt 2 Þ ¼ MSEmin ðt 4 Þg ¼ MSEmin ðt aj Þ ¼ fð1 f Þ=ngðRi S/ Þ P 0; ði ¼ 1; 2; . . . ; 10; j ¼ 1; 2; . . . ; 5Þ:
ð1:40Þ
, ratio estimator t1 due to Naik and Gupta [15] It is observed from (1.38), (1.39) and (1.40) that the usual unbiased estimator y and ratio-type estimators tsi, (i = 1, 2, . . . , 10) due to Singh et al. [13] are inferior to the usual linear regression estimator tReg, class of estimators t2 (at their optimum condition) due to Jhajj et al. [9], class of estimators t4 (at their optimum condition) and classes of estimators t aj (at their optimum conditions), (j = 1, 2, . . . , 5) due to Abd-Elfattah et al. [1]. Thus the procedure adopted by the authors Naik and Gupta [15], Jhajj et al. [9], Singh et al. [13] and Abd-Elfattah et al. [1] are unable to give estimators better than the usual linear regression estimators tReg. It poses a question. Is there any procedure which can give estimator better than the usual linear regression estimator tReg? Indeed Shabbir and Gupta [10] have given the procedure to answer this question. But the derivation of the result concerning MSE expression of the ratio-type estimator t3 proposed by them is not correct. So the entire results of the paper by Shabbir and Gupta [10] are erroneous. In this paper we have made an effort to propose the class of estimators for the population mean Y which includes the ratio-type estimator t3 envisaged by Shabbir and Gupta [10] and Abd-Elfattah et al. [1] class of estimators t a5 . We have obtained the bias and MSE expressions of the proposed class of estimators, up to first order of approximation. Asymptotic optimum estimator (AOE) in the proposed class of estimators is identified alongwith its MSE formula. The correct MSE expression of Shabbir and Gupta [10] ratio-type estimator t3 is also given. Two phase sampling version of the proposed class of estimators is also given alongwith its properties. Numerical illustration is given in support of the present study.
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2. The suggested class of estimators We define the following class of estimators for population mean Y as
!a wP þ dg b ½d1 þ d2 ðP PÞ t¼y ; b þ dg wP
ð2:1Þ
where w and g are either real numbers or function of known parameters of the auxiliary attribute /, such as Cp, b2(/) qPb and kP. The scalar a takes values 1, (for product-type estimator) and +1 (for ratio-type estimator); d is an integer which takes b þ dgÞ and (wP + dg) are non-negative and (d1, d2) are suitably values +1 and 1 for designing the estimators such that ðw P chosen constants such that the sum of the constants (d1, d2) need not be unity (i.e. (d1 + d2) – 1). Motivation of proposing the class of estimators t is taking from Upadhyaya and Singh [11], Kadilar and Cingi [3–5] and Shabbir and Gupta [10]. The class of estimators t reduces to following set of known estimators/classes of estimators for different values of constants (d1, d2, w, g, d, a): (The usual unbiased estimator), (i) for ðd1 ; d2 ; aÞ ¼ ð1; 0; 0Þ; t ! y (ii) for (w, g, a, d) = (1, 0, 1, 1), t ? t3 [10], (iii) for ðd1 ; d2 ; a; dÞ ¼ ð1; 0; 1; 1Þ; t ! ta5 [1]. To obtain the bias and MSE of suggested class of estimators t, we write
¼ Yð1 þ e0 Þ; y b ¼ Pð1 þ e1 Þ; P such that
Eðe0 Þ ¼ Eðe1 Þ ¼ 0 and up to first order of approximation, we have
E e20 ¼ fð1 f Þ=ngC 2y ; E e21 ¼ fð1 f Þ=ngC 2P ; Eðe0 e1 Þ ¼ fð1 f Þ=ngqPb C P C y : Expressing the suggested class of estimators t in terms of e’s, we have
t ¼ Yð1 þ e0 Þðd1 d2 Pe1 Þð1 þ me1 Þa ;
ð2:2Þ
where m = [(wP)/(wP + dg)]. We assume that jme1j < 1, so that (1 + me1)a is expandable. Expanding the right hand side of (2.2), multiplying out and neglecting terms of e’s having power greater than two, we have
ða þ 1Þ t ffi Y d1 ð1 þ e0 Þ ðd1 am þ d2 PÞðe1 þ e0 e1 Þ þ am d1 m þ d2 P e21 2 or
ða þ 1Þ ðt YÞ ffi Y d1 ð1 þ e0 Þ ðd1 am þ d2 PÞðe1 þ e0 e1 Þ þ am d1 m þ d2 P e21 1 : 2
ð2:3Þ
Taking expectation on both sides of (2.3), we get the bias of the suggested class of estimators t to the first order of approximation as
ð1 f Þ 2 d1 ða þ 1Þ C P am BðtÞ ¼ Y ðd1 1Þ þ m þ d2 P ðd1 am þ d2 PÞkP : n 2
ð2:4Þ
Squaring both sides of (2.3) and neglecting terms of e’s having power greater than two, we have
h 2 ðt YÞ2 ffi Y 2 1 þ d1 ð1 þ e0 Þ2 þ ðamd1 þ d2 PÞ2 e21 2d1 ð1 þ e0 Þþ2ðd1 am þ d2 PÞðe1 þ e0 e1 Þ d1 ða þ 1Þ d ða þ 1Þ 2am m þ d2 P e21 2d1 ðamd1 þ d2 PÞðe1 þ 2e0 e1 Þ þ 2amd1 1 m þ d2 P e21 2 2 n o h 2 2 ¼ Y 2 1 þ d1 1 þ 2e0 2ame1 þ e20 þ að2a þ 1Þm2 e21 4ame0 e1 þd2 P 2 e21 2d1 d2 Pðe1 þ 2e0 e1 Þ 2amPe21 aða þ 1Þ 2 2 2d1 1 þ e0 amðe1 þ e0 e1 Þ þ m e1 2d2 P e1 þ e0 e1 ame21 : ð2:5Þ 2
H.P. Singh, R.S. Solanki / Applied Mathematics and Computation 218 (2012) 7798–7812
7803
Taking expectation of both sides of (2.5), we get the MSE of the suggested class of estimators t to the first order of approximation as
h i 2 2 MSEðtÞ ¼ Y 2 1 þ d1 A þ d2 B þ 2d1 d2 C 2d1 D 2d2 E ;
ð2:6Þ
where
h n oi A ¼ 1 þ fð1 f Þ=ng C 2y þ amC 2P ðð2a þ 1Þm 4kP Þ ; B ¼ fð1 f Þ=ngP2 C 2P ; C ¼ f2ð1 f Þ=ngPC 2P ðam kP Þ; h i D ¼ 1 þ fð1 f Þ=ngamC 2P ffða þ 1Þ=2gm kP g ; E ¼ fð1 f Þ=ngPC 2P ðam kP Þ: The MSE of the suggested class of estimators t is minimized for
d1 ¼ ½ðBD CEÞ=ðAB C 2 Þ ¼ d1 ;
ðsayÞ;
d2 ;
ðsayÞ:
2
d2 ¼ ½ðAE CDÞ=ðAB C Þ ¼
Thus the resulting minimum MSE of class t is obtained as
" MSEmin ðtÞ ¼ Y 2 1
ðBD2 2CDE þ AE2 Þ ðAB C 2 Þ
# ð2:7Þ
:
Thus we established the following theorem. Theorem 2.1. To the first order of approximation,
"
MSEðtÞ P Y
2
1
ðBD2 2CDE þ AE2 Þ
#
ðAB C 2 Þ
with equality holding if
d1 ¼ d1
and d2 ¼ d2 :
Remark 2.1. For d1 = 1, the suggested class of estimators t reduces to the class of estimators as
b ½1 þ d2 ðP PÞ t ¼ y
wP þ dg b þ dg wP
!a
ð2:8Þ
:
Putting d1 = 1 in (2.5) and (2.7), we get the bias and MSE of the class of estimators t⁄, respectively as
ð1 f Þ 2 ða þ 1Þm YC P am kP þ Pd2 ðam kP Þ ; n 2 h i 2 2 MSEðt Þ ¼ Y 1 þ A 2D þ d2 B þ 2d2 ðC EÞ ;
Bðt Þ ¼
ð2:9Þ ð2:10Þ
where
h i ð1 þ A 2DÞ ¼ fð1 f Þ=ng C 2y þ amC 2P ðam 2kP Þ ; ðC EÞ ¼ fð1 f Þ=ngPC 2p ðam kP Þ: Minimization of MSE(t⁄) with respect to d2, yields the optimum values of d2 as
d2 ¼ fðC EÞ=Bg ¼ ½ðam kP Þ=P ¼ d ;
ðsayÞ:
⁄
Thus the resulting minimum MSE of class t is obtained as
MSEmin ðt Þ ¼ fð1 f Þ=ngS2y 1 q2Pb ¼ MSEðtReg Þ: Thus we state the following theorem. Theorem 2.2. To the first order of approximation,
MSEðt Þ P fð1 f Þ=ngS2y 1 q2Pb
ð2:11Þ
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H.P. Singh, R.S. Solanki / Applied Mathematics and Computation 218 (2012) 7798–7812
with equality holding if
d2 ¼ d : Remark 2.2. For [(w, g, a, d) = (1, 0, 1, 1)] in (2.4) and (2.6), we get the bias and MSE expressions of the Shabbir and Gupta [10] ratio-type estimator t3 to the first order of approximation, respectively as
Bðt 3 Þ ¼ Y½ðd1 1Þ þ fð1 f Þ=ngC 2P ðd1 þ d2 PÞð1 kP Þ; h i 2 2 MSEðt 3 Þ ¼ Y 2 1 þ d1 A þ d2 B þ 2d1 d2 C 2d1 D 2d2 E ;
ð2:12Þ ð2:13Þ
where
h n oi A ¼ 1 þ fð1 f Þ=ng C 2y þ C 2P ð3 4kP Þ ; B ¼ B ¼ fð1 f Þ=ngP 2 C 2P ; C ¼ f2ð1 f Þ=ngPC 2P ð1 kP Þ; D ¼ ½1 þ fð1 f Þ=ngC 2P ð1 kP Þ; E ¼ fð1 f Þ=ngPC 2P ð1 kP Þ: The MSE of the ratio-type estimator t3 is minimized for
d1 ¼ ½ðB D C E Þ=ðA B C 2 Þ ¼ d10 ;
ðsayÞ;
d20 ;
ðsayÞ:
2
d2 ¼ ½ðA E C D Þ=ðA B C Þ ¼ Explicit expressions of
d10
and
d20
are respectively obtained as
h i ¼ 1 fð1 f Þ=ngC 2P ð1 kP Þð1 2kP Þ h n oi = 1 þ fð1 f Þ=ng C 2y C 2P ð1 2kP Þ2 ; h n oi d20 ¼ ðk 1Þ 1 fð1 f Þ=ng C 2y þ C 2P ð1 2kP Þ h n oi = 1 þ fð1 f Þ=ng C 2y C 2P ð1 2kP Þ2 :
d10
Thus the resulting minimum MSE of the ratio-type estimator t3 is given by
h i " # 2 Þ 2 ð1 f Þ C 2y þ C 2P 1 kP kP ð1f C P ð1 kP Þ2 C 2y C 2P n ðB D2 2C D E þ A E2 Þ 2 h n o i ¼Y 1 : MSEmin ðt3 Þ ¼ Þ ðA B C 2 Þ C 2y C 2P ð1 2kP Þ2 n 1 þ ð1f n
ð2:14Þ
We note that (2.13) and (2.14) is the correct MSE (minimum MSE) expression of Shabbir and Gupta [10] ratio-type estimator t3. The MSE (minimum MSE) expression of ratio-type estimatort3 obtained by Shabbir and Gupta [[10] p.3, Eq. (2.4) and (2.7)] is not correct. From (1.4), (1.20), (1.25), (1.37), (2.11) and (2.14) we have
MSEðt Reg Þ½¼ MSEmin ðt 2 Þ ¼ MSEmin ðt 4 Þg ¼ MSEmin ðt aj Þ ¼ MSEmin t Þ MSEmin ðt 3 Þ ¼
Y 2 ½B ðA D Þ þ C ðE C Þ2 B ðA B C 2 Þ
P 0;
ðj ¼ 1; 2; . . . ; 5Þ:
ð2:15Þ
From (2.15), it is observed that the ratio-type estimator t3 [10] is more efficient than the usual linear regression estimator tReg, Jhajj et al. [9] class of estimators t2, class of estimators t4, Abd-Elfattah et al. [1] classes of estimators taj ; ðj ¼ 1; 2; . . . ; 5Þ , , ratio estimator t1 due to Naik and Gupta [15] and ratio-type class of estimators t⁄ and hence the usual unbiased estimator y estimators tsi, (i = 1, 2, . . . , 10) due to Singh et al. [13] at their optimum conditions. In addition to many, the estimators cited in Tables 1 and 2 are members of the suggested class of estimators t. ðjÞ Similarly some product-type estimators t pi ; ði ¼ 1; 2; . . . ; 20; j ¼ 1; 2Þ can be generated from the suggested class of estimators t.
3. Empirical study In this section we compare the performance of different estimators considered here using two population data sets as earlier considered by Shabbir and Gupta [10] and Abd-Elfattah et al. [1]. The description of population data sets are as follows.
H.P. Singh, R.S. Solanki / Applied Mathematics and Computation 218 (2012) 7798–7812 Table 1 Some members (ratio-type) of the class t, when d[ = 1]. Ratio-type estimator (a = 1)
w
g
ð1Þ 2 b b þ S/ Þ ½d1 þ d2 ðP PÞ½ðNP tr1 ¼ y þ S/ Þ=ðNP P ð1Þ 2 b b þ f Þ ½d1 þ d2 ðP PÞ½ðNP tr2 ¼ y þ f Þ=ðNP P ð1Þ b b ½d1 þ d2 ðP PÞ½fb tr3 ¼ y 2 ð/ÞP þ kP g=fb2 ð/Þ P þ ð1Þ 2 b b þ kP Þ ½d1 þ d2 ðP PÞ½ðNP tr4 ¼ y þ kP Þ=ðNP P ð1Þ b b þ 1Þ ½d1 þ d2 ðP PÞ½ðNP tr5 ¼ y þ 1Þ=ðN P ð1Þ b b þ C P Þ ½d1 þ d2 ðP PÞ½ðNP þ C P Þ=ðN P tr6 ¼ y ð1Þ b bþ ½d1 þ d2 ðP PÞ½ðNP tr7 ¼ y þ Pb Þ=ðN P Pb Þ ð1Þ b b þ S/ Þ ½d1 þ d2 ðP PÞ½ðNP tr8 ¼ y þ S/ Þ=ðN P ð1Þ b b þ f Þ ½d1 þ d2 ðP PÞ½ðNP tr9 ¼ y þ f Þ=ðN P ð1Þ b b þ gÞ ½d1 þ d2 ðP PÞ½ðNP þ gÞ=ðN P tr10 ¼ y ð1Þ b b þ kP Þ ½d1 þ d2 ðP PÞ½ðNP þ kP Þ=ðN P tr11 ¼ y ð1Þ b bþ ½d1 þ d2 ðP PÞ½ðnP tr12 ¼ y þ Pb Þ=ðn P Pb Þ ð1Þ b b tr13 ¼ y½d1 þ d2 ðP PÞ½ðnP þ S/ Þ=ðn P þ S/ Þ
NP
S/
q
kP g
q
NP
f
b2(/ )
kP
NP
kP
N
1
N
CP
N
qPb
N
S/
N
f
N
g = (1 f)
N
kP
n
qPb
n
S/
b b þ f Þ ½d1 þ d2 ðP PÞ½ðnP tr14 ¼ y þ f Þ=ðn P ð1Þ b b þ gÞ ½d1 þ d2 ðP PÞ½ðnP ¼y þ gÞ=ðn P t
n
f
n
g = (1 f)
ð1Þ b b þ kP Þ ½d1 þ d2 ðP PÞ½ðnP þ kP Þ=ðn P tr16 ¼ y ð1Þ b b þ Pg t ¼ y½d1 þ d2 ðP PÞ½fb ð/ÞP þ Pg=fb ð/Þ P
n
kP
b2(/)
P
q
q
ð1Þ
r15
r17
2
2
ð1Þ 2 b b þ PÞ ½d1 þ d2 ðP PÞ½ðNP þ PÞ=ðNP P tr18 ¼ y ð1Þ b b þ PÞ ½d1 þ d2 ðP PÞ½ðNP ¼y þ PÞ=ðN P t
NP
P
N
P
ð1Þ b b þ PÞ ½d1 þ d2 ðP PÞ½ðnP þ PÞ=ðn P tr20 ¼ y
n
P
r19
Table 2 Some members (ratio-type) of the class t, when d [1]. Ratio-type estimator (a = 1)
w
g
ð2Þ 2 b b S/ Þ ½d1 þ d2 ðP PÞ½ðNP tr1 ¼ y S/ Þ=ðNP P ð2Þ 2 b b f Þ ½d1 þ d2 ðP PÞ½ðNP tr2 ¼ y f Þ=ðNP P ð2Þ b b ½d1 þ d2 ðP PÞ½fb tr3 ¼ y 2 ð/ÞP kP g=fb2 ð/Þ P kP g ð2Þ 2 b b kP Þ ½d1 þ d2 ðP PÞ½ðNP tr4 ¼ y kP Þ=ðNP P ð2Þ b b 1Þ ½d1 þ d2 ðP PÞ½ðNP tr5 ¼ y 1Þ=ðN P ð2Þ b b C P Þ ½d1 þ d2 ðP PÞ½ðNP C P Þ=ðN P tr6 ¼ y ð2Þ b b ½d1 þ d2 ðP PÞ½ðNP tr7 ¼ y Pb Þ=ðN P Pb Þ ð2Þ b b S/ Þ ½d1 þ d2 ðP PÞ½ðNP tr8 ¼ y S/ Þ=ðN P ð2Þ b b f Þ ½d1 þ d2 ðP PÞ½ðNP tr9 ¼ y f Þ=ðN P ð2Þ b b gÞ ½d1 þ d2 ðP PÞ½ðNP gÞ=ðN P tr10 ¼ y ð2Þ b b kP Þ ½d1 þ d2 ðP PÞ½ðNP kP Þ=ðN P tr11 ¼ y ð2Þ b b ½d1 þ d2 ðP PÞ½ðnP tr12 ¼ y Pb Þ=ðn P Pb Þ ð2Þ b b tr13 ¼ y½d1 þ d2 ðP PÞ½ðnP S/ Þ=ðn P S/ Þ ð2Þ b b f Þ ½d1 þ d2 ðP PÞ½ðnP tr14 ¼ y f Þ=ðn P ð2Þ b b gÞ ½d1 þ d2 ðP PÞ½ðnP gÞ=ðn P tr15 ¼ y ð2Þ b b kP Þ ½d1 þ d2 ðP PÞ½ðnP kP Þ=ðn P tr16 ¼ y ð2Þ b b Pg tr17 ¼ y½d1 þ d2 ðP PÞ½fb2 ð/ÞP Pg=fb2 ð/Þ P ð2Þ 2 b b PÞ ½d1 þ d2 ðP PÞ½ðNP tr18 ¼ y PÞ=ðNP P ð2Þ b b PÞ ½d1 þ d2 ðP PÞ½ðNP PÞ=ðN P tr19 ¼ y ð2Þ b b PÞ ½d1 þ d2 ðP PÞ½ðnP PÞ=ðn P tr20 ¼ y
NP
S/
q
q
q
q
NP
f
b2 (/)
kP
NP
kP
N
1
N
CP
N
qPb
N
S/
N
f
N
g = (1 f)
N
kP
n
qPb
n
S/
n
f
n
g = (1 f)
n
kP
b2(/)
P
NP
P
N
P
n
P
Population I [Source: Sukhatme and Sukhatme [12], p. 256]. y = Number of villages in the circles. / = A circle consisting more than five villages. N ¼ 89; n ¼ 23; Y ¼ 3:36; P ¼ 0:124; qPb ¼ 0:766; C y ¼ 0:601; C P ¼ 2:678; b2 ð/Þ ¼ 6:162. Population II [Source: Sukhatme and Sukhatme [12], p. 256].
7805
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H.P. Singh, R.S. Solanki / Applied Mathematics and Computation 218 (2012) 7798–7812
y = Number of villages in the circles. / = A circle consisting more than five villages. N ¼ 89; n ¼ 23; Y ¼ 1102; P ¼ 0:124; qPb ¼ 0:624; C y ¼ 0:65; C P ¼ 2:678; b2 ð/Þ ¼ 6:162: , Naik and Gupta [15] ratio We have computed the percent relative efficiencies (PREs) of the usual unbiased estimator y estimator t1, Jhajj et al. [9] class of estimators t2, Shabbir and Gupta [10] ratio-type estimatort3, Singh et al. [13] ratio-type estimators tsi(i = 1, 2, . . . , 10), class of estimators t4, Abd-Elfattah et al. [1] classes of estimators taj ðj ¼ 1; 2; . . . ; 5Þ, ratio-type ðjÞ estimators t a5ðjÞ ðj ¼ 1; 2; . . . ; 9Þ, class of estimators t⁄, usual linear regression estimator tReg and ratio-type estimators tri (which are members of suggested class of estimators (t) (i = 1, 2, . . . , 20; j = 1, 2) with respect to the usual unbiased estimator (at their optimum conditions) and displayed in Tables 3–5. y It is observed from Tables 3–5 that. The usual linear regression estimator tReg (or the class of estimators t2 due to Jhajj et al. [9], the class of estimators t4, the classes of estimators t aj ðj ¼ 1; 2; . . . ; 5Þ due to Abd-Elfattah et al. [1] and class of estimators t⁄ at their optimum conditions) , ratio estimator t1 [15], ratio-type estimators tsi [13], (i = 1, 2, . . . , 10) is more efficient than the usual unbiased estimator y and ta5ðjÞ ; ðj ¼ 1; 2; . . . ; 9Þ in both population data sets I and II. The performance of the estimators t1, ts1, ts4, ts6, ts8 and ta5ð8Þ in both the population data sets I and II. are even worse than the usual unbiased estimator y ðjÞ The ratio-type estimators tri (which are members of suggested class of estimators t), (i = 1, 2, . . . , 20; j = 1, 2) perform better than the usual linear regression estimators tReg and hence than the estimators/classes of estimators ; t1 ; t2 ; t4 ; taj ; ðj ¼ 1; 2; . . . ; 5Þ; t ; t si ; ði ¼ 1; 2; . . . ; 10Þ and t a5ðjÞ ; ðj ¼ 1; 2; . . . ; 9Þ. Further we note that the proposed estimay ð1Þ tors t ri ; ði ¼ 1; 2; . . . ; 20Þ are inferior to Shabbir and Gupta [10] ratio-type estimator t3 (with corrected version of MSE). ð1Þ This shows that the suggested class of estimators t with d = 1 (i.e. tri ; i ¼ 1; 2; . . . ; 20Þ play no significant role in improving ð2Þ the efficiency of the estimators. Table 4 clearly indicates that the estimators t ri ; ði ¼ 1; 2; . . . ; 20Þ are superior to all the
Table 3 . PREs of different estimators with respect to y Estimator
ð1Þ tr1 ð1Þ tr2 ð1Þ tr3 ð1Þ tr4 ð1Þ tr5 ð1Þ tr6 ð1Þ tr7 ð1Þ tr8 ð1Þ tr9 ð1Þ tr10
Þ PREð; y
Estimator
Population I
Population II
300.18
186.62
308.21
189.09
302.64
188.63
320.19
193.60
328.43
194.98
300.18
186.62
334.15
197.61
346.58
199.93
348.97
200.56
334.78
196.75
Þ PREð; y Population I
Population II
ð1Þ tr11 ð1Þ tr12 ð1Þ tr13 ð1Þ tr14 ð1Þ tr15 ð1Þ tr16 ð1Þ tr17 ð1Þ tr18 ð1Þ tr19 ð1Þ tr20
351.89
201.51
334.15
197.61
322.22
193.22
328.43
194.98
297.92
185.91
336.81
197.90
Estimator
Þ PREð; y
312.85
190.48
328.43
194.98
353.57
201.76
342.04
198.72
Table 4 . PREs of different estimators with respect to y Estimator
ð2Þ tr1 ð2Þ tr2 ð2Þ tr3 ð2Þ tr4 ð2Þ tr5 ð2Þ tr6 ð2Þ tr7 ð2Þ tr8 ð2Þ tr9 ð2Þ tr10 ⁄
Þ PREð; y Population I
Population II
730.92
262.76
541.80
239.15
648.68
242.48
443.18
218.93
410.57
215.24
730.92
262.76
395.04
209.90
372.03
206.39
368.71
205.57
393.56
211.46
Indicates the largest efficiency.
Population I ð2Þ tr11 ð2Þ tr12 ð2Þ tr13 ð2Þ tr14 ð2Þ tr15 ð2Þ tr16 ð2Þ tr17 ð2Þ tr18 ð2Þ tr19 ð2Þ tr20
Population II
364.99
204.44
928.50⁄
250.83
433.61
220.07
410.57
215.24
848.50
273.48⁄
389.09
209.41
491.59
230.96
410.57
215.24
363.01
204.16
379.15
208.10
7807
H.P. Singh, R.S. Solanki / Applied Mathematics and Computation 218 (2012) 7798–7812 Table 5 . PREs of different estimators with respect to y Þ PREð; y
Estimator
⁄
Estimator
Population I
Population II
y t1 t2 [or t4or t aj or t⁄ or tReg ]
100 7.13 241.99
100 7.79 163.77
t3
Þ PREð; y Population I
Population II
ts9 ts10 ta5ð1Þ
239.34 125.21 135.72
163.07 92.84 124.12
358.13
202.93
ta5ð2Þ
114.54
110.48
ts1
4.93
5.69
ta5ð3Þ
215.97
143.87
ts2
237.55
162.01
ta5ð4Þ
142.28
127.97
ts3
221.18
155.31
ta5ð5Þ
230.24
162.84
ts4
71.86
69.11
ta5ð6Þ
133.08
79.25
ts5
214.98
152.67
ta5ð7Þ
126.67
115.08
ts6 ts7
44.86 229.07
37.62 160.25
ta5ð8Þ ta5ð9Þ
39.33 110.99
30.18 106.51
ts8
18.64
17.38
(j = 1,2, . . . , 5).
estimators presented in the paper including Shabbir and Gupta [10] ratio-type estimator t3 for both the population data ð2Þ sets I and II. It follows that the proposed class of estimators t with d = 1 (i.e. tri ; i ¼ 1; 2; . . . ; 20Þ play significant role in improving the efficiency of the estimators. However this conclusion should not extrapolated, in general. ð2Þ The estimator tr12 (based on n, P and qPb) gives the largest efficiency over other competitors in population I, while the ð2Þ estimator t r15 (based on n, P and g) has the largest efficiency over other competitors in population II. Thus taking into consideration the above discussions, we have concluded that the proposal of the class of estimators t is justified. 4. Two phase sampling In Section 2 we have discussed the problem of estimating the population mean Y of the study variable y assuming that the information about the proportion P of population units possessing the auxiliary attribute /, highly correlated with study variable y, is known in advance. However in many practical situations it may happen that the proportion P is not known, before the start of the survey. In such situations, we generally apply the double (two phase) sampling design to furnish an efficient b 0 denote the proportion of units possessing attribute / in the first phase sample of estimator for the population mean Y. Let P b denote the proportion of units possessing attribute / in the second phase sample of size n[
b 0 = PÞ: b ð P t1d ¼ y
ð4:1Þ
The MSE of the ratio estimator t1d, up to first order approximation is given by
MSEðt 1d Þ ¼ Y 2
ð1 f Þ 2 1 1 2 Cy þ 0 C P ð1 2kP Þ : n n n
ð4:2Þ
The class of estimators of population mean Y in two phase sampling due to Jhajj et al. [9] is given by
; u0 Þ; t2d ¼ hðy
ð4:3Þ
b P b 0 Þ and hðy ; u0 Þ is the function of ðy ; u0 Þ such that ½hðy ; 1Þ ¼ Y; 8Y and the function hðy ; u0 Þ satisfies certain regwhere u ½¼ ð P= ularity conditions similar to those given in Jhajj et al. [[9] (Sec. 3, p. 45)]. The minimum MSE of the class of estimators t2d, up to terms of order n1, is given by 0
MSEmin ðt 2d Þ ¼
1 1 2 1 1 2 S ; 0 Sy 1 q2Pb þ n n n0 N y
ð4:4Þ
which equals to the approximate MSE of the two phase sampling usual linear regression estimator, defined as
b 0 PÞ; b ^/ ð P þb tRegd ¼ ½y i.e., MSEmin(t2d) = MSE(tRegd). Shabbir and Gupta [10] suggested the ratio-type estimator for population mean Y in two phase sampling as
ð4:5Þ
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H.P. Singh, R.S. Solanki / Applied Mathematics and Computation 218 (2012) 7798–7812
b 0 PÞð b P b 0 = PÞ; b ½w1 þ w2 ð P t3d ¼ y
ð4:6Þ
where (w1, w2) are suitably chosen constants whose sum is not necessarily equal to ‘unity’. Two phase sampling version of Singh et al. [13] ratio-type estimators tsi, (i = 1, 2, . . . , 10) are given by
b 0 PÞfð b b 0 Þ=ð PÞg; b ^/ ð P þb tsd1 ¼ ½y P b0
b0
b0
b0
ð4:7Þ
b b þ b ð/Þg; ^/ ð P PÞ½f þb tsd2 ¼ ½y P þ b2 ð/Þg=f P 2
ð4:8Þ
b b þ C P Þg; ^/ ð P PÞfð þb tsd3 ¼ ½y P þ C P Þ=ð P b0
ð4:9Þ
b b ^/ ð P PÞ½fb þb tsd4 ¼ ½y 2 ð/Þ P þ C P g=fb2 ð/Þ P þ C P g; b0
b0
ð4:10Þ
b0
b b ^/ ð P PÞ½fC þb tsd5 ¼ ½y P P þ b2 ð/Þg=fC P P þ b2 ð/Þg;
ð4:11Þ
b0
ð4:12Þ
b b ^/ ð P PÞfðC þb tsd6 ¼ ½y P P þ qPb Þ=ðC p P þ qPb Þg; b0
b0
b ^/ ð P PÞfð þb qPb P þ C P Þ=ðqPb Pb þ C P Þg; tsd7 ¼ ½y b0 b0
ð4:13Þ
b0
b b ^/ ð P PÞ½fb þb tsd8 ¼ ½y 2 ð/Þ P þ qPb g=fb2 ð/Þ P þ qPb g;
ð4:14Þ
b0
ð4:15Þ
b b q P þ b ð/Þg=fq P ^/ ð P PÞ½f þb tsd9 ¼ ½y 2 Pb Pb þ b2 ð/Þg; b0
b0
b0
b b þ q Þg: ^/ ð P PÞfð þb P þ qPb Þ=ð P tsd10 ¼ ½y Pb
ð4:16Þ
To the first order of approximation, the MSEs of the ratio-type estimator tsdi are given by
MSEðt sdi Þ ¼
o 1 1 1 1 n 2 S2 ; 0 Sy 1 q2Pb þ S2/ R2i þ n n n0 N y
ð4:17Þ
where Ri, (i = 1, 2, . . . , 10) are same as defined in Section 1. Keeping the form of the ratio-type estimators tsdi, (i = 1, 2, . . . , 10); one may suggest the class of estimators for population mean Y as
b 0 PÞfðw b b 0 þ gÞ=ðw P b þ gÞg; ^/ ð P þb t4d ¼ ½y P
ð4:18Þ
where (w, g) are same as defined for the class of estimators t4 in Section 1. Note that the ratio-type estimators tsdi , (i = 1, 2, . . . , 10); are members of the class of estimators t4d for different values of w and g. To the first order of approximation, the MSE of class of the estimators t4d is given by
MSEðt 4d Þ ¼
o 1 1 1 1 n 2 0 Sy 1 q2pb þ R2 S2/ þ S2 ; n n n0 N y
ð4:19Þ
where R⁄ is same as defined earlier. The MSE of the class of estimators t4d is minimized for R⁄ = 0. Thus the resulting minimum MSE of t4d is given by
MSEmin ðt4d Þ ¼
1 1 2 1 1 2 0 Sy 1 q2Pb þ S ; n n n0 N y
ð4:20Þ
which equals to the approximate MSE of the usual linear regression estimator tRegd (i.e. MSEmin(t4d) = MSE(tRegd)). Motivated by Abd-Elfattah et al. [1], one may define the classes of estimators in two phase sampling as
"
# b 0 PÞg b b 0 PÞg b ^/ ð P ^/ ð P þb þb fy b 0 þ b ð/Þg ; b 0 þ m2 fy fP P 2 b b þ b ð/Þg P fP 2 " # b 0 PÞg b b 0 PÞg b ^/ ð P ^/ ð P þb þb fy fy 0 0 b b ðP þ CP Þ ; P þ m2 ¼ m2 b b þ b ð/Þg P fP 2 " # b 0 PÞg b b 0 PÞg b ^/ ð P ^/ ð P þb þb fy b 0 þ m2 fy b0 þ CP g ; fb2 ð/Þ P P ¼ m1 b b þ b ð/Þg P fP 2 " # 0 b b b b ^ ^ fy þ b/ ð P PÞg b 0 fy þ b/ ð P 0 PÞg 0 b fC P P þ b2 ð/Þg ; P þ m2 ¼ m1 b b þ b ð/Þg P fP 2
tad1 ¼ m1
ð4:21Þ
tad2
ð4:22Þ
tad3 tad4
ð4:23Þ
ð4:24Þ
where (m1, m2) are weights that satisfies the condition (m1 + m2) = 1. The common minimum MSE of the classes of estimators tadj , to the first order of approximation is given by
MSEmin ðtadj Þ ¼
1 1 2 1 1 2 S ¼ MSEðt Regd Þ; 0 Sy 1 q2Pb þ n n n0 N y
The following two phase sampling estimators
ðj ¼ 1; 2; . . . 4Þ:
ð4:25Þ
H.P. Singh, R.S. Solanki / Applied Mathematics and Computation 218 (2012) 7798–7812
7809
b 0 þ C P Þ=ð P b þ C P Þg; fð P tad5ð1Þ ¼ y
ð4:26Þ
b 0 þ b ð/Þg=f P b þ b ð/Þg; ½f P tad5ð2Þ ¼ y 2 2
ð4:27Þ
b 0 þ C P g=fb ð/Þ P b þ C P g; ½fb2 ð/Þ P tad5ð3Þ ¼ y 2
ð4:28Þ
b0
b þ b ð/Þg; ½fC P P þ b2 ð/Þg=fC P P tad5ð4Þ ¼ y 2
ð4:29Þ
b 0 þ q Þ=ð P b þ q Þg; fð P tad5ð5Þ ¼ y Pb Pb
ð4:30Þ
b0
b þ q Þg; fðC P P þ qPb Þ=ðC P P tad5ð6Þ ¼ y Pb
ð4:31Þ
b 0 þ C P Þ=ðq P b fðqPb P tad5ð7Þ ¼ y Pb þ C P Þg;
ð4:32Þ
b 0 þ q g=fb ð/Þ P b þ q g; ½fb2 ð/Þ P ¼y 2 Pb Pb
ð4:33Þ
b 0 þ b ð/Þg=fq P b ½fqPb P tad5ð9Þ ¼ y 2 Pb þ b2 ð/Þg;
ð4:34Þ
tad5ð8Þ
etc. are members of the following ratio-type class of estimators
b 0 þ gÞ=ðw P b þ gÞg; fðw P tad5 ¼ y
ð4:35Þ
where (w, g) are same as defined for the class t4 in Section 1. To the first order of approximation, the MSEs of the ratio-type estimators tad5ðjÞ ; ðj ¼ 1; 2; . . . ; 9Þ are given by
MSEft ad5ðjÞ g ¼ Y 2
1f 2 1 1 2 Cy þ 0 C P mj mj 2kP ; n n n
ð4:36Þ
where, mj ; ðj ¼ 1; 2; . . . ; 9Þ are same as defined in Section 1. To the first order of approximation, the minimum MSE of class tad5 is given by
MSEmin ðt ad5 Þ ¼
1 1 2 1 1 2 0 Sy 1 q2Pb þ S ¼ MSEmin ðtRegd Þ: 0 n n n N y
ð4:37Þ
5. The Suggested class of estimators in two phase The double sampling version of the suggested class of estimators t Section 2 for the population mean Y is defined by
b 0 PÞ b ½w1 þ w1 ð P td ¼ y
!a b 0 þ dg wP ; b þ dg wP
ð5:1Þ
where (w, g) are either real numbers or function of known parameters of the auxiliary attribute / such as Cp, b2(/) qPb and kp. The scalar a takes values 1 (for product-type estimator) and +1 (for ratio-type estimator); d is an integer which takes values b 0 þ dgÞ and ðw P b þ dgÞ are non-negative and (w1, w2) are suitably chosen +1 and 1 for designing the estimators such that ðw P constants whose sum is not necessarily equal to unity. To the first order of approximation, the bias and MSE of the class of estimators td are respectively obtained as
1 1 2 w1 ða þ 1Þm Bðtd Þ ¼ Y ðw1 1Þ þ 0 C P am þ w2 P ðw1 am þ w2 PÞkP ; n n 2 MSEðt d Þ ¼ Y 2 1 þ w21 A1 þ w22 B1 þ 2w1 w2 C 1 2w1 D1 2w2 E1 ; where
ð1 f Þ 2 1 1 A1 ¼ 1 þ Cy þ 0 amC 2P fð2a þ 1Þm 4kP g ; n n n
1 1 2 2 P CP ; B1 ¼ n n0
1 1 C 1 ¼ 2P 0 ðam kP ÞC 2P ; n n
1 1 ða þ 1Þ D1 ¼ 1 þ 0 amC 2P m kP ; n n 2
1 1 E1 ¼ PC 2P ðam kP Þ: n n0 The MSE of the class of estimators td is minimized for
ð5:2Þ ð5:3Þ
7810
H.P. Singh, R.S. Solanki / Applied Mathematics and Computation 218 (2012) 7798–7812
h i w1 ¼ ðB1 D1 C 1 E1 Þ= A1 B1 C 21 ¼ w1 ; h i w2 ¼ ðA1 E1 C 1 D1 Þ= A1 B1 C 21 ¼ w2 ;
ðsayÞ; ðsayÞ:
Thus the resulting minimum MSE of the class td is obtained as
3 B1 D21 2C 1 D1 E1 þ A1 E21 5: MSEmin ðtd Þ ¼ Y 2 41 A1 B1 C 21 2
ð5:4Þ
Now we established the following theorem. Theorem 5.1. To the first order of approximation,
3 B1 D21 2C 1 D1 E1 þ A1 E21 5 MSEðt d Þ P Y 2 41 A1 B1 C 21 2
with equality holding if
w1 ¼ w1
and w2 ¼ w2 :
Remark 5.1. For w1 = 1, the class of estimators td reduces to the class of estimators
!a b0 b 0 PÞ b w P þ dg : ½1 þ w2 ð P td ¼ y b þ dg wP
ð5:5Þ
It can be easily shown to the first order of approximation that the minimum MSE of the class t d is
MSEmin t d ¼
1 1 2 1 1 2 0 Sy 1 q2Pb þ S ¼ MSEðtRegd Þ: n n n0 N y
ð5:6Þ
In addition to many, the estimators cited in Tables 6 and 7 are members of the proposed class of estimators td. ðjÞ Similarly some product-type estimators tdpi ; ði ¼ 1; 2; . . . ; 20; j ¼ 1; 2Þ can be generated from the suggested class of estimators.
6. Empirical study In this section we compare the performance of different estimators considered here in two phase sampling using the same data sets as we discussed before. The description of population data sets are as follows.
Table 6 Some members (ratio-type) of the class td, when d [1]. Ratio-type estimator (a = 1)
w
ð1Þ b 0 PÞ½fb b b0 b ½w1 þ w2 ð P tdr1 ¼ y 2 ð/Þ P þ kP g=fb2 ð/Þ P þ ð1Þ 0 0 b b b b ½w1 þ w2 ð P PÞ½ðN P þ 1Þ=ðN P þ 1Þ tdr2 ¼ y ð1Þ b 0 PÞ½ðN b b 0 þ C P Þ=ðN P b þ C P Þ ½w1 þ w2 ð P P tdr3 ¼ y ð1Þ b 0 PÞ½ðN b b0 þ bþ ½w1 þ w2 ð P tdr4 ¼ y P Þ=ðN P Pb Pb Þ ð1Þ b 0 PÞ½ðN b b 0 þ S/ Þ=ðN P b þ S/ Þ ½w1 þ w2 ð P tdr5 ¼ y P ð1Þ b 0 PÞ½ðN b b 0 þ f Þ=ðN P b þ f Þ ½w1 þ w2 ð P tdr6 ¼ y P ð1Þ 0 0 b b b b þ gÞ tdr7 ¼ y½w1 þ w2 ð P PÞ½ðN P þ gÞ=ðN P ð1Þ b 0 PÞ½ðN b b 0 þ kP Þ=ðN P b þ kP Þ ½w1 þ w2 ð P P tdr8 ¼ y ð1Þ b 0 PÞ½ðn b b0 þ bþ ½w1 þ w2 ð P tdr9 ¼ y P Þ=ðn P Pb Pb Þ ð1Þ b 0 PÞ½ðn b b 0 þ S/ Þ=ðn P b þ S/ Þ ½w1 þ w2 ð P tdr10 ¼ y P ð1Þ b 0 PÞ½ðn b b 0 þ f Þ=ðn P b þ f Þ ½w1 þ w2 ð P tdr11 ¼ y P
b2(/)
kP
N
1
N
CP
q
q
ð1Þ tdr12 ð1Þ tdr13
q
q
b 0 PÞ½ðn b b 0 þ gÞ=ðn P b þ gÞ ½w1 þ w2 ð P ¼y P b 0 PÞ½ðn b b 0 þ kP Þ=ðn P b þ kP Þ ½w1 þ w2 ð P ¼y P
kP g
g
N
qPb
N
S/
N
f
N
g = (1 f)
N
kP
n
qPb
n
S/
n
f
n
g = (1 f)
n
kP
7811
H.P. Singh, R.S. Solanki / Applied Mathematics and Computation 218 (2012) 7798–7812 Table 7 Some members (ratio-type) of the class td when, d[=1]. Ratio-type estimator (a = 1)
w
ð2Þ b 0 PÞ½fb b b0 b ½w1 þ w2 ð P tdr1 ¼ y 2 ð/Þ P kP g=fb2 ð/Þ P ð2Þ b 0 PÞ½ðN b b 0 1Þ=ðN P b 1Þ ½w1 þ w2 ð P tdr2 ¼ y P ð2Þ b 0 PÞ½ðN b b 0 C P Þ=ðN P b C P Þ ½w1 þ w2 ð P P tdr3 ¼ y ð2Þ b 0 PÞ½ðN b b0 b ½w1 þ w2 ð P tdr4 ¼ y P Pb Þ=ðN P Pb Þ ð2Þ b 0 PÞ½ðN b b 0 S/ Þ=ðN P b S/ Þ ½w1 þ w2 ð P tdr5 ¼ y P ð2Þ b 0 PÞ½ðN b b 0 f Þ=ðN P b f Þ ½w1 þ w2 ð P tdr6 ¼ y P ð2Þ b 0 PÞ½ðN b b 0 gÞ=ðN P b gÞ ½w1 þ w2 ð P P tdr7 ¼ y ð2Þ b 0 PÞ½ðN b b 0 kP Þ=ðN P b kP Þ ½w1 þ w2 ð P P tdr8 ¼ y ð2Þ b 0 PÞ½ðn b b0 b ½w1 þ w2 ð P tdr9 ¼ y P Pb Þ=ðn P Pb Þ ð2Þ b 0 PÞ½ðn b b 0 S/ Þ=ðn P b S/ Þ ½w1 þ w2 ð P tdr10 ¼ y P ð2Þ b 0 PÞ½ðn b b 0 f Þ=ðn P b f Þ ½w1 þ w2 ð P tdr11 ¼ y P
b2(/)
kp
N
1
N
CP
q
q
kP g
q
q
ð2Þ b 0 PÞ½ðn b b 0 gÞ=ðn P b gÞ ½w1 þ w2 ð P P tdr12 ¼ y ð2Þ b 0 PÞ½ðn b b 0 kP Þ=ðn P b kP Þ ½w1 þ w2 ð P P tdr13 ¼ y
g
N
qPb
N
S/
N
f
N
g = (1 f)
N
kP
n
qPb
n
S/
n
f
n
g = (1 f)
n
kP
Table 8 . PREs of different estimators with respect to y Estimator
ð1Þ tdr1 ð1Þ tdr2 ð1Þ tdr3 ð1Þ tdr4 ð1Þ tdr5 ð1Þ tdr6 ð1Þ tdr7 ð1Þ tdr8 ð1Þ tdr9 ð1Þ tdr10 ð1Þ tdr11 ð1Þ tdr12 ð1Þ tdr13 ⁄
Þ PREð; y Population I
Population II
177.79
144.39
181.92
146.20
177.36
143.79
182.76
146.92
184.48
147.53
184.80
147.70
182.85
146.69
185.19
147.94
176.77
144.10
180.98
145.71
181.92
146.20
176.96
143.57
183.14
146.99
Estimator
Þ PREð; y Population I
Population II
ð2Þ tdr1 ð2Þ tdr2 ð2Þ tdr3 ð2Þ tdr4 ð2Þ tdr5 ð2Þ tdr6 ð2Þ tdr7 ð2Þ tdr8 ð2Þ tdr9 ð2Þ tdr10 ð2Þ tdr11 ð2Þ tdr12 ð2Þ tdr13
207.12
156.66
191.83
151.23
210.21
159.96
190.25
150.01
187.68
149.16
187.29
148.96
190.09
150.37
186.84
148.68
215.49⁄
158.08
193.98
152.29
191.83
151.23
213.63
161.52⁄
189.61
149.89
Indicates the largest efficiency.
Population I (Source: Sukhatme and Sukhatme [12], p. 256). y = Number of villages in the circles. / = A circle consisting more than five villages. N ¼ 89; n ¼ 23; n0 ¼ 45; Y ¼ 3:36; P ¼ 0:124; qPb ¼ 0:766; C y ¼ 0:601; C P ¼ 2:678; b2 ð/Þ ¼ 6:162: Population II (Source: Sukhatme and Sukhatme [12], p. 256). y = Number of villages in the circles. / = A circle consisting more than five villages. N ¼ 89; n ¼ 23; n0 ¼ 45; Y ¼ 1102; P ¼ 0:124; qPb ¼ 0:624; C y ¼ 0:65; C P ¼ 2:678; b2 ð/Þ ¼ 6:162. , Naik and Gupta [15] ratio We have computed the percent relative efficiencies (PREs) of the usual unbiased estimator y estimator t1d, Jhajj et al. [9] class of estimators t2d, Shabbir and Gupta [10] ratio-type estimatort3d, Singh et al. [13] ratio-type estimators tsdi, (i = 1, 2, . . . , 10); class of estimators t4d, Abd-Elfattah et al. [1] classes of estimators t adj ; ðj ¼ 1; 2; . . . ; 5Þ; ratiotype estimators tad5ðjÞ ; ðj ¼ 1; 2; . . . ; 9Þ; class of estimators t d , usual linear regression estimator tRegd and ratio-type estimators ðjÞ t dri (which are members of suggested class of estimators td) (i = 1, 2, . . . , 13; j = 1, 2), with respect to the usual unbiased esti (at their optimum conditions) and displayed in Tables 8 and 9. mator y
7812
H.P. Singh, R.S. Solanki / Applied Mathematics and Computation 218 (2012) 7798–7812
Table 9 . PREs of different estimators with respect to y Estimator
Þ PREð; y
Estimator
Population I
Population II
y t1d t2d [or t4d or tadj or td or tRegd]
100 10.43 163.09
100 11.36 134.54
tsd9 tsd10 tad5ð1Þ
t3d
Þ PREð; y Population I
Population II
162.29 115.30 121.00
175.97 114.41 114.69
185.99
148.30
tad5ð2Þ
109.13
106.67
tsd1
7.30
8.38
tad5ð3Þ
154.80
125.16
tsd2
161.74
133.75
tad5ð4Þ
124.36
116.83
tsd3
156.54
130.68
tad5ð5Þ
159.47
134.12
tsd4
79.48
77.24
tad5ð6Þ
119.60
85.28
tsd5
154.46
129.44
tad5ð7Þ
116.12
109.45
tsd6
55.24
47.77
tad5ð8Þ
49.58
39.61
tsd7
159.10
132.96
tad5ð9Þ
106.99
104.20
tsd8
25.79
24.19
It is observed from Tables 8 and 9 that the trends of the suggested estimators as well as other estimators presented in Section 4 are same as observed for the estimators in single phase sampling. However the performance of ratio-type estimator ð2Þ ð2Þ tdr9 appears to be the best among all the estimators (in two phase) discussed here in population I, while the estimator tdr12 is proved to be the best among the entire estimators (of two phase) in population II. Thus the proposal of the class of estimators td is justified in two phase (double) sampling. 7. Conclusion We have developed the generalized version of Shabbir and Gupta [10] ratio-type estimator which includes the estimators suggested by Naik and Gupta [15], Singh et al. [13], Abd-Elfattah et al. [1] and usual unbiased estimator. Asymptotic optimum estimator (AOE) in the proposed class of estimators is identified alongwith its mean square error formula. It has been shown that the AOE of the proposed class of estimators is more efficient than the estimators/classes of estimators due to Naik and Gupta [15], Jhajj et al. [9], Shabbir and Gupta [10], Singh et al. [13], Abd-Elfattah et al. [1] and usual linear regression estimator. Theoretical results are also supported through two natural population data sets earlier used by Shabbir and Gupta [10] and Abd-Elfattah et al. [1]. Double sampling version of the suggested class is also discussed along with its properties. Acknowledgement Authors wish to thank the Editor-in-Chief and two anonymous referees for their helpful comments that aided in improving this article. References [1] A.M. Abd-Elfattah, E.A. El-Sherpieny, S.M. Mohamed, O.F. Abdou, Improvement in estimating the population mean in simple random sampling using information on auxiliary attribute, Applied Mathematics and Computation 215 (2010) 4198–4202. [2] B.V.S. Sisodia, V.K. Dwivedi, A modified ratio estimator using coefficient of an auxiliary variable, Journal of the Indian Society of Agricultural Statistics 33 (1981) 13–18. [3] C. Kadilar, H. Cingi, Ratio estimators for the population variance in simple and stratified random sampling, Applied Mathematics and Computation 173 (2) (2006) 1047–1059. [4] C. Kadilar, H. Cingi, An improvement in estimating the population mean by using the correlation coefficient, Hacettepe Journal of Mathematics and Statistics 35 (1) (2006) 103–109. [5] C. Kadilar, H. Cingi, Improvement in estimating the population mean in simple random sampling, Applied Mathematics Letters 19 (2006) 75–79. [6] D.C. Roy, A regression-type estimator in two phase sampling using two auxiliary variable, Pakistan Journal of Statistics 19 (3) (2003) 281–290. [7] H.P. Singh, N. Agnihotri, A general procedure of estimating population mean using auxiliary information in sample surveys, Statistics in Transition-new Series 8 (1) (2008) 71–87. [8] H.P. Singh, R. Tailor, Use of known correlation coefficient in estimating the finite population mean, Statistics in Transition 6 (2003) 555–560. [9] H.S. Jhajj, M.K. Sharma, L.K. Grover, A family of estimators of population mean using information on auxiliary attribute, Pakistan Journal of Statistics 22 (1) (2006) 43–50. [10] J. Shabbir, S. Gupta, On Estimating the finite population mean with known population proportion of an auxiliary variable, Pakistan Journal of Statistics 23 (1) (2007) 1–9. [11] L.N. Upadhyaya, H.P. Singh, Use of transformed auxiliary variable in estimating the finite population mean, Biometrical Journal 41 (5) (1999) 627–636. [12] P.V. Sukhatme, B.V. Sukhatme, Sampling Theory of Surveys with Applications, Asia Publishing House, New Delhi, India, 1970. [13] R. Singh, P. Chauhan, N. Sawan, F. Smarandache, Ratio estimators in simple random sampling using information on auxiliary attribute, Pakistan Journal of Statistics and Operation Research 4 (1) (2008) 47–53. [14] S.K. Ray, R.K. Singh, Difference-cum-ratio type estimators, Journal of the Indian Statistical Association 19 (1981) 147–151. [15] V.D. Naik, P.C. Gupta, A note on estimating of mean with known population of an auxiliary character, Journal of Indian Society of Agricultural Statistics 48 (2) (1996) 151–158.