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On estimation of population mean using known coefficient of variation
Microelectron.Reliab.,Vol. 37, No. 5, pp. 841-843, 1997
(~I
Copyright © 1997 Published by Elsevier Science Ltd Printed in Great Britain. All rights reserved 0026-2714/97 $17.00+.00
Pergamon
PII: S0026-2714(96)00113-8
TECHNICAL NOTE ON ESTIMATION OF POPULATION MEAN USING K N O W N C O E F F I C I E N T O F VARIATION S. A. H. RIZVI and R. K A R A N S I N G H Department of Statistics, Lucknow University, Lucknow 226 007, India
(Received for publication 23 May 1996) Abstract--For the estimation of population mean, a generalized class of estimators using known coefficient of variation Cr of the study variable y is proposed, its bias and mean square error (MSE) are found and its comparative study with the usual mean per unit estimator has been done. As an illustration, an empirical study is also included. Copyright © 1997 Elsevier Science Ltd.
1. INTRODUCTION
Y2 = )7 1 +
Let Y~be the values on variable y under study for the ith (i = 1, 2 . . . . . N) unit of the population of size N, = ( l / N ) ~ v = 1 Yi be the population mean and 1 ~ (Y~-
(1)
i=1
C~ = P2
#,*
= )7{1 +
k(u - 1)} (4)
( C~r 1) = ) 7 + k ( u - 1) )73=)7+ k \~yy--
~)~
P'=N
- 1
/C
\k,
)7,=)7~Cy/'
+
k 2fCs~ \ -c,- - - 1 )
=
(5)
)TUk' + k2(u - 1)
(6)
g3 and
Based on a simple random sample of size n with the sample values YR, );2,-.., Y, on y, let l~
)7=-ni=lYi'
2 1
sy
(n
_
where k, kl and k 2 are the characterizing scalars to be chosen suitably in the sense of minimizing the mean square error of the estimators.
~-~ (yi _ )7)2" 1)i=1
Further, let Csr be the sample coefficient of variation of y, and for simplicity, let the population size N be large enough as compared to n so that the finite population correction terms may be ignored. For estimating the p o p u l a t i o n mean Y, a generalized class of estimators is proposed as )7o=g()7, u)
Let s 2y - S y 2 + e I
)7--- Y + e o , so that
(2)
where u = CsylCy and g()7, u) satisfying the validity conditions of Taylor's series expansion and having first-, second- and third-order partial derivatives bounded, is a bounded function of ()7, u) such that g(Y, 1 ) = Y and the first-order partial derivative @()7, u)/6)7 with respect to )7 is unity. Some special cases of the generalized estimator )Tg are )7Csr k )71= (~...-r) : )TUk
2. BIASAND MEANSQUAREERROR
E(eo)=E(et)=O, E(e~)=P-~2(fl21y) - 1), n where
=#4 E(eoeO
(3) 841
#3
(8)
842
Technical Note
Further. let g'2 -- 6g(f, u)] , g~ -- 62g(~, u)] Ou J(£ 1) ~U2 (Y,1) be the first- and second-order partial derivatives at (Y, l) with respect to u, respectively, and
g';2-
+ eo{(1 + ~-~1 e "xl/2/ (1 + e~) - , s,/ \
1}9';2
+ 3! (-9 -
g(y*, u*)
6y + (u - 1)
~eg eo =e°+ (Y2-Tq - 8S~ + ..... ' ~'e 2
g~
e2
eoe I
+ 2 ( 4 s ~ e P~
+
= g(Y, I) + (-9- Y)g'l + (u - 1)g i 1 + 2! {(-9 - F ) M + (u - 1)=01
(2s~
}
~ s ,~ + . . . . .
?<4
Y. = O(Y,u)
eoe~ 2YS~
}
6ySu J ( r , ,
be the second-order partial derivative at (Y, 1) with respect to (y, u). Now, expanding g(-9, u) in third-order Taylor's series about the point (Y, 1), we have
el 2Syz
~:;
} ?
+
. . . . . g';~
+ 3! (y - f') 615 + (u - 1) 6uu g(-9*' u*).
(9)
+ 2(y - Y)(u - 1)0'/2 }
' {
+3!
(9-
Y)2 6
C~y
= Y+(-9- h + ( c
Taking expectation to the terms of order O(n-~), we have
6} 3 1)~uu 0(-9",u*)
8} + ( u -
)
2s"3
- 1 gl
l ~#](f12(,,-
,{
+ 3! (-9 - Y) 6y + (u - 1) = ;+(-9-F)+
(~s,,) ~-I
+ ~(
:u}
1)
#2
4s ~,
+ ~
_#3 2~g,~ vs,;
+ @-
g(-9*, u*)
gl
or
Bias(Yo) = E(-go) + 2 (\S,/Y
- ln[{C: _ 21 ~l,y, Cy _ 81 (fl2(y) -- 1)}gi
1{:
kS,,/Y + 3~ (-9 - L ~ + (u - l) ~
}
+ }
( & l . - 1) + c~ - ~ . . , c . g~
+ y
?l(y)CyI C y 912"
g(y*, u*)
(10)
= Y + e ° + { (S~+e')l/2/(~+e°)SylY - 1}9'2 Further, from equation (9), the mean square error of -90 up to terms of order O(n-1) is given by
1 !S2y+el)l/2/(Y+eo) 1 g~ +2 Sy/~" -
MSE(-9o) =
{(S2+e~)~/2/(f'+ eo) l }g'/2 + eo St~? -+ 3~ (-9 - h 6~ + (u -
(-9,--h=eo+
+~
1+~-~} s./
~) a-~; /l+ \
l + selVT1 u \ +
E( L _ ~q2
eo+
o(-9*, u*)
I = E_eo 2 + ~'_eg (~2 + ~e~
- 1 g;
+ 2(eoe,
\2s, -t
¢;
e~,
]
)o2j.
eoet~ PS~j (g~)2 (11)
Substituting the values from equation (8) in
Technical Note
gives the optimum value of the characterizing scalar k minimizing the mean square error to be
equation (11) and simplifying, we have
MSEG)= in[
. ~#2(/32(,)-1)
+ 2~'"3
843
/J2 F?2
{
/~2] ,-]
C2 + ~ (f12(,) -- 1) - ?l(,)Cr and the minimum mean square error of 375 for this optimum value of k is given by equation (14). (d) For a normal distribution, equation (16) reduces to
+
(12) which is minimized for
kopt - { C2+½}
1+
and the minimum mean square error of 375 for this optimum value of k is given by
g~=
--2
2
MSE(375)min - Y Cr n
= ~_2
(13)
= MSE(37)
--2
2
Y Cr
(17)
n{C 2 + ½}
-
1+
(18)
n
where ~1 =
{'
C ; ~t_ 4 (J~2(y) - - 1) - - ~ l ( y ) C y
}
which shows that the estimator 37s utilizing a known coefficient of variation is more efficient in the sense of having a lesser mean square error than the mean per unit estimator 37 in the normal parent.
,
4. EMPIRICAL STUDY
and the minimum mean square error of 37gis given by MSE(Yg)mln
-2 2
p2 Ctl/
f
H
(.
1
c; + 4 (/Le,) - 1) -
yie,~Cyj t
o
Considering the data given in Cochran [1] dealing with paralytic polio cases 'Placebo' (y) group, computations of required values of/# have been done and comparison among different estimators is made for a simple random sample of size n with replacement. For the data considered here, we have P = 2.58,
P2 = 9.8894,
#3=47.015235,
~4 = 421.96088.
(14)
Using the above values, we get 3. CONCLUDING REMARKS
(a) From equation (14), it is clear that for a negatively skewed distribution, the proposed estimator 37~is more efficient than the mean per unit estimator 37. (b) Again, from equation (14), for a symmetrical distribution, 37g is more efficient than 37. (c) In particular for the estimator 375 = y(2 - uk),
(15)
we have g~ = kY which when equated to equation (13) having
{,
}'
MSE(37) = 0.290865 and MSE(37g)mi, = 0.1586581 whence, the percent relative efficiency (PRE) of the proposed estimator yg over the usual estimator 37 is PRE(37g) = 183.32817 showing that 37gis more efficient than the usual mean per unit estimator in the sense of having a lesser mean square error.
REFERENCES
1. Cochran, W. G., Sampling Techniques, 3rd edn. Wiley, New York, 1977.
(~I
Copyright © 1997 Published by Elsevier Science Ltd Printed in Great Britain. All rights reserved 0026-2714/97 $17.00+.00
Pergamon
PII: S0026-2714(96)00113-8
TECHNICAL NOTE ON ESTIMATION OF POPULATION MEAN USING K N O W N C O E F F I C I E N T O F VARIATION S. A. H. RIZVI and R. K A R A N S I N G H Department of Statistics, Lucknow University, Lucknow 226 007, India
(Received for publication 23 May 1996) Abstract--For the estimation of population mean, a generalized class of estimators using known coefficient of variation Cr of the study variable y is proposed, its bias and mean square error (MSE) are found and its comparative study with the usual mean per unit estimator has been done. As an illustration, an empirical study is also included. Copyright © 1997 Elsevier Science Ltd.
1. INTRODUCTION
Y2 = )7 1 +
Let Y~be the values on variable y under study for the ith (i = 1, 2 . . . . . N) unit of the population of size N, = ( l / N ) ~ v = 1 Yi be the population mean and 1 ~ (Y~-
(1)
i=1
C~ = P2
#,*
= )7{1 +
k(u - 1)} (4)
( C~r 1) = ) 7 + k ( u - 1) )73=)7+ k \~yy--
~)~
P'=N
- 1
/C
\k,
)7,=)7~Cy/'
+
k 2fCs~ \ -c,- - - 1 )
=
(5)
)TUk' + k2(u - 1)
(6)
g3 and
Based on a simple random sample of size n with the sample values YR, );2,-.., Y, on y, let l~
)7=-ni=lYi'
2 1
sy
(n
_
where k, kl and k 2 are the characterizing scalars to be chosen suitably in the sense of minimizing the mean square error of the estimators.
~-~ (yi _ )7)2" 1)i=1
Further, let Csr be the sample coefficient of variation of y, and for simplicity, let the population size N be large enough as compared to n so that the finite population correction terms may be ignored. For estimating the p o p u l a t i o n mean Y, a generalized class of estimators is proposed as )7o=g()7, u)
Let s 2y - S y 2 + e I
)7--- Y + e o , so that
(2)
where u = CsylCy and g()7, u) satisfying the validity conditions of Taylor's series expansion and having first-, second- and third-order partial derivatives bounded, is a bounded function of ()7, u) such that g(Y, 1 ) = Y and the first-order partial derivative @()7, u)/6)7 with respect to )7 is unity. Some special cases of the generalized estimator )Tg are )7Csr k )71= (~...-r) : )TUk
2. BIASAND MEANSQUAREERROR
E(eo)=E(et)=O, E(e~)=P-~2(fl21y) - 1), n where
=#4 E(eoeO
(3) 841
#3
(8)
842
Technical Note
Further. let g'2 -- 6g(f, u)] , g~ -- 62g(~, u)] Ou J(£ 1) ~U2 (Y,1) be the first- and second-order partial derivatives at (Y, l) with respect to u, respectively, and
g';2-
+ eo{(1 + ~-~1 e "xl/2/ (1 + e~) - , s,/ \
1}9';2
+ 3! (-9 -
g(y*, u*)
6y + (u - 1)
~eg eo =e°+ (Y2-Tq - 8S~ + ..... ' ~'e 2
g~
e2
eoe I
+ 2 ( 4 s ~ e P~
+
= g(Y, I) + (-9- Y)g'l + (u - 1)g i 1 + 2! {(-9 - F ) M + (u - 1)=01
(2s~
}
~ s ,~ + . . . . .
?<4
Y. = O(Y,u)
eoe~ 2YS~
}
6ySu J ( r , ,
be the second-order partial derivative at (Y, 1) with respect to (y, u). Now, expanding g(-9, u) in third-order Taylor's series about the point (Y, 1), we have
el 2Syz
~:;
} ?
+
. . . . . g';~
+ 3! (y - f') 615 + (u - 1) 6uu g(-9*' u*).
(9)
+ 2(y - Y)(u - 1)0'/2 }
' {
+3!
(9-
Y)2 6
C~y
= Y+(-9- h + ( c
Taking expectation to the terms of order O(n-~), we have
6} 3 1)~uu 0(-9",u*)
8} + ( u -
)
2s"3
- 1 gl
l ~#](f12(,,-
,{
+ 3! (-9 - Y) 6y + (u - 1) = ;+(-9-F)+
(~s,,) ~-I
+ ~(
:u}
1)
#2
4s ~,
+ ~
_#3 2~g,~ vs,;
+ @-
g(-9*, u*)
gl
or
Bias(Yo) = E(-go) + 2 (\S,/Y
- ln[{C: _ 21 ~l,y, Cy _ 81 (fl2(y) -- 1)}gi
1{:
kS,,/Y + 3~ (-9 - L ~ + (u - l) ~
}
+ }
( & l . - 1) + c~ - ~ . . , c . g~
+ y
?l(y)CyI C y 912"
g(y*, u*)
(10)
= Y + e ° + { (S~+e')l/2/(~+e°)SylY - 1}9'2 Further, from equation (9), the mean square error of -90 up to terms of order O(n-1) is given by
1 !S2y+el)l/2/(Y+eo) 1 g~ +2 Sy/~" -
MSE(-9o) =
{(S2+e~)~/2/(f'+ eo) l }g'/2 + eo St~? -+ 3~ (-9 - h 6~ + (u -
(-9,--h=eo+
+~
1+~-~} s./
~) a-~; /l+ \
l + selVT1 u \ +
E( L _ ~q2
eo+
o(-9*, u*)
I = E_eo 2 + ~'_eg (~2 + ~e~
- 1 g;
+ 2(eoe,
\2s, -t
¢;
e~,
]
)o2j.
eoet~ PS~j (g~)2 (11)
Substituting the values from equation (8) in
Technical Note
gives the optimum value of the characterizing scalar k minimizing the mean square error to be
equation (11) and simplifying, we have
MSEG)= in[
. ~#2(/32(,)-1)
+ 2~'"3
843
/J2 F?2
{
/~2] ,-]
C2 + ~ (f12(,) -- 1) - ?l(,)Cr and the minimum mean square error of 375 for this optimum value of k is given by equation (14). (d) For a normal distribution, equation (16) reduces to
+
(12) which is minimized for
kopt - { C2+½}
1+
and the minimum mean square error of 375 for this optimum value of k is given by
g~=
--2
2
MSE(375)min - Y Cr n
= ~_2
(13)
= MSE(37)
--2
2
Y Cr
(17)
n{C 2 + ½}
-
1+
(18)
n
where ~1 =
{'
C ; ~t_ 4 (J~2(y) - - 1) - - ~ l ( y ) C y
}
which shows that the estimator 37s utilizing a known coefficient of variation is more efficient in the sense of having a lesser mean square error than the mean per unit estimator 37 in the normal parent.
,
4. EMPIRICAL STUDY
and the minimum mean square error of 37gis given by MSE(Yg)mln
-2 2
p2 Ctl/
f
H
(.
1
c; + 4 (/Le,) - 1) -
yie,~Cyj t
o
Considering the data given in Cochran [1] dealing with paralytic polio cases 'Placebo' (y) group, computations of required values of/# have been done and comparison among different estimators is made for a simple random sample of size n with replacement. For the data considered here, we have P = 2.58,
P2 = 9.8894,
#3=47.015235,
~4 = 421.96088.
(14)
Using the above values, we get 3. CONCLUDING REMARKS
(a) From equation (14), it is clear that for a negatively skewed distribution, the proposed estimator 37~is more efficient than the mean per unit estimator 37. (b) Again, from equation (14), for a symmetrical distribution, 37g is more efficient than 37. (c) In particular for the estimator 375 = y(2 - uk),
(15)
we have g~ = kY which when equated to equation (13) having
{,
}'
MSE(37) = 0.290865 and MSE(37g)mi, = 0.1586581 whence, the percent relative efficiency (PRE) of the proposed estimator yg over the usual estimator 37 is PRE(37g) = 183.32817 showing that 37gis more efficient than the usual mean per unit estimator in the sense of having a lesser mean square error.
REFERENCES
1. Cochran, W. G., Sampling Techniques, 3rd edn. Wiley, New York, 1977.