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Quartile ranked set sampling for estimating the distribution function
Journal of the Egyptian Mathematical Society (2015) xxx, xxx–xxx
Egyptian Mathematical Society
Journal of the Egyptian Mathematical Society www.etms-eg.org www.elsevier.com/locate/joems
ORIGINAL ARTICLE
Quartile ranked set sampling for estimating the distribution function Amer Ibrahim Al-Omari Al al-Bayt University, Faculty of Science, Department of Mathematics, Mafraq, Jordan Received 24 January 2014; revised 5 July 2014; accepted 15 April 2015
KEYWORDS Distribution function; Quartile ranked set sampling; Simple random sampling; Ranked set sampling; Relative efficiency
Abstract Quartile ranked set sampling (QRSS) method is suggested by Muttlak (2003) for estimating the population mean. In this article, the QRSS procedure is considered to estimate the distribution function of a random variable. The proposed QRSS estimator is compared with its counterparts based on simple random sampling (SRS) and ranked set sampling (RSS) schemes. It is found that the suggested estimator of the distribution function of a random variable X for a given x is biased and more efficient than its competitors using SRS and RSS. 2010 MATHEMATICS SUBJECT CLASSIFICATION:
62G30; 62D05; 62G05; 62G07
ª 2015 The Author. Production and hosting by Elsevier B.V. on behalf of Egyptian Mathematical Society. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/ licenses/by-nc-nd/4.0/).
1. Introduction Let X1 ; X2 ; . . . ; Xn be a random sample of size n having probability density function (pdf) fðxÞ and cumulative distribution function (cdf) FðxÞ, with a finite mean l and variance r2 . Let X11i ; X12i ; . . . ; X1ni ; X21i ; X22i ; . . . ; X2ni ; . . . ; Xn1i ; Xn2i ; . . . ; Xnni be n independent simple random samples each of size n in the ith cycle ði ¼ 1; 2; . . . ; mÞ. Let FSRS ðxÞ denote the empirical distribution function of a simple random sample X1 ; X2 ; . . . ; Xnm from FðxÞ. Bahadur [2] showed that FSRS ðxÞ has the following properties: E-mail address: [email protected] Peer review under responsibility of Egyptian Mathematical Society.
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1. F SRS ðxÞ is an unbiased estimator of F ðxÞ for a given x. 1 2. Var½F SRS ðxÞ ¼ mn F ðxÞ½1 F ðxÞ. 3. F SRS ðxÞ is a consistent estimator of F ðxÞ. The RSS was first suggested by McIntyre [3] as a method for estimating the mean of pasture and forage yields. The RSS is a useful method when the sampling units can be easily ranked than quantified. McIntyre proposed the ranked set sample mean as an estimator of the population mean and showed that the RSS mean estimator is unbiased and is more efficient than the SRS counterpart. The RSS can be described as follows: randomly select n sets each of size n from the target population. Then, visually rank the units within each sample with respect to the variable of interest. From the first set of n units the smallest ranked unit is selected. From the second set of n units the second smallest ranked unit is selected. The process is continued until the largest ranked unit is measured from the nth set. To increase the
http://dx.doi.org/10.1016/j.joems.2015.01.006 1110-256X ª 2015 The Author. Production and hosting by Elsevier B.V. on behalf of Egyptian Mathematical Society. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Please cite this article in press as: A.I. Al-Omari, Quartile ranked set sampling for estimating the distribution function, Journal of the Egyptian Mathematical Society (2015), http://dx.doi.org/10.1016/j.joems.2015.01.006
2
A.I. Al-Omari Table 1
The relative precision of FQRSS ðxÞ with respect to FSRS ðxÞ and bias values of FQRSS ðxÞ for 4 6 n 6 11.
FðxÞ 0.01 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 0.99
RP Bias RP Bias RP Bias RP Bias RP Bias RP Bias RP Bias RP Bias RP Bias RP Bias RP Bias
n¼4
n¼6
n¼8
n ¼ 10
n¼5
n¼7
n¼9
n ¼ 11
0.5270 0.0090 0.6852 0.0718 0.9939 0.0973 1.6994 0.0849 2.9902 0.0475 4.3694 0.0005 3.0824 0.0481 1.7065 0.0848 1.0080 0.0960 0.6628 0.0736 0.5126 0.0098
7.9487 0.0093 1.4566 0.0429 1.3260 0.0254 1.6332 0.0044 2.2115 0.0041 2.5640 0.0008 2.2121 0.0040 1.6410 0.0051 1.3748 0.0277 1.4455 0.0428 7.3353 0.0092
5.3923 0.0088 1.1587 0.0066 1.0919 0.0482 1.5037 0.0727 3.3200 0.0505 7.3573 0.0000 3.2736 0.0508 1.5109 0.0733 1.1220 0.0483 1.1659 0.0063 4.9833 0.0086
9.1690 0.0099 1.2156 0.0650 1.2896 0.0396 1.7397 0.0098 2.9599 0.0224 4.8399 0.0005 2.9261 0.0216 1.7521 0.0088 1.2898 0.0391 1.2054 0.0650 9.4109 0.0099
11.0248 0.0096 1.7024 0.0655 1.3138 0.0802 1.2961 0.0673 1.3908 0.0367 1.4534 0.0003 1.3843 0.0354 1.2757 0.0662 1.2997 0.0812 1.6756 0.0659 11.8336 0.0096
7.0763 0.0092 1.4095 0.0356 1.4388 0.0142 1.9075 0.0062 2.5254 0.0103 2.8914 0.0005 2.5467 0.0105 1.8497 0.0067 1.4229 0.0136 1.4006 0.0352 7.3257 0.0093
4.5962 0.0084 1.1780 0.0006 1.1553 0.0527 1.7570 0.0679 3.3480 0.0447 5.5273 0.0004 3.3917 0.0443 1.7220 0.0686 1.1603 0.0536 1.1633 0.0004 4.4370 0.0084
8.2674 0.0099 1.1780 0.0592 1.3873 0.0257 1.9325 0.0201 3.1859 0.0265 4.8417 0.0003 3.2440 0.0256 1.9539 0.0194 1.3658 0.0256 1.1870 0.0592 8.6399 0.0099
sample size, the whole process can be repeated m times to obtain a set of size nm units. Let Xjð1:nÞi ; Xjð2:nÞi ; . . . ; Xjðn:nÞi be the order statistics of the jth sample Xj1i ; Xj2i ; . . . ; Xjni ðj ¼ 1; 2; . . . ; nÞ in the ith cycle ði ¼ 1; 2; . . . ; mÞ. Then, the measured units X1ð1:nÞi ; X2ð2:nÞi ; . . . ; Xnðn:nÞi are denoted to the RSS. David and Nagaraja [4] showed that the cdf and the pdf of the jth order statistic Xð j:nÞ are given by n X n ½FðxÞi ½1 FðxÞni ; 1 < x < 1; Fð j:nÞ ðxÞ ¼ i i¼j and fð j:nÞ ðxÞ ¼
n! ½FðxÞj1 ½1 FðxÞnj fðxÞ: ð j 1Þ!ðn jÞ!
The mean and the variance of Xð j:nÞ are given by 2 R1 R1 lð j:nÞ ¼ 1 xfð j:nÞ ðxÞdx and r2ð j:nÞ ¼ 1 x lð j:nÞ fð j:nÞ ðxÞdx, respectively. Takahasi and Wakimoto [5] independently introduced the same method of RSS with mathematical theory and showed that n n 1X 1X fð j:nÞ ðxÞ; l ¼ l ; and n j¼1 n j¼1 ð j:nÞ n n 2 1X 1X r2ð j:nÞ þ lð j:nÞ l : r2 ¼ n j¼1 n j¼1
fðxÞ ¼
For a fixed x, Stokes and Sager [6] suggested an estimator for FðxÞ using RSS as FRSS ðxÞ ¼
m X n 1 X I Xjð j:nÞi 6 x ; mn i¼1 j¼1
where IðÞ is an indicator function. They proved the following: 1. F RSS ðxÞ is an unbiased estimator for F ðxÞ. P 2. Var½F RSS ðxÞ ¼ mn1 2 nj¼1 F ð j:nÞ ðxÞ 1 F ð j:nÞ ðxÞ .
½F RSS ðxÞ pðxÞE ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi converges in distribution to the standard 3. F RSS Var½F RSS ðxÞ
normal as m ! 1 when x and n are fixed. For more about estimation of the distribution function in ranked set sampling methods see Stokes and Sager [6], Samawi and Al-Sagheer [7], Kim and Kim [8], Al-Saleh and Samuh [9], and Ghosh and Tiwari [10]. The rest of this paper is organized as follows. In Section 2, we introduced the suggested estimation of the distribution function using QRSS method. The performance of the new estimator against its SRS and RSS counterparts is given in Section 3. Section 4, is devoted for some inferences about FðxÞ. In Section 5, some concluding remarks are provided. 2. Estimation of FðxÞ using QRSS The quartile ranked set sampling procedure as suggested by Muttlak [1] can be summarized as follows. Randomly select n samples each of size n units from the target population and rank the units within each sample with respect to the variable of interest. If the sample size n is even, select and measure from the first n=2 samples the Q1 ðn þ 1Þth smallest ranked unit of each sample, i.e., the first quartile, and from the second n=2 samples the Q3 ðn þ 1Þth smallest ranked unit of each sample, i.e., the third quartile. Note that, we always take the nearest integer of Q1 ðn þ 1Þth and Q3 ðn þ 1Þth where Q1 ¼ 25%, and Q3 ¼ 75%. If the sample size n is odd, select and measure from the first ðn 1Þ=2 samples the Q1 ðn þ 1Þth smallest ranked unit of each sample and from the other ðn 1Þ=2 samples the Q3 ðn þ 1Þth smallest ranked unit of each sample, and from one sample the median for that sample. The cycle can be repeated m times if needed to get a sample of size nm units. If the sample size n is even, in the ith cycle ði ¼ 1; 2; . . . ; mÞ, let XjðQ1 ðnþ1Þ:nÞi be the ðQ1 ðn þ 1ÞÞth smallest ranked unit of the jth sample j ¼ 1; 2; . . . ; n2 , and XjðQ3 ðnþ1Þ:nÞi be the ðQ3 ðn þ 1ÞÞth smallest ranked unit of the jth sample j ¼ nþ2 ; nþ4 ;...;n . 2 2
Please cite this article in press as: A.I. Al-Omari, Quartile ranked set sampling for estimating the distribution function, Journal of the Egyptian Mathematical Society (2015), http://dx.doi.org/10.1016/j.joems.2015.01.006
Estimation of the distribution function using QRSS
3
Note that, the measured units X1ðQ1 ðnþ1Þ:nÞi ; X2ðQ1 ðnþ1Þ:nÞi ; . . . ; Xn2ðQ1 ðnþ1Þ:nÞi are independent and identically distributed (iid) random variables, and also Xnþ2 ; Xnþ4 ;...; 2 ðQ3 ðnþ1Þ:nÞi 2 ðQ3 ðnþ1Þ:nÞi XnðQ3 ðnþ1Þ:nÞi are iid. However, all units X1ðQ1 ðnþ1Þ:nÞi ; X2ðQ1 ðnþ1Þ:nÞi ; ; Xnþ4 ; . . . ; XnðQ3 ðnþ1Þ:nÞi are . . . ; Xn2ðQ1 ðnþ1Þ:nÞi ; Xnþ2 2 ðQ3 ðnþ1Þ:nÞi 2 ðQ3 ðnþ1Þ:nÞi mutually independent but not identically distributed. These measured units are denoted by QRSSE. For odd sample size, let XjðQ1 ðnþ1Þ:nÞi be the Q1 ðn þ 1Þth smallest ranked unit of the jth sample j ¼ 1; 2; . . . ; n1 , 2 , Xjðnþ1:nÞi be the median of the jth sample of the rank j ¼ nþ1 2
with variance 1 FQ1 ðxÞ 1 FQ1 ðxÞ þ FQ3 ðxÞ 1 FQ3 ðxÞ : Var FQRSSE ðxÞ ¼ 2nm ð2Þ
And the suggested QRSSO estimator of FðxÞ is given by n1
m X 2 1 1 X FQRSSO ðxÞ ¼ I XjðQ1 ðnþ1Þ:nÞi 6 x þ I Xnþ1ðnþ1:nÞi 6 x 2 2 nm i¼1 j¼1 nm
þ
2
and XjðQ3 ðnþ1Þ:nÞi be the Q3 ðn þ 1Þth smallest ranked unit of the jth sample j ¼ nþ3 ; nþ5 ; . . . ; n . Note that, the only mea2 2 are iid, sured units X1ðQ1 ðmþ1Þ:mÞi ; X2ðQ1 ðnþ1Þ:nÞi ; . . . ; Xn1 2 ðQ1 ðnþ1Þ:nÞi nþ5 and also Xnþ3 ; X ; . . . ; X are iid. nðq ðnþ1Þ:nÞi ðQ ðnþ1Þ:nÞi ðQ ðnþ1Þ:nÞi 3 3 3 2 2 However, all units X1ðQ1 ðmþ1Þ:mÞi ; X2ðQ1 ðnþ1Þ:nÞi ; . . . ; Xn1 ; 2 ðQ1 ðnþ1Þ:nÞi nþ5 Xnþ1ðnþ1:nÞi ; Xnþ3 ; X ; . . . ; X are nðq3 ðnþ1Þ:nÞi 2 ðQ3 ðnþ1Þ:nÞi 2 ðQ3 ðnþ1Þ:nÞi 2
2
mutually independent but not identically distributed. These measured units are denoted by QRSSO. Following Stokes and Sager [6], let V0 ¼ ðV1 ; V2 ; . . . ; Vn Þ be a multinomial random vector with nm trials and P ¼ 1n ; 1n ; . . . ; 1n be a probability vector, and assume that the nm random variables were found by first observing V and then selecting Vj units randomly from a population with pdf fð j:nÞ ðxÞ ð j ¼ 1; 2; . . . ; nÞ. Let Y1 ; Y2 ; . . . ; Ynm denote to the obtained mn units. Also, following Stokes and Sager [6] we have the following theorem based on QRSS method. Theorem 1. Based on the same conditions of Theorem 1 in Stokes and Sager [6] we have the following: (1) For even sample size, fY 1 ; Y 2 ; . . . ; Y nm jV ¼ nm nm g has the same probability structure ; 0; 0; . . . ; 0; 2 2 as X jðQ1 ðnþ1Þ:nÞi ; X lðQ3 ðnþ1Þ:nÞi ; j ¼ 1; 2; . . . ; n2 ; l ¼ nþ2 ; 2 nþ4 ; . . . ; n; i ¼ 1; 2; . . . ; mg; a QRSSE from the same 2 population. n ; (2) For odd sample size, Y 1 ; Y 2 ; . . . ; Y nm jV ¼ v1 ¼ ðn1Þm 2 o ðn1Þm ; 0; 0; . . . ; 0; vðnþ1Þ ¼ m; 0; 0; . . . ; 0; vn ¼ 2
m X n 1 X I XjðQ3 ðnþ1Þ:nÞi 6 x ; nm i¼1 nþ3 j¼
ð3Þ
2
with variance 1 n 1 Var FQRSSO ðxÞ ¼ 2 FQ1 ðxÞ 1 FQ1 ðxÞ nm 2 þ FQ3 ðxÞ 1 FQ3 ðxÞ
þ FQ2 ðxÞ 1 FQ2 ðxÞ ;
ð4Þ
where
n 1 3n 3 ; FQ1 ðxÞ ¼ B FðxÞ; ; 4 4 nþ1 nþ1 ; ; FQ2 ðxÞ ¼ B FðxÞ; 2 2 3n 3 n 1 ; ; FQ3 ðxÞ ¼ B FðxÞ; 4 4 and Bðw; a; bÞ is the incomplete beta function defined as Rw Bðw; a; bÞ ¼ 0 ua1 ð1 uÞb1 du. Assume that n is odd, and nþ1 is odd, and let 2 Q1 ¼ Xðnþ3Þ ; Q2 ¼ Xðnþ1Þ , and Q3 ¼ Xð3nþ1Þ , then 4
2
4
n1 3n3 1 fQ1 ðxÞ ¼ n1 3n3 ½FðxÞ 4 ½1 FðxÞ 4 fðxÞ; B 4 ; 4
ð5Þ
n1 1 fQ2 ðxÞ ¼ nþ1 nþ1 ½FðxÞð1 FðxÞÞ 2 fðxÞ; B 2 ; 2
ð6Þ
and 3n3 n1 1 fQ3 ðxÞ ¼ 3n3 n1 ½FðxÞ 4 ½1 FðxÞ 4 fðxÞ; B 4 ; 4
ð7Þ
2
where Bðd; #Þ is the Beta function defined as has the same probability structure as n1 nþ1 XjðQ1 ðnþ1Þ:nÞi ; Xtðt:nÞi XlðQ3 ðnþ1Þ:nÞi ; j ¼ 1; 2; . . . ; ; t : 2 2
nþ3 nþ5 ; ; . . . ; n; i ¼ 1; 2; . . . ; m ; l¼ 2 2
Bðd; #Þ ¼
CðdÞCð#Þ ðd 1Þ!ð# 1Þ! : ¼ Cðd þ #Þ ðd þ # 1Þ!
ð8Þ
Some properties of the QRSSE and QRSSO estimators of the distribution function are provided in the following propositions. Proposition 1.
a QRSSO from the same population. Proof. The proof is directly and similar to the proof of Theorem 1 in Stokes and Sager [6]. h The suggested QRSSE estimator of FðxÞ is given by
(1) E F QRSSE ðxÞ ¼ 12 F Q1 ðxÞ þ F Q3 ðxÞ . n1 (2) E F QRSSO ðxÞ ¼ 2n F Q1 ðxÞ 1 F Q1 ðxÞ þ F Q3 ðxÞ 1 F Q3 ðxÞ g þ 1n F Q2 ðxÞ; where F Q1 ðxÞ; F Q2 ðxÞ; F Q3 ðxÞ, are defined above.
n
FQRSSE ðxÞ ¼
m X m X n 2 1 X 1 X I XjðQ1 ðnþ1Þ:nÞi 6 x I XjðQ3 ðnþ1Þ:nÞi 6 x ; nm i¼1 j¼1 nm i¼1 nþ2 j¼
2
ð1Þ
Proposition 2. FQRSSO ðxÞ and FQRSSE ðxÞ are biased estimators of FðxÞ, with bias given by
Please cite this article in press as: A.I. Al-Omari, Quartile ranked set sampling for estimating the distribution function, Journal of the Egyptian Mathematical Society (2015), http://dx.doi.org/10.1016/j.joems.2015.01.006
4
A.I. Al-Omari
1 Bias FQRSSE ðxÞ ¼ FQ1 ðxÞ þ FQ3 ðxÞ FQRSSE ðxÞ; 2 and n 1 FQ1 ðxÞ 1 FQ1 ðxÞ þ FQ3 ðxÞ 1 FQ3 ðxÞ Bias FQRSSO ðxÞ ¼ 2n 1 þ FQ2 ðxÞ FQRSSO ðxÞ: n
It is easy to prove Propositions 1 and 2 based on some algebra. From this proposition we can see that the suggested estimators are biased, but based on the calculations given in Section 3, the bias is close to zero.
distribution function with respect to SRS and RSS estimators based on the same number of measured units. We considered different sample sizes 4 6 n 6 11. The relative precisions (RP) of FRSS ðxÞ and FQRSS ðxÞ with respect to FSRS ðxÞ are defined as: Var½FSRS ðxÞ ; and Var½FRSS ðxÞ Var½FSRS ðxÞ ; RP FQRSS ðxÞ; FSRS ðxÞ ¼ MSE FQRSS ðxÞ RP½FRSS ðxÞ; FSRS ðxÞ ¼
ð9Þ
As Zi ’s are independent and identically with finite mean and variance random variables, then based on the central limit thepffiffiffi m½Zi EðZi Þ orem we can conclude pffiffiffiffiffiffiffiffiffiffiffi converges in distribution to
where MSE is the mean squared error of an estimator. The results are presented in Tables 1 and 2 using QRSS and RSS, respectively. The results in bold fonts in both tables are the best RP values of the distribution function estimators QRSS and RSS with respect to SRS for different values of FðxÞ. Based on QRSS from Table 1, the largest RP values are observed as FðxÞ goes to zero or 1. For example, with n ¼ 5 and FðxÞ ¼ 0:1, 0.99, the RP values are 11.0248 and 11.8336, respectively. Otherwise, the RP values increase when FðxÞ is close to 0.5. In all cases the bias is negligible and close to zero. From Table 2 we observe that FRSS ðxÞ is more efficient than FSRS ðxÞ. However, FQRSS ðxÞ is better than FRSS ðxÞ for most of cases considered in this study. As an example, when n ¼ 9 and FðxÞ ¼ 0:4, the relative precisions of RSS and QRSSO are 2.6347 and 3.3480, respectively. The optimal values of the sample size using QRSSE and QRSSO methods for estimating FðxÞ are summarized in Table 3. From Table 3 it can be noted that QRSSO is more efficient than QRSSE for estimating FðxÞ. Also, the sample size n ¼ 5 has most occurrence among other sample sizes considered in this study.
the standard normal distribution. The second part can be proved similarly by assuming
4. Inferences about the distribution function
Proposition 3. For fixed n and m ! 1, we have the following: F QRSSE ðxÞE½F QRSSE ðxÞ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi converges in distribution to N ð0; 1Þ. Var½F QRSSE ðxÞ F QRSSO ðxÞE½F QRSSO ðxÞ 2. pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi converges in distribution to Nð0; 1Þ. Var½F QRSSO ðxÞ
1.
Proof. Following Samawi and Al-Sagheer [7] and Kim and Kim [8], the proof of the first part can be done by assuming n
n 2 1X 1X Zi ¼ I XjðQ1 ðnþ1Þ:nÞi 6 x þ I XjðQ3 ðnþ1Þ:nÞi 6 x ; i n j¼1 n nþ2 j¼
2
¼ 1; 2; . . . ; m:
VarðZi Þ
Zi ¼
n1 2 1 1X I XjðQ1 ðnþ1Þ:nÞi 6 x þ I Xnþ1ðnþ1:nÞi 6 x 2 2 n j¼1 n
þ
n 1X I XjðQ3 ðnþ1Þ:nÞi 6 x : n nþ3 j¼
2
3. Numerical comparisons Numerical comparisons are considered in this section to investigate the performance of the suggested QRSS estimator of the Table 2
In this section, a pointwise estimate of FðxÞ is considered and some inferences about the population distribution are presented using QRSS. As m gets large, then based on Proposition 3, an approximate 100ð1 aÞ% confidence interval for E½FR ðxÞ; R ¼ QRSSE; QRSSO; RSS; SRS, is given by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d½FR ðxÞ < E½FR ðxÞ < FR ðxÞ FR ðxÞ Za=2 Var qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d½FR ðxÞ; ð10Þ þ Za=2 Var where Za=2 is the upper 100ða=2Þ% quantile of the Nð0; 1Þ, and
The relative precision of FRSS ðxÞ with respect to FSRS ðxÞ for 4 6 n 6 11.
FðxÞ
n¼4
n¼6
n¼8
n ¼ 10
n¼5
n¼7
n¼9
n ¼ 11
0.01 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 0.99
0.9886 1.3071 1.5471 1.6880 1.7891 1.8487 1.7841 1.6816 1.5507 1.2972 1.1206
1.0388 1.4832 1.8193 2.0402 2.2099 2.2449 2.2061 2.0456 1.8057 1.4716 1.0458
1.1355 1.6539 2.0421 2.3306 2.4696 2.5601 2.4787 2.3528 2.0610 1.6241 1.0523
1.1018 1.7929 2.2984 2.6122 2.7623 2.8271 2.7775 2.6052 2.3004 1.8179 1.0764
1.0851 1.3798 1.6973 1.8933 1.9969 2.0644 1.9545 1.9025 1.6731 1.3743 1.0146
1.0644 1.5550 1.9610 2.2060 2.2982 2.3500 2.3337 2.1952 1.9757 1.5503 1.0781
1.0712 1.7035 2.2084 2.4760 2.6347 2.6801 2.6349 2.4917 2.1988 1.7450 1.1562
1.0749 1.9029 2.3711 2.7136 2.8966 2.9600 2.9724 2.7661 2.4008 1.8920 1.1301
Please cite this article in press as: A.I. Al-Omari, Quartile ranked set sampling for estimating the distribution function, Journal of the Egyptian Mathematical Society (2015), http://dx.doi.org/10.1016/j.joems.2015.01.006
Estimation of the distribution function using QRSS Table 3
5
The best values of the sample size using QRSS for estimating FðxÞ.
FðxÞ
0.01
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.90
0.99
n
5
5
7
11
9
8
9
11
7
5
5
( n 2 X 1 F^jðQ1 ðnþ1Þ:nÞ ðxÞ 1 F^jðQ1 ðnþ1Þ:nÞ ðxÞ 2 ðm 1Þn j¼1 ) n X ^ ^ FjðQ ðnþ1Þ:nÞ ðxÞ 1 FjðQ ðnþ1Þ:nÞ ðxÞ ð11Þ þ
d FQRSSE ðxÞ ¼ Var
3
3
j¼nþ2 2
1 d FQRSSO ðxÞ ¼ Var ðm 1Þn2 8 n1 9 2 > X > > > > ^ ^ ^ ^ > > FjðQ1 ðnþ1Þ:nÞ ðxÞ 1 FjðQ1 ðnþ1Þ:nÞ ðxÞ þ FQ2 ðxÞ 1 FQ2 ðxÞ > > > < = j¼1 ; ð12Þ n X > > > > > > ^ ^ > > FjðQ3 ðnþ1Þ:nÞ ðxÞ 1 FjðQ3 ðnþ1Þ:nÞ ðxÞ þ > > : ; nþ3 j¼
2
and Upper BoundðUBÞ ¼ FR ðxÞ þ Za=2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d½FR ðxÞ: Var
Finally, an approximate 100ð1 aÞ% confidence for FðxÞ can be obtained by solving the equations 2LB ¼ hðFÞ and 2UB ¼ hðFÞ, numerically, or by any suitable method, where based on QRSSE we have hðFÞ ¼ FQ1 ðxÞ þ FQ3 ðxÞ; and using QRSSO it will be 1 hðFÞ ¼ ðn 1Þ FQ1 ðxÞ 1 FQ1 ðxÞ n þ FQ3 ðxÞ 1 FQ3 ðxÞ þ 2FQ2 ðxÞ : It is of interest to note here that any possible solution for last two equations should be singular since hðFÞ is increasing function in FðxÞ.
n 1 X Var½FRSS ðxÞ ¼ Fð j:nÞ ðxÞ 1 Fð j:nÞ ðxÞ ; 2 mn j¼1
and
5. Conclusion
d½FSRS ðxÞ ¼ Var
1 ^ ^ FðxÞ 1 FðxÞ ; nm 1
where based on QRSS, RSS, and SRS, respectively, we have m 1X F^iðt:nÞ ðxÞ ¼ I Xjðt:nÞi 6 x ; t ¼ Q1 ðn þ 1Þ; Q3 ðn þ 1Þ; m i¼1
nþ1 ; j ¼ 1; 2; . . . ; n; 2
ð13Þ
m 1X F^ið j:nÞ ðxÞ ¼ I Xjð j:nÞi 6 x ; j ¼ 1; 2; . . . ; n; m i¼1
ð14Þ
Acknowledgments
and 1 F^SRS ðxÞ ¼ nm
In this study, QRSS procedure is considered to estimate the distribution function of a random variable. The QRSS is compared with RSS and SRS in estimating the distribution function. It is found that QRSS is more efficient than SRS and also it is more efficient than RSS in most cases considered in this study. The RP values are preferred as FðxÞ goes to zero or 1 and increase when FðxÞ is close to 0.5. The bias of the QRSS estimator of the distribution function is negligible.
nm X
IðYi 6 xÞ:
ð15Þ
i¼1
Based on the 100ð1 aÞ% confidence interval of E½FR ðxÞ; R ¼ QRSSE; QRSSO; RSS; SRS, a 100ð1 aÞ% confidence interval for FðxÞ can be found as 1 0 FR ðxÞ E½FR ðxÞ C B P@Za=2 6 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 6 Z1a=2 A ¼ 1 a: d Var ½FR ðxÞ
ð16Þ
Solving (16) for E½FR ðxÞ; R ¼ QRSSE; QRSSO; RSS; SRS to get qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d½FR ðxÞ 6 E½FR ðxÞ 6 FR ðxÞ P FR ðxÞ Z1a=2 Var qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d½FR ðxÞ ¼ 1 a; þ Za=2 Var and the limits will be Lower BoundðLBÞ ¼ FR ðxÞ Z1a=2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d½FR ðxÞ; Var
The author is thankful to the editor and the anonymous reviewers for their valuable comments and suggestions that significantly improved the original version of the article. References [1] H.A. Muttlak, Investigating the use of quartile ranked set samples for estimating the population mean, Appl. Math. Comput. 146 (2003) 437–443. [2] R.R. Bahadur, A note on quantiles in large samples, Ann. Math. Stat. 37 (1996) 577–580. [3] G.A. McIntyre, A method for unbiased selective sampling using ranked sets, Aust. J. Agric. Res. 3 (1952) 385–390. [4] H.A. David, H.N. Nagaraja, Order Statistics, 3rd ed., John Wiley & Sons, Inc., Hoboken, New Jersey, 2003. [5] K. Takahasi, K. Wakimoto, On the unbiased estimates of the population mean based on the sample stratified by means of ordering, Ann. Inst. Stat. Math. 20 (1968) 1–31. [6] S.L. Stocks, T. Sager, Characterization of a ranked set sample with application to estimating distribution functions, J. Am. Stat. Assoc. 83 (1988) 374–381.
Please cite this article in press as: A.I. Al-Omari, Quartile ranked set sampling for estimating the distribution function, Journal of the Egyptian Mathematical Society (2015), http://dx.doi.org/10.1016/j.joems.2015.01.006
6 [7] H. Samawi, O.M. Al-Sageer, On the estimation of the distribution function using extreme and median ranked set sampling, Biometrical J. 43 (3) (2001) 357–373. [8] D.H. Kim, D.W. Kim, G.H. Kim, On the estimation of the distribution function using extreme median ranked set sampling, JKDAS 7 (2) (2005) 429–439.
A.I. Al-Omari [9] M.F. Al-Saleh, M.H. Samuh, On multistage ranked set sampling for distribution and median estimation, Comput. Stat. Data Anal. 52 (4) (2008) 2066–2078. [10] K. Ghosh, R.C. Tiwari, Estimating the distribution function using k-tuple ranked set samples, J. Stat. Plann. Infer. 138 (4) (2008) 929–949.
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Journal of the Egyptian Mathematical Society www.etms-eg.org www.elsevier.com/locate/joems
ORIGINAL ARTICLE
Quartile ranked set sampling for estimating the distribution function Amer Ibrahim Al-Omari Al al-Bayt University, Faculty of Science, Department of Mathematics, Mafraq, Jordan Received 24 January 2014; revised 5 July 2014; accepted 15 April 2015
KEYWORDS Distribution function; Quartile ranked set sampling; Simple random sampling; Ranked set sampling; Relative efficiency
Abstract Quartile ranked set sampling (QRSS) method is suggested by Muttlak (2003) for estimating the population mean. In this article, the QRSS procedure is considered to estimate the distribution function of a random variable. The proposed QRSS estimator is compared with its counterparts based on simple random sampling (SRS) and ranked set sampling (RSS) schemes. It is found that the suggested estimator of the distribution function of a random variable X for a given x is biased and more efficient than its competitors using SRS and RSS. 2010 MATHEMATICS SUBJECT CLASSIFICATION:
62G30; 62D05; 62G05; 62G07
ª 2015 The Author. Production and hosting by Elsevier B.V. on behalf of Egyptian Mathematical Society. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/ licenses/by-nc-nd/4.0/).
1. Introduction Let X1 ; X2 ; . . . ; Xn be a random sample of size n having probability density function (pdf) fðxÞ and cumulative distribution function (cdf) FðxÞ, with a finite mean l and variance r2 . Let X11i ; X12i ; . . . ; X1ni ; X21i ; X22i ; . . . ; X2ni ; . . . ; Xn1i ; Xn2i ; . . . ; Xnni be n independent simple random samples each of size n in the ith cycle ði ¼ 1; 2; . . . ; mÞ. Let FSRS ðxÞ denote the empirical distribution function of a simple random sample X1 ; X2 ; . . . ; Xnm from FðxÞ. Bahadur [2] showed that FSRS ðxÞ has the following properties: E-mail address: [email protected] Peer review under responsibility of Egyptian Mathematical Society.
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1. F SRS ðxÞ is an unbiased estimator of F ðxÞ for a given x. 1 2. Var½F SRS ðxÞ ¼ mn F ðxÞ½1 F ðxÞ. 3. F SRS ðxÞ is a consistent estimator of F ðxÞ. The RSS was first suggested by McIntyre [3] as a method for estimating the mean of pasture and forage yields. The RSS is a useful method when the sampling units can be easily ranked than quantified. McIntyre proposed the ranked set sample mean as an estimator of the population mean and showed that the RSS mean estimator is unbiased and is more efficient than the SRS counterpart. The RSS can be described as follows: randomly select n sets each of size n from the target population. Then, visually rank the units within each sample with respect to the variable of interest. From the first set of n units the smallest ranked unit is selected. From the second set of n units the second smallest ranked unit is selected. The process is continued until the largest ranked unit is measured from the nth set. To increase the
http://dx.doi.org/10.1016/j.joems.2015.01.006 1110-256X ª 2015 The Author. Production and hosting by Elsevier B.V. on behalf of Egyptian Mathematical Society. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Please cite this article in press as: A.I. Al-Omari, Quartile ranked set sampling for estimating the distribution function, Journal of the Egyptian Mathematical Society (2015), http://dx.doi.org/10.1016/j.joems.2015.01.006
2
A.I. Al-Omari Table 1
The relative precision of FQRSS ðxÞ with respect to FSRS ðxÞ and bias values of FQRSS ðxÞ for 4 6 n 6 11.
FðxÞ 0.01 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 0.99
RP Bias RP Bias RP Bias RP Bias RP Bias RP Bias RP Bias RP Bias RP Bias RP Bias RP Bias
n¼4
n¼6
n¼8
n ¼ 10
n¼5
n¼7
n¼9
n ¼ 11
0.5270 0.0090 0.6852 0.0718 0.9939 0.0973 1.6994 0.0849 2.9902 0.0475 4.3694 0.0005 3.0824 0.0481 1.7065 0.0848 1.0080 0.0960 0.6628 0.0736 0.5126 0.0098
7.9487 0.0093 1.4566 0.0429 1.3260 0.0254 1.6332 0.0044 2.2115 0.0041 2.5640 0.0008 2.2121 0.0040 1.6410 0.0051 1.3748 0.0277 1.4455 0.0428 7.3353 0.0092
5.3923 0.0088 1.1587 0.0066 1.0919 0.0482 1.5037 0.0727 3.3200 0.0505 7.3573 0.0000 3.2736 0.0508 1.5109 0.0733 1.1220 0.0483 1.1659 0.0063 4.9833 0.0086
9.1690 0.0099 1.2156 0.0650 1.2896 0.0396 1.7397 0.0098 2.9599 0.0224 4.8399 0.0005 2.9261 0.0216 1.7521 0.0088 1.2898 0.0391 1.2054 0.0650 9.4109 0.0099
11.0248 0.0096 1.7024 0.0655 1.3138 0.0802 1.2961 0.0673 1.3908 0.0367 1.4534 0.0003 1.3843 0.0354 1.2757 0.0662 1.2997 0.0812 1.6756 0.0659 11.8336 0.0096
7.0763 0.0092 1.4095 0.0356 1.4388 0.0142 1.9075 0.0062 2.5254 0.0103 2.8914 0.0005 2.5467 0.0105 1.8497 0.0067 1.4229 0.0136 1.4006 0.0352 7.3257 0.0093
4.5962 0.0084 1.1780 0.0006 1.1553 0.0527 1.7570 0.0679 3.3480 0.0447 5.5273 0.0004 3.3917 0.0443 1.7220 0.0686 1.1603 0.0536 1.1633 0.0004 4.4370 0.0084
8.2674 0.0099 1.1780 0.0592 1.3873 0.0257 1.9325 0.0201 3.1859 0.0265 4.8417 0.0003 3.2440 0.0256 1.9539 0.0194 1.3658 0.0256 1.1870 0.0592 8.6399 0.0099
sample size, the whole process can be repeated m times to obtain a set of size nm units. Let Xjð1:nÞi ; Xjð2:nÞi ; . . . ; Xjðn:nÞi be the order statistics of the jth sample Xj1i ; Xj2i ; . . . ; Xjni ðj ¼ 1; 2; . . . ; nÞ in the ith cycle ði ¼ 1; 2; . . . ; mÞ. Then, the measured units X1ð1:nÞi ; X2ð2:nÞi ; . . . ; Xnðn:nÞi are denoted to the RSS. David and Nagaraja [4] showed that the cdf and the pdf of the jth order statistic Xð j:nÞ are given by n X n ½FðxÞi ½1 FðxÞni ; 1 < x < 1; Fð j:nÞ ðxÞ ¼ i i¼j and fð j:nÞ ðxÞ ¼
n! ½FðxÞj1 ½1 FðxÞnj fðxÞ: ð j 1Þ!ðn jÞ!
The mean and the variance of Xð j:nÞ are given by 2 R1 R1 lð j:nÞ ¼ 1 xfð j:nÞ ðxÞdx and r2ð j:nÞ ¼ 1 x lð j:nÞ fð j:nÞ ðxÞdx, respectively. Takahasi and Wakimoto [5] independently introduced the same method of RSS with mathematical theory and showed that n n 1X 1X fð j:nÞ ðxÞ; l ¼ l ; and n j¼1 n j¼1 ð j:nÞ n n 2 1X 1X r2ð j:nÞ þ lð j:nÞ l : r2 ¼ n j¼1 n j¼1
fðxÞ ¼
For a fixed x, Stokes and Sager [6] suggested an estimator for FðxÞ using RSS as FRSS ðxÞ ¼
m X n 1 X I Xjð j:nÞi 6 x ; mn i¼1 j¼1
where IðÞ is an indicator function. They proved the following: 1. F RSS ðxÞ is an unbiased estimator for F ðxÞ. P 2. Var½F RSS ðxÞ ¼ mn1 2 nj¼1 F ð j:nÞ ðxÞ 1 F ð j:nÞ ðxÞ .
½F RSS ðxÞ pðxÞE ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi converges in distribution to the standard 3. F RSS Var½F RSS ðxÞ
normal as m ! 1 when x and n are fixed. For more about estimation of the distribution function in ranked set sampling methods see Stokes and Sager [6], Samawi and Al-Sagheer [7], Kim and Kim [8], Al-Saleh and Samuh [9], and Ghosh and Tiwari [10]. The rest of this paper is organized as follows. In Section 2, we introduced the suggested estimation of the distribution function using QRSS method. The performance of the new estimator against its SRS and RSS counterparts is given in Section 3. Section 4, is devoted for some inferences about FðxÞ. In Section 5, some concluding remarks are provided. 2. Estimation of FðxÞ using QRSS The quartile ranked set sampling procedure as suggested by Muttlak [1] can be summarized as follows. Randomly select n samples each of size n units from the target population and rank the units within each sample with respect to the variable of interest. If the sample size n is even, select and measure from the first n=2 samples the Q1 ðn þ 1Þth smallest ranked unit of each sample, i.e., the first quartile, and from the second n=2 samples the Q3 ðn þ 1Þth smallest ranked unit of each sample, i.e., the third quartile. Note that, we always take the nearest integer of Q1 ðn þ 1Þth and Q3 ðn þ 1Þth where Q1 ¼ 25%, and Q3 ¼ 75%. If the sample size n is odd, select and measure from the first ðn 1Þ=2 samples the Q1 ðn þ 1Þth smallest ranked unit of each sample and from the other ðn 1Þ=2 samples the Q3 ðn þ 1Þth smallest ranked unit of each sample, and from one sample the median for that sample. The cycle can be repeated m times if needed to get a sample of size nm units. If the sample size n is even, in the ith cycle ði ¼ 1; 2; . . . ; mÞ, let XjðQ1 ðnþ1Þ:nÞi be the ðQ1 ðn þ 1ÞÞth smallest ranked unit of the jth sample j ¼ 1; 2; . . . ; n2 , and XjðQ3 ðnþ1Þ:nÞi be the ðQ3 ðn þ 1ÞÞth smallest ranked unit of the jth sample j ¼ nþ2 ; nþ4 ;...;n . 2 2
Please cite this article in press as: A.I. Al-Omari, Quartile ranked set sampling for estimating the distribution function, Journal of the Egyptian Mathematical Society (2015), http://dx.doi.org/10.1016/j.joems.2015.01.006
Estimation of the distribution function using QRSS
3
Note that, the measured units X1ðQ1 ðnþ1Þ:nÞi ; X2ðQ1 ðnþ1Þ:nÞi ; . . . ; Xn2ðQ1 ðnþ1Þ:nÞi are independent and identically distributed (iid) random variables, and also Xnþ2 ; Xnþ4 ;...; 2 ðQ3 ðnþ1Þ:nÞi 2 ðQ3 ðnþ1Þ:nÞi XnðQ3 ðnþ1Þ:nÞi are iid. However, all units X1ðQ1 ðnþ1Þ:nÞi ; X2ðQ1 ðnþ1Þ:nÞi ; ; Xnþ4 ; . . . ; XnðQ3 ðnþ1Þ:nÞi are . . . ; Xn2ðQ1 ðnþ1Þ:nÞi ; Xnþ2 2 ðQ3 ðnþ1Þ:nÞi 2 ðQ3 ðnþ1Þ:nÞi mutually independent but not identically distributed. These measured units are denoted by QRSSE. For odd sample size, let XjðQ1 ðnþ1Þ:nÞi be the Q1 ðn þ 1Þth smallest ranked unit of the jth sample j ¼ 1; 2; . . . ; n1 , 2 , Xjðnþ1:nÞi be the median of the jth sample of the rank j ¼ nþ1 2
with variance 1 FQ1 ðxÞ 1 FQ1 ðxÞ þ FQ3 ðxÞ 1 FQ3 ðxÞ : Var FQRSSE ðxÞ ¼ 2nm ð2Þ
And the suggested QRSSO estimator of FðxÞ is given by n1
m X 2 1 1 X FQRSSO ðxÞ ¼ I XjðQ1 ðnþ1Þ:nÞi 6 x þ I Xnþ1ðnþ1:nÞi 6 x 2 2 nm i¼1 j¼1 nm
þ
2
and XjðQ3 ðnþ1Þ:nÞi be the Q3 ðn þ 1Þth smallest ranked unit of the jth sample j ¼ nþ3 ; nþ5 ; . . . ; n . Note that, the only mea2 2 are iid, sured units X1ðQ1 ðmþ1Þ:mÞi ; X2ðQ1 ðnþ1Þ:nÞi ; . . . ; Xn1 2 ðQ1 ðnþ1Þ:nÞi nþ5 and also Xnþ3 ; X ; . . . ; X are iid. nðq ðnþ1Þ:nÞi ðQ ðnþ1Þ:nÞi ðQ ðnþ1Þ:nÞi 3 3 3 2 2 However, all units X1ðQ1 ðmþ1Þ:mÞi ; X2ðQ1 ðnþ1Þ:nÞi ; . . . ; Xn1 ; 2 ðQ1 ðnþ1Þ:nÞi nþ5 Xnþ1ðnþ1:nÞi ; Xnþ3 ; X ; . . . ; X are nðq3 ðnþ1Þ:nÞi 2 ðQ3 ðnþ1Þ:nÞi 2 ðQ3 ðnþ1Þ:nÞi 2
2
mutually independent but not identically distributed. These measured units are denoted by QRSSO. Following Stokes and Sager [6], let V0 ¼ ðV1 ; V2 ; . . . ; Vn Þ be a multinomial random vector with nm trials and P ¼ 1n ; 1n ; . . . ; 1n be a probability vector, and assume that the nm random variables were found by first observing V and then selecting Vj units randomly from a population with pdf fð j:nÞ ðxÞ ð j ¼ 1; 2; . . . ; nÞ. Let Y1 ; Y2 ; . . . ; Ynm denote to the obtained mn units. Also, following Stokes and Sager [6] we have the following theorem based on QRSS method. Theorem 1. Based on the same conditions of Theorem 1 in Stokes and Sager [6] we have the following: (1) For even sample size, fY 1 ; Y 2 ; . . . ; Y nm jV ¼ nm nm g has the same probability structure ; 0; 0; . . . ; 0; 2 2 as X jðQ1 ðnþ1Þ:nÞi ; X lðQ3 ðnþ1Þ:nÞi ; j ¼ 1; 2; . . . ; n2 ; l ¼ nþ2 ; 2 nþ4 ; . . . ; n; i ¼ 1; 2; . . . ; mg; a QRSSE from the same 2 population. n ; (2) For odd sample size, Y 1 ; Y 2 ; . . . ; Y nm jV ¼ v1 ¼ ðn1Þm 2 o ðn1Þm ; 0; 0; . . . ; 0; vðnþ1Þ ¼ m; 0; 0; . . . ; 0; vn ¼ 2
m X n 1 X I XjðQ3 ðnþ1Þ:nÞi 6 x ; nm i¼1 nþ3 j¼
ð3Þ
2
with variance 1 n 1 Var FQRSSO ðxÞ ¼ 2 FQ1 ðxÞ 1 FQ1 ðxÞ nm 2 þ FQ3 ðxÞ 1 FQ3 ðxÞ
þ FQ2 ðxÞ 1 FQ2 ðxÞ ;
ð4Þ
where
n 1 3n 3 ; FQ1 ðxÞ ¼ B FðxÞ; ; 4 4 nþ1 nþ1 ; ; FQ2 ðxÞ ¼ B FðxÞ; 2 2 3n 3 n 1 ; ; FQ3 ðxÞ ¼ B FðxÞ; 4 4 and Bðw; a; bÞ is the incomplete beta function defined as Rw Bðw; a; bÞ ¼ 0 ua1 ð1 uÞb1 du. Assume that n is odd, and nþ1 is odd, and let 2 Q1 ¼ Xðnþ3Þ ; Q2 ¼ Xðnþ1Þ , and Q3 ¼ Xð3nþ1Þ , then 4
2
4
n1 3n3 1 fQ1 ðxÞ ¼ n1 3n3 ½FðxÞ 4 ½1 FðxÞ 4 fðxÞ; B 4 ; 4
ð5Þ
n1 1 fQ2 ðxÞ ¼ nþ1 nþ1 ½FðxÞð1 FðxÞÞ 2 fðxÞ; B 2 ; 2
ð6Þ
and 3n3 n1 1 fQ3 ðxÞ ¼ 3n3 n1 ½FðxÞ 4 ½1 FðxÞ 4 fðxÞ; B 4 ; 4
ð7Þ
2
where Bðd; #Þ is the Beta function defined as has the same probability structure as n1 nþ1 XjðQ1 ðnþ1Þ:nÞi ; Xtðt:nÞi XlðQ3 ðnþ1Þ:nÞi ; j ¼ 1; 2; . . . ; ; t : 2 2
nþ3 nþ5 ; ; . . . ; n; i ¼ 1; 2; . . . ; m ; l¼ 2 2
Bðd; #Þ ¼
CðdÞCð#Þ ðd 1Þ!ð# 1Þ! : ¼ Cðd þ #Þ ðd þ # 1Þ!
ð8Þ
Some properties of the QRSSE and QRSSO estimators of the distribution function are provided in the following propositions. Proposition 1.
a QRSSO from the same population. Proof. The proof is directly and similar to the proof of Theorem 1 in Stokes and Sager [6]. h The suggested QRSSE estimator of FðxÞ is given by
(1) E F QRSSE ðxÞ ¼ 12 F Q1 ðxÞ þ F Q3 ðxÞ . n1 (2) E F QRSSO ðxÞ ¼ 2n F Q1 ðxÞ 1 F Q1 ðxÞ þ F Q3 ðxÞ 1 F Q3 ðxÞ g þ 1n F Q2 ðxÞ; where F Q1 ðxÞ; F Q2 ðxÞ; F Q3 ðxÞ, are defined above.
n
FQRSSE ðxÞ ¼
m X m X n 2 1 X 1 X I XjðQ1 ðnþ1Þ:nÞi 6 x I XjðQ3 ðnþ1Þ:nÞi 6 x ; nm i¼1 j¼1 nm i¼1 nþ2 j¼
2
ð1Þ
Proposition 2. FQRSSO ðxÞ and FQRSSE ðxÞ are biased estimators of FðxÞ, with bias given by
Please cite this article in press as: A.I. Al-Omari, Quartile ranked set sampling for estimating the distribution function, Journal of the Egyptian Mathematical Society (2015), http://dx.doi.org/10.1016/j.joems.2015.01.006
4
A.I. Al-Omari
1 Bias FQRSSE ðxÞ ¼ FQ1 ðxÞ þ FQ3 ðxÞ FQRSSE ðxÞ; 2 and n 1 FQ1 ðxÞ 1 FQ1 ðxÞ þ FQ3 ðxÞ 1 FQ3 ðxÞ Bias FQRSSO ðxÞ ¼ 2n 1 þ FQ2 ðxÞ FQRSSO ðxÞ: n
It is easy to prove Propositions 1 and 2 based on some algebra. From this proposition we can see that the suggested estimators are biased, but based on the calculations given in Section 3, the bias is close to zero.
distribution function with respect to SRS and RSS estimators based on the same number of measured units. We considered different sample sizes 4 6 n 6 11. The relative precisions (RP) of FRSS ðxÞ and FQRSS ðxÞ with respect to FSRS ðxÞ are defined as: Var½FSRS ðxÞ ; and Var½FRSS ðxÞ Var½FSRS ðxÞ ; RP FQRSS ðxÞ; FSRS ðxÞ ¼ MSE FQRSS ðxÞ RP½FRSS ðxÞ; FSRS ðxÞ ¼
ð9Þ
As Zi ’s are independent and identically with finite mean and variance random variables, then based on the central limit thepffiffiffi m½Zi EðZi Þ orem we can conclude pffiffiffiffiffiffiffiffiffiffiffi converges in distribution to
where MSE is the mean squared error of an estimator. The results are presented in Tables 1 and 2 using QRSS and RSS, respectively. The results in bold fonts in both tables are the best RP values of the distribution function estimators QRSS and RSS with respect to SRS for different values of FðxÞ. Based on QRSS from Table 1, the largest RP values are observed as FðxÞ goes to zero or 1. For example, with n ¼ 5 and FðxÞ ¼ 0:1, 0.99, the RP values are 11.0248 and 11.8336, respectively. Otherwise, the RP values increase when FðxÞ is close to 0.5. In all cases the bias is negligible and close to zero. From Table 2 we observe that FRSS ðxÞ is more efficient than FSRS ðxÞ. However, FQRSS ðxÞ is better than FRSS ðxÞ for most of cases considered in this study. As an example, when n ¼ 9 and FðxÞ ¼ 0:4, the relative precisions of RSS and QRSSO are 2.6347 and 3.3480, respectively. The optimal values of the sample size using QRSSE and QRSSO methods for estimating FðxÞ are summarized in Table 3. From Table 3 it can be noted that QRSSO is more efficient than QRSSE for estimating FðxÞ. Also, the sample size n ¼ 5 has most occurrence among other sample sizes considered in this study.
the standard normal distribution. The second part can be proved similarly by assuming
4. Inferences about the distribution function
Proposition 3. For fixed n and m ! 1, we have the following: F QRSSE ðxÞE½F QRSSE ðxÞ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi converges in distribution to N ð0; 1Þ. Var½F QRSSE ðxÞ F QRSSO ðxÞE½F QRSSO ðxÞ 2. pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi converges in distribution to Nð0; 1Þ. Var½F QRSSO ðxÞ
1.
Proof. Following Samawi and Al-Sagheer [7] and Kim and Kim [8], the proof of the first part can be done by assuming n
n 2 1X 1X Zi ¼ I XjðQ1 ðnþ1Þ:nÞi 6 x þ I XjðQ3 ðnþ1Þ:nÞi 6 x ; i n j¼1 n nþ2 j¼
2
¼ 1; 2; . . . ; m:
VarðZi Þ
Zi ¼
n1 2 1 1X I XjðQ1 ðnþ1Þ:nÞi 6 x þ I Xnþ1ðnþ1:nÞi 6 x 2 2 n j¼1 n
þ
n 1X I XjðQ3 ðnþ1Þ:nÞi 6 x : n nþ3 j¼
2
3. Numerical comparisons Numerical comparisons are considered in this section to investigate the performance of the suggested QRSS estimator of the Table 2
In this section, a pointwise estimate of FðxÞ is considered and some inferences about the population distribution are presented using QRSS. As m gets large, then based on Proposition 3, an approximate 100ð1 aÞ% confidence interval for E½FR ðxÞ; R ¼ QRSSE; QRSSO; RSS; SRS, is given by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d½FR ðxÞ < E½FR ðxÞ < FR ðxÞ FR ðxÞ Za=2 Var qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d½FR ðxÞ; ð10Þ þ Za=2 Var where Za=2 is the upper 100ða=2Þ% quantile of the Nð0; 1Þ, and
The relative precision of FRSS ðxÞ with respect to FSRS ðxÞ for 4 6 n 6 11.
FðxÞ
n¼4
n¼6
n¼8
n ¼ 10
n¼5
n¼7
n¼9
n ¼ 11
0.01 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 0.99
0.9886 1.3071 1.5471 1.6880 1.7891 1.8487 1.7841 1.6816 1.5507 1.2972 1.1206
1.0388 1.4832 1.8193 2.0402 2.2099 2.2449 2.2061 2.0456 1.8057 1.4716 1.0458
1.1355 1.6539 2.0421 2.3306 2.4696 2.5601 2.4787 2.3528 2.0610 1.6241 1.0523
1.1018 1.7929 2.2984 2.6122 2.7623 2.8271 2.7775 2.6052 2.3004 1.8179 1.0764
1.0851 1.3798 1.6973 1.8933 1.9969 2.0644 1.9545 1.9025 1.6731 1.3743 1.0146
1.0644 1.5550 1.9610 2.2060 2.2982 2.3500 2.3337 2.1952 1.9757 1.5503 1.0781
1.0712 1.7035 2.2084 2.4760 2.6347 2.6801 2.6349 2.4917 2.1988 1.7450 1.1562
1.0749 1.9029 2.3711 2.7136 2.8966 2.9600 2.9724 2.7661 2.4008 1.8920 1.1301
Please cite this article in press as: A.I. Al-Omari, Quartile ranked set sampling for estimating the distribution function, Journal of the Egyptian Mathematical Society (2015), http://dx.doi.org/10.1016/j.joems.2015.01.006
Estimation of the distribution function using QRSS Table 3
5
The best values of the sample size using QRSS for estimating FðxÞ.
FðxÞ
0.01
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.90
0.99
n
5
5
7
11
9
8
9
11
7
5
5
( n 2 X 1 F^jðQ1 ðnþ1Þ:nÞ ðxÞ 1 F^jðQ1 ðnþ1Þ:nÞ ðxÞ 2 ðm 1Þn j¼1 ) n X ^ ^ FjðQ ðnþ1Þ:nÞ ðxÞ 1 FjðQ ðnþ1Þ:nÞ ðxÞ ð11Þ þ
d FQRSSE ðxÞ ¼ Var
3
3
j¼nþ2 2
1 d FQRSSO ðxÞ ¼ Var ðm 1Þn2 8 n1 9 2 > X > > > > ^ ^ ^ ^ > > FjðQ1 ðnþ1Þ:nÞ ðxÞ 1 FjðQ1 ðnþ1Þ:nÞ ðxÞ þ FQ2 ðxÞ 1 FQ2 ðxÞ > > > < = j¼1 ; ð12Þ n X > > > > > > ^ ^ > > FjðQ3 ðnþ1Þ:nÞ ðxÞ 1 FjðQ3 ðnþ1Þ:nÞ ðxÞ þ > > : ; nþ3 j¼
2
and Upper BoundðUBÞ ¼ FR ðxÞ þ Za=2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d½FR ðxÞ: Var
Finally, an approximate 100ð1 aÞ% confidence for FðxÞ can be obtained by solving the equations 2LB ¼ hðFÞ and 2UB ¼ hðFÞ, numerically, or by any suitable method, where based on QRSSE we have hðFÞ ¼ FQ1 ðxÞ þ FQ3 ðxÞ; and using QRSSO it will be 1 hðFÞ ¼ ðn 1Þ FQ1 ðxÞ 1 FQ1 ðxÞ n þ FQ3 ðxÞ 1 FQ3 ðxÞ þ 2FQ2 ðxÞ : It is of interest to note here that any possible solution for last two equations should be singular since hðFÞ is increasing function in FðxÞ.
n 1 X Var½FRSS ðxÞ ¼ Fð j:nÞ ðxÞ 1 Fð j:nÞ ðxÞ ; 2 mn j¼1
and
5. Conclusion
d½FSRS ðxÞ ¼ Var
1 ^ ^ FðxÞ 1 FðxÞ ; nm 1
where based on QRSS, RSS, and SRS, respectively, we have m 1X F^iðt:nÞ ðxÞ ¼ I Xjðt:nÞi 6 x ; t ¼ Q1 ðn þ 1Þ; Q3 ðn þ 1Þ; m i¼1
nþ1 ; j ¼ 1; 2; . . . ; n; 2
ð13Þ
m 1X F^ið j:nÞ ðxÞ ¼ I Xjð j:nÞi 6 x ; j ¼ 1; 2; . . . ; n; m i¼1
ð14Þ
Acknowledgments
and 1 F^SRS ðxÞ ¼ nm
In this study, QRSS procedure is considered to estimate the distribution function of a random variable. The QRSS is compared with RSS and SRS in estimating the distribution function. It is found that QRSS is more efficient than SRS and also it is more efficient than RSS in most cases considered in this study. The RP values are preferred as FðxÞ goes to zero or 1 and increase when FðxÞ is close to 0.5. The bias of the QRSS estimator of the distribution function is negligible.
nm X
IðYi 6 xÞ:
ð15Þ
i¼1
Based on the 100ð1 aÞ% confidence interval of E½FR ðxÞ; R ¼ QRSSE; QRSSO; RSS; SRS, a 100ð1 aÞ% confidence interval for FðxÞ can be found as 1 0 FR ðxÞ E½FR ðxÞ C B P@Za=2 6 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 6 Z1a=2 A ¼ 1 a: d Var ½FR ðxÞ
ð16Þ
Solving (16) for E½FR ðxÞ; R ¼ QRSSE; QRSSO; RSS; SRS to get qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d½FR ðxÞ 6 E½FR ðxÞ 6 FR ðxÞ P FR ðxÞ Z1a=2 Var qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d½FR ðxÞ ¼ 1 a; þ Za=2 Var and the limits will be Lower BoundðLBÞ ¼ FR ðxÞ Z1a=2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d½FR ðxÞ; Var
The author is thankful to the editor and the anonymous reviewers for their valuable comments and suggestions that significantly improved the original version of the article. References [1] H.A. Muttlak, Investigating the use of quartile ranked set samples for estimating the population mean, Appl. Math. Comput. 146 (2003) 437–443. [2] R.R. Bahadur, A note on quantiles in large samples, Ann. Math. Stat. 37 (1996) 577–580. [3] G.A. McIntyre, A method for unbiased selective sampling using ranked sets, Aust. J. Agric. Res. 3 (1952) 385–390. [4] H.A. David, H.N. Nagaraja, Order Statistics, 3rd ed., John Wiley & Sons, Inc., Hoboken, New Jersey, 2003. [5] K. Takahasi, K. Wakimoto, On the unbiased estimates of the population mean based on the sample stratified by means of ordering, Ann. Inst. Stat. Math. 20 (1968) 1–31. [6] S.L. Stocks, T. Sager, Characterization of a ranked set sample with application to estimating distribution functions, J. Am. Stat. Assoc. 83 (1988) 374–381.
Please cite this article in press as: A.I. Al-Omari, Quartile ranked set sampling for estimating the distribution function, Journal of the Egyptian Mathematical Society (2015), http://dx.doi.org/10.1016/j.joems.2015.01.006
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Please cite this article in press as: A.I. Al-Omari, Quartile ranked set sampling for estimating the distribution function, Journal of the Egyptian Mathematical Society (2015), http://dx.doi.org/10.1016/j.joems.2015.01.006