Estimators in simple random sampling: Searls approach

Transcription

1 Songklanakarin J Sci Technol 35 ( Nov - Dec Original Article Estimators in simple random sampling: Searls approach Jirawan Jitthavech* and Vichit Lorchirachoonkul School of Applied Statistics National Institute of Development Administration Bang Kapi Bangkok 1040 Thailand Received 1 March 013; Accepted 14 August 013 Abstract This paper investigates four new estimators in simple random sampling biased sample mean ratio estimator and two linear regression estimators using known coefficients of variation of the stud variable and auiliar variable The properties of the new estimators are obtained omparisons among the new estimators three traditional estimators Searls sample mean Kouncu and Kadilar (KK estimators Sisodia and Dwivedi (SD estimator and Shabbir and Gupta (SG estimator are undertaken The analsis shows that the two proposed linear regression estimators are more efficient than the new biased sample mean and at least as efficient as the three traditional estimators and SD estimator At least one of the proposed linear regression estimators is alwas more efficient than the new ratio estimator and Searls sample mean From the numerical results using two data sets published in the literature the proposed linear regression estimators are clearl more efficient than all seven eisting estimators Kewords: estimator selection criteria linear regression estimator bias MSE relative efficienc simple random sampling 1 Introduction In simple random sampling where the population size is N and the sample size is n Searls (1964 proposed a biased sample mean with a smaller MSE than the traditional sample mean obtained b utilizing a known coefficient of variation without the finite population correction factor 1 n 1 N However in this paper Searls sample mean using the finite population correction factor is defined as w 1 (1 The bias and MSE of Searls sample mean respectivel are shown to be Y Bias( w ( 1 * orresponding author address: jirawan@asnidaacth Y MSE( w (3 1 It is obvious from Equation 3 that the relative efficienc of Searls sample mean w with respect to the traditional sample mean is equal to(1 1 Without the finite population correction the results in Equation 1-3 become the same results as in the original Searls work (Searls 1964 However Searls sample mean should be used under the condition 1 to avoid a large unacceptable bias in the estimate When we can find an auiliar variable which is highl correlated with the stud variable Y the use of known auiliar information in the ratio estimator can reduce the MSE of the sample mean The traditional combined ratio estimator is defined as r (4 The bias and MSE of r to first order of approimation respectivel are given b ochran (1977

2 750 J Jitthavech & V Lorchirachoonkul / Songklanakarin J Sci Technol 35 ( Bias( r Y (5 MSE( r Y (6 which is less than MSE( if Motivated b Searls (1964 several researchers proposed estimators using the known coefficient of variation of the auiliar variable for estimating the population mean when the stud variable is highl correlated with the auiliar variable for eample the Sisodia and Dwivedi (SD ratio estimator (Sisodia and Dwivedi 1981 which is defined as: SD The bias and MSE of SD to first order of approimation respectivel are given b Bias( SD Y MSE( SD Y (9 B Equation 6 and 9 the relative efficienc of respect to r is greater than unit when 1 With respect to the relative efficienc of than unit when (7 (8 SD with (10 SD is greater (11 Gupta and Shabbir (008 improved the estimation of population mean b proposing the ratio estimator 1 ( p w w (1 where and are either constants or functions of the known parameters of auiliar variable w 1 and w are constants The bias and MSE of p to first order of approimation respectivel are given b Kouncu and Kadilar (010 (KK estimator Bias( p ( w1 1 Y w 1Y ( w (13 (1 MSE( MSE( lr p (14 (1 MSE( lr Y where MSE( lr Y (1 (ochran 1977 and 1 The values of w 1 and w which minimize MSE( p are given b 1 w1 (15 1 [ (1 ] Y (1 ( w (16 [1 (1 ] It is clear from Equation 14 that MSE( p is less than MSE( lr Table 1 gives si values of and as suggested b Kouncu and Kadilar (010 where ( is the kurtosis of the auiliar variable Recentl Shabbir and Gupta (011 proposed an eponential ratio tpe estimator (SG estimator 1 ( ep SG k k (N 1 (17 where 1 8(1 N k1 and 1 (1 Y 1 1 k k1 (1 N 1 N The bias and MSE of SG to first order of approimation respectivel are given b 3 Bias( 1 1 k SG Y k k1 (1 8(1 N N (1 N (18 Table 1 Some members of Kouncu and Kadilar (KK estimators in simple random sampling KK Estimator p(0 p(1 p( p(3 p(4 p ( ( 1 ( (

3 J Jitthavech & V Lorchirachoonkul / Songklanakarin J Sci Technol 35 ( (1 MSE( 1 N SG Y 4(1 N 1 (1 (19 This paper is an attempt to etend the powerful Searls approach to the traditional estimators using auiliar variable in simple random sampling In the net two sections we suggest a new biased sample mean a new ratio estimator and two new regression estimators in simple random sampling Section 4 presents the comparison among the new estimators the traditional estimators and the KK estimators p( i i = (Kouncu and Kadilar 010 SD estimator and SG estimator The numerical results for estimators are presented in Section 5 using two data sets from the published literature Finall the last section is a summar and conclusions New ratio estimator and new biased sample mean When the stud variable Y is highl correlated with the auiliar variable a ratio estimate is commonl used for estimating the population mean A new ratio estimator using coefficients of variation of Y and can be written as rw w w (0 where w (1 w (1 and is the population mean of the auiliar variable The coefficients of variation and are assumed to be known The new ratio estimator rw in Equation 0 can be epressed in terms of the traditional ratio estimator r as 1 rw r (1 1 The bias and MSE of rw to first degree of approimation respectivel are given b 1 ( Bias( rw Bias( r Y 1 1 ( 1 MSE( rw MSE( r (3 1 From Equation 3 it can be concluded that the relative efficienc of the new ratio estimator rw with respect to the traditional ratio estimator r is greater than unit if < When onl is used in the estimation the new ratio estimator in Equation 0 becomes rw w (4 w where a new biased sample mean w using the coefficient of variation of the auiliar variable is defined as w 1 (5 It is obvious from Equation 4 and 5 that rw is actuall the traditional ratio estimator since w and w use the same weight 1 The bias and MSE of w can be epressed in the similar form as Equation and 3 respectivel The bias of w can also be written in the similar manner but the MSE of w to first order of approimation is given b Y MSE( w (6 (1 B Equation 3 and 6 it can be verified that the relative efficienc of w with respect to w is greater than unit if 3 New linear regression estimators B following the definition of the traditional linear regression estimator (ochran 1977 the first new linear regression estimator is defined as lrw w bw ( w (7 The value of b w that minimizes the variance of lrw can be shown to be 1 b w b (8 1 where b S S (ochran 1977 B substituting bw from Equation 8 into Equation 7 the bias of lrw can be written as Bias( Bias( (1 lrw w (9 The MSE of lrw to first order of approimation is given b MSE( MSE( lr lrw (30 (1 Again when onl is used in the estimation the new linear regression estimator in Equation 7 becomes lrw w bw ( w (31 Similarl it can be shown that bw b Bias( lrw Bias( w (1 (3

4 75 J Jitthavech & V Lorchirachoonkul / Songklanakarin J Sci Technol 35 ( MSE( MSE( lr lrw (33 (1 From Equation 30 and 33 it can be concluded that both new linear regression estimators lrw and lrw are alwas more efficient than the traditional regression estimator lr 4 Estimator comparisons Four new estimators w rw lrw and lrw have been presented in the above sections The first part is limited to the relative efficienc comparison among four new estimators The selected new estimators are compared with the traditional estimators Searls sample mean KK estimators SD estimator and SG estimator in the second part The relative efficiencies of w rw lrw and lrw with respect to and are shown in Table The comparison conditions in Table can be derived directl from the MSE formulae in Section and 3 The estimator w is eliminated from the future comparison since the relative efficienc of w with respect to lrw is alwas less than unit Lemma 1: If (1 (1 then lrw is more efficient than 4 rw unless (1 / (1 Otherwise lrw is more efficient than rw providing that (1 (1 (1 1 1 (1 (1 (1 and as efficient as rw providing that takes on a particular value such as (1 (1 (1 1 1 (1 (1 (1 which is subject to 1 Lemma : If (1 then lrw is more efficient than rw for 0 1; otherwise lrw is as efficient as rw when (1 The proofs of Lemma 1 and are shown in Appendi From Lemma it can be concluded that the relative efficienc of the linear regression estimator lrw with Table Relative efficienc comparisons among the proposed estimators: Estimators w rw w 1 lrw rw (1 1 if 1 4 (1 1 lrw (1 1 if 1 (1 1 if if 1 lrw 1for

5 J Jitthavech & V Lorchirachoonkul / Songklanakarin J Sci Technol 35 ( respect to the ratio estimator rw is not less than unit Therefore in this stud onl two new estimators lrw and lrw are proposed However since these two proposed estimators have been shown to be biased lrw and lrw should be used under 1 and 1 respectivel to avoid the unacceptable large bias Net the two proposed estimators lrw and lrw are compared with seven eisting estimators consisting of three traditional estimators r lr Searls sample mean w KK estimators p( i SD estimator SD and SG estimator SG It can be concluded from Table 3 that the proposed linear regression estimators lrw and lrw are alwas more efficient than the traditional estimators and lr Again as shown in Table 3 lrw is alwas more efficient than Searls sample mean w Lemma 3: If (1 then lrw is more efficient than r for 0 1; otherwise lrw is as efficient as r when (1 Lemma 4: If then lrw is more efficient (1 than r for 0 1; otherwise lrw is as efficient as r when (1 Table 3 Relative efficiencies of proposed estimators with respect to eisting estimators Estimators Eisting Proposed Estimators lrw lrw 1 for for 0 1 r 1 if (1 (1 1 (1 1 1 if (1 (1 1 (1 1 lr 1 for for 0 1 w (1 1 if for 0 1 p ( i 1 if 1 (1 ( 1 if 1 (1 ( SD 1 if (1 (1 1 (1 1 1 if (1 (1 1 (1 1 SG (1 1 if (1 ( 1 4(1 N 4 ( (1 N (1 1 if (1 1 4 (1 N 4 ( (1 N

6 754 J Jitthavech & V Lorchirachoonkul / Songklanakarin J Sci Technol 35 ( Lemma 5: If then lrw is more efficient than KK estimators p( i i 01 5 for 0 1 Otherwise lrw is more efficient than KK estimators when 1 (1 ( and as efficient as KK estimators when 1 (1 ( Lemma 6: If (1 ( then lrw is more efficient than KK estimators p( i i 01 5 for 0 1 Otherwise lrw is more efficient than KK esti- mators when 1 (1 ( and as efficient as KK estimators when 1 (1 ( subject to 1 Lemma 6: If (1 ( then lrw is more efficient than KK estimators p( i i 01 5 for 0 1 Otherwise lrw is more efficient than KK estimators when 1 (1 ( and as efficient as KK estimators when 1 (1 ( subject to 1 Lemma 7: If more efficient than (1 then lrw is SD for 0 1; otherwise lrw is as efficient as SD when (1 Lemma 8: If and (1 then lrw is more efficient than SD otherwise lrw is as efficient as SD (1 1 N Lemma 9: If 1 for 0 1; when = then lrw in a large population is more efficient than SG for 0 1 Lemma 10: If 64(1 N ( then lrw in a large population is more efficient than SG for 0 1 If ( then lrw in a large population is more efficient than SG when and as efficient as SG when ( 1 ( 1 The proofs of Lemma 3-10 are shown in Appendi The above analsis shows that the relative efficiencies of the two proposed linear regression estimators lrw and lrw with respect to three traditional estimators r lr and SD estimator SD respectivel are not less than unit The proposed linear regression estimator lrw is more efficient than Searls sample mean w but lrw is more efficient than w if Lemma and 10 give the conditions that the two proposed linear regression estimators lrw and lrw are more efficient than KK estimators p ( i and SG estimator SG 5 Numerical eamples For comparison of estimators consider two data sets from published literature Eample 1 (see Gupta and Shabbir 008: The stud variable Y is the level of apple production (in 1000 tons and the auiliar variable is the number of apple trees in 104 villages in the East Anatolia Region in 1999 The required data are summarized as follows: N = 104 n = 0 Y = 654 = = 1866 = 1653 ( = 0865 In Eample and From Table 3 it can be concluded that lrw and lrw are alwas more efficient than and that lrw is alwas more efficient than w Since the conditions in the first if parts of Lemma and 10 are obviousl satisfied lrw and lrw are more efficient than SD and SG Similarl lrw and lrw are more efficient than r b Lemma 3 and 4 and lrw is more efficient than p( i i 01 5 b Lemma 5 Since (1 (1 in this eample is greater than unit it can be concluded from Table 3 that lrw is more efficient than w The values of and the RHS of the condition in Lemma 6 summarized in Table 4 lead to the conclusion that the condition in the first part of Lemma 6 is satisfied for all values of in this eample Therefore lrw is more efficient than p ( i i 01 5 b Lemma 6 In summar lrw and lrw are more efficient than three traditional estimators Searls sample mean KK estimators SD estimator and SG estimator In Eample 1 lrw is more efficient than lrw since In other

7 J Jitthavech & V Lorchirachoonkul / Songklanakarin J Sci Technol 35 ( words lrw is the most efficient estimator among the compared estimators Eample (see Kouncu and Kadilar 009: The stud variable Y is the number of teachers and the auiliar variable is the number of students in primar and secondar schools in 93 districts of Turke in 007 The population statistics are given b N = 93 n = 180 Y = = = = ( = In Eample and B the similar reasoning as in Eample 1 lrw and lrw are more efficient than the three traditional estimators Searls sample mean KK estimators SD estimator and SG estimator But in Eample lrw is the most efficient estimator among compared estimators since The MSEs relative efficiencies and relative absolute biases of estimators with respect to the traditional ratio estimator r of Eample 1 and are summarized in Table 5 The numerical results are consistent with the above analsis The relative efficienc of lrw in Eample 1 is highest equal to 1303% and the relative efficienc of lrw in Eample is highest equal to 188% The relative efficiencies of both proposed linear regression estimators are clearl higher than the traditional estimators Searls sample mean KK SD and SG estimators In both eamples the relative absolute bias of lrw is in the order as the traditional ratio estimator Table 4 The values of and the RHS of the condition in Lemma 6 KK Estimator p(0 p(1 p( p(3 p(4 p (5 E (1 ( E (1 ( Table 5 Relative efficiencies of estimators with respect to r and their relative absolute biases Estimator Eample 1 Eample MSE Relative Relative MSE Relative Relative Efficienc % Abs Bias% Efficienc % Abs Bias% lrw lrw rw r lr p( p( p( p( p( p( SD SG

8 756 J Jitthavech & V Lorchirachoonkul / Songklanakarin J Sci Technol 35 ( Summar and onclusions In this paper the powerful concept of Searls biased sample mean has been etended to cover the sample mean the ratio and linear regression estimators in simple random sampling Four new estimators using the coefficient of variation of the auiliar variable w rw lrw and lrw have been investigated and their properties are also obtained The selection criteria based on relative efficienc among these new estimators r lr p ( i i = 01 5 SD and SG are shown in Table to 3 and Lemma 1 to 10 In the relative efficienc comparisons it is found that the new linear regression estimator lrw is more efficient than the new sample mean w The relative efficienc of lrw with respect to the new ratio estimator rw is not less than unit The linear regression estimator lrw is more efficient than another linear regression estimator lrw if This leads us to propose onl two new linear regression estimators The relative efficiencies of the two proposed linear regression estimators with respect to the three traditional estimators r and lr and SD estimator SD respectivel are shown to be at least equal to unit The proposed linear regression estimator lrw is alwas more efficient than Searls sample mean w but another linear regression estimator lrw is more efficient than Searls sample mean w if 1 (1 (1 or Y and are sufficientl high correlated The estimator selection criteria among the proposed estimators KK estimators p( i and SG estimator SG are given in Lemma and 10 In two published data sets the relative efficiencies of lrw and lrw are clearl higher than those of other estimators as shown in Table5 The relative biases of lrw and lrw are 88% and 03% in Eample1 and 005% and 018% in Eample respectivel There is not much difference between all KK estimators and SG estimator as shown in Table 5 References ochran WG 1977 Sampling Techniques John Wile and Sons New York USA Gupta S and Shabbir J 008 On Improvement in Estimating the Population Mean in Simple Random Sampling Journal of Applied Statistics Kouncu N and Kadilar 009 Efficient Estimators for the Population Mean Hacettepe Journal of Mathematics and Statistics Kouncu N and Kadilar 010 On Improvement in Estimating Population Mean in Stratified Random Sampling Journal of Applied Statistics Searls DT 1964 The Utilization of a Known oefficient of Variation in the Estimation Procedure Journal of the American Statistical Association Shabbir J and Gupta S 011 On Estimating Finite Population Mean in Simple and Stratified Sampling ommunications in Statistics - Theor and Methods Sisodia BVS and Dwivedi VK 1981 A Modified Ratio Estimator using Known oefficient of Variation of Auiliar Variable Journal of the Indian Societ of Agricultural Statistics

9 J Jitthavech & V Lorchirachoonkul / Songklanakarin J Sci Technol 35 ( Proof of Lemma 1: From Table lrw is at least as efficient as rw if Appendi (1 (1 (1 1 1 (A-1 (1 (1 (1 onsider the first case where (1 (1 Under this condition if (1 (1 then the RHS in Equation A-1 is negative It can be concluded that lrw is more efficient than rw for 0 1 If (1 (1 then the RHS is equal to zero From Equation A-1 lrw is more efficient than rw unless (1 (1 When (1 (1 lrw is as efficient as rw If (1 (1 4 it is obvious that (1 (1 4 When (1 (1 the minimum of the LHS in Equation A-1 for 0 1 occurs at 1 and is equal to LHS 4 (1 1 1 min (1 onsider the difference between LHS 1 min and the RHS in Equation A-1: (1 (1 ( LHS 1 0 min (1 (1 (1 Therefore if (1 (1 it can be concluded that in Equation A-1 the LHS is greater than the RHS for 0 1 In other words if (1 (1 lrw is more efficient than rw for 0 1 Net consider the case where (1 (1 Under this condition the RHS in Equation A-1 is equal to zero From Equation A-1 it can be concluded that lrw is more efficient than rw unless When lrw is as efficient as rw The final case is (1 (1 Under this condition it follows immediatel that 4 In other words if (1 (1 it implies that (1 (1 and (1 (1 Therefore from Equation A-1 lrw is more efficient than rw if the inequalit of Equation A-1 is a strictl inequalit and is as efficient as rw if takes on a particular value such that the inequalit (A-1 becomes an equalit In summar if (1 (1 then lrw is more efficient than 4 rw unless (1 / (1 Otherwise lrw is more efficient than rw provided that Equation A-1 is a strictl inequalit and as efficient as rw provided that takes on a particular value such that the inequalit of Equation A-1 becomes an equalit Proof of Lemma : From Table the condition that lrw is at least as efficient as rw can be written as (1 (1 1 (1 1 (A-

10 758 J Jitthavech & V Lorchirachoonkul / Songklanakarin J Sci Technol 35 ( The inequalit of Equation A- is the same as the inequalit of Equation A-1 if the term (1 is replaced b the (1 term The proof of Lemma is much simpler than the proof of Lemma 1 since (1 (1 is onl greater than zero It can be shown in the similar manner as in the first part of the proof of Lemma 1 that if (1 lrw is more efficient than rw for 0 1; otherwise lrw is as efficient as rw when (1 Proof of Lemma 3: From Table 3 lrw is more efficient than r if (1 (1 1 (1 1 Since the inequalit of Equation A-3 is eactl the same as the inequalit of Equation A- the conclusion is the same as Lemma Proof of Lemma 4: From Table 3 lrw is more efficient than r if (A-3 (1 (1 1 (1 1 (A-4 The inequalit of Equation A-4 is in the same form as the inequalit of Equation A- if the term 1 is replaced b 1 Therefore it can be concluded that if (1 lrw is as r when (1 Proof of Lemma 5: From Table 3 lrw is at least as efficient as p( i if lrw is more efficient than r for 0 1; otherwise 1 (1 ( (A-5 1 If it can be shown that the RHS in Equation A-5 is less than or equal to zero Therefore it can be 1 concluded that if lrw is more efficient than KK estimators p( i i 01 5 for 0 1 Otherwise lrw is more efficient than KK estimators when 1 (1 ( and as efficient as KK estimators when 1 (1 ( Proof of Lemma 6: From Table 3 the condition that lrw is at least as efficient as p( i is 1 (1 ( (A-6

11 J Jitthavech & V Lorchirachoonkul / Songklanakarin J Sci Technol 35 ( If (1 ( the RHS in Equation A-6 is either negative or zero Therefore lrw is more efficient than KK estimators p( i i 01 5 for 0 1 Otherwise lrw is more efficient than KK estimators when the inequalit of Equation A-6 is a strictl inequalit and as efficient as KK estimators when the inequalit of Equation A-6 becomes an equalit Proof of Lemma 7: From Table 3 the condition that lrw is at least as efficient as SD is (1 (1 1 (1 1 The inequalit of Equation A-7 is eactl the same as the inequalit of Equation A- if the term (A-7 is replaced b the term It can be shown in the similar manner that if (1 lrw is more efficient than SD for 0 1; otherwise lrw is as efficient as SD when (1 Proof of Lemma 8: Again from Table 3 the condition that lrw is at least as efficient as SD is (1 (1 1 (1 1 (A-8 The inequalit of Equation A-8 is in the same form as the inequalit of Equation A-7 if the term 1 is replaced b 1 Therefore it can be concluded that if (1 lrw is more efficient than SD for 0 1; otherwise lrw is as efficient as SD when (1 Proof of Lemma 9: From Table 3 the condition that lrw is at least as efficient as SG is 4 (1 (1 ( (1 N 64 (1 N (A-9 1 N If 1 the terms (1 4 (1 N and ( (1 N can be approimatel equal to zero for large N

12 760 J Jitthavech & V Lorchirachoonkul / Songklanakarin J Sci Technol 35 ( Under such condition and large N it can be concluded that the LHS in Equation A-9 is greater than zero for 0 1 Therefore lrw in a large population is more efficient than SG for 0 1 if Proof of Lemma 10: From Table 3 lrw is at least as efficient as SG if 1 N 1 4 (1 (1 (1 ( 1 (A (1 N 64 (1 N When N is large the term (1 4(1 N is insignificant when compared with Under this condition the LHS in Equation A-10 can be approimatel reduced to (1 ( 1 (A 11 which is greater than zero for 0 1 if ( When N is large the RHS in Equation A-10 is approimatel equal to zero if 8 (1 N Therefore if 64(1 N ( then lrw in a large population is more efficient than SG for 0 1 When ( the condition 64(1 N is also satisfied for large N Therefore if ( lrw in a large population is more efficient than SG when ( 1 and as efficient as SG estimator SG when ( 1