Calibration approach based estimators of finite population mean in two - stage stratified random sampling

In the present paper, an attempt has been made to conduct a limited simulation study to examine the relative performance of calibration approach based estimators. The real data has been taken from Appendix-C of Sarndal et al., (2003). The study variate y and the auxiliary variate x are the population of the Sweden in year 1985 and year 1975 which is divided in 284 municipalities (MU284). The results have been found that the calibration estimators have outperformed the usual estimator of finite population mean in two-stage stratified random sampling.

Int.J.Curr.Microbiol.App.Sci (2018) 7(1): 1808-1815 International Journal of Current Microbiology and Applied Sciences ISSN: 2319-7706 Volume Number 01 (2018) Journal homepage: http://www.ijcmas.com Original Research Article https://doi.org/10.20546/ijcmas.2018.701.219 Calibration Approach Based Estimators of Finite Population Mean in Two - Stage Stratified Random Sampling Sandeep Kumar* Department of Agricultural Statistics, Narendra Deva University of Agriculture and Technology, Kumarganj, Faizabad-224229 (UP), India *Corresponding author ABSTRACT Keywords Finite population, Auxiliary information, Two-stage stratified random sampling, Calibration estimators, Population mean Article Info Accepted: 14 December 2017 Available Online: 10 January 2018 Deville and Sarndal (1992) developed calibration estimator by using the auxiliary information to obtain a better estimate of the population total of study variate and auxiliary variable x While calibration approach does not assume any explicit relationship between y and x but it assume only that X , the population total of x is known following Deville and Sarndal (1992) calibration approach, calibration estimators of finite population mean in two-stage stratified random sampling have been developed The variances and unbiased estimator of their variances have been derived In the present paper, an attempt has been made to conduct a limited simulation study to examine the relative performance of calibration approach based estimators The real data has been taken from Appendix-C of Sarndal et al., (2003) The study variate y and the auxiliary variate x are the population of the Sweden in year 1985 and year 1975 which is divided in 284 municipalities (MU284) The results have been found that the calibration estimators have outperformed the usual estimator of finite population mean in two-stage stratified random sampling Introduction The auxiliary information has been effectively used in sample surveys at selection, stratification and estimation stage for bringing about the improvement in the estimate of population parameters The different fundamental approaches are used in finite population survey sampling These are design based approaches, model based approach and model assisted approach Under design based approach, the most common unbiased estimator of finite population total Y of study variable y is the well-known Horvitz-Thompson (HT) estimator is given by YˆHT   d i yi (1) is With variance   V YˆHT    Dij  yi  y j  (2) i 1 j 1 1808 N N Int.J.Curr.Microbiol.App.Sci (2018) 7(1): 1808-1815 Where Dij   ij   i j  , d i  /  i ,  i being the inclusion of probability of ith unit in the sample s which has been drawn from the finite population of size N by a probability sampling design P() and y i is the observed value of y corresponding to the ith unit selected in sample s Deville and Sarndal (1992) developed calibration estimator of finite population total is given by   ˆC   ˆ   ˆ    ˆ  ,  ˆ  d q x y d q x is is i i i i i i where i stage unit (fsu) level, that means the total of x NI i.e X   xi , at i th fsu level is known The i 1 usual estimator of population mean of the study variate y in two-stage stratified random sampling has been developed in section-2 The calibration estimators of the population mean in two-stage stratified random sampling by calibrating the design weight have been developed in the section-3 The variances and variance estimators of the calibration estimators have been developed in the section4 and a limited simulation study has been conducted in the section-5 (3) The usual estimator of population mean in two-stage stratified random sampling Which is equivalent to GREG estimator of Y (See, Cassel et al., 1976) An approximate variance of YˆC of Y for a large sample is given by (Deville and Sarndal, 1992) Consider the finite population (U  U1 ,U ,U , ,U N ) consists of N first stage units (fsu) and is stratified into L strata such that U t consists of N t fsu and L  t 1   N Where Ei  yi  xi , and   d q x y i 1 N i i d q x i 1 i i   We also consider that each fsu in tth stratum (t  1,2,3, , L) has M t number of second stage units (ssu) Now the following terms are define as N N V YˆC    Dij d i Ei  d j E j  (4) i 1 j 1 i t Ytij  value of the characteristic y under study i on jth ssu ( j  1,2,3, , M t ) corresponding , to the ith fsu (i  1,2,3, , N t ) in the tth stratum i An attempt has been made in the present paper to develop calibration estimator of finite population mean in two-stage stratified random sampling Following the calibration approach of Deville and Sarndal (1992) & calibration based estimator in two-stage sampling Aditya et al., (2016) when auxiliary information on a single auxiliary variable x related to study variable y is known at first t tij    t j 1 t th th of i fsu in t stratum Yti  t  i 1 ti , mean of ssu’s t t  tij , the population mean  t  t i 1 j 1 of y in tth stratum Yt  1809 Int.J.Curr.Microbiol.App.Sci (2018) 7(1): 1808-1815 L   t 1 t t t , the population mean of Y L   t t 1 t Where S in the population S tiy t tij  ti  and   t  1  j 1 S tby  1  V  yts     Stby  nt  nt  t  twy t  t S i 1 tiy An unbiased estimator of V ( yts ) is obtained as 1  Vˆ ( yts )    stby  t  nt  t  Nt ti  t 2  (  t  1) i 1 Now, consider that a sample of size nt fsu’s th out of N t fsu’s is selected from t stratum and  Where stwy sub-samples of size mt out of M t ssu’s from the selected nt fsu’s are drawn by SRSWOR (simple random sampling without replacement) This process is carried out independently in each stratum We further define mt  ytij , sample mean from ith selected mt j 1 fsu (i  1,2,3, , nt ) in tth stratum nt y ti , the overall sample mean in tth  nt i 1 stratum yts = mt s   ytij  yti  mt  j 1 s i 1 tiy L t 1 Where Wt  t such that  L W t 1 t 1 The variance of y s is obtained as L V  y s   Wt 2V  yts  t 1 L W t 1 tiy nt y s   Wt yts (7) = nt  1    stwy (6) m  t   t An unbiased estimator of  in stratified two stage random sampling is given by yti = t  1    S tby  nt  nt  t   1     S twy  (8)  mt  t   An unbiased estimator of V  y s  is obtained as nt  yti  yts  stby   nt  i 1 L Vˆ  y s   Wt 2Vˆ  yts  t 1 Obviously, the y ts is an unbiased estimator of t .The variance of  1    Stwy (5)  mt  t  yts is obtained as L  1   = Wt   stby t t 1  nt   1810  1     stwy  (9)  mt  t   Int.J.Curr.Microbiol.App.Sci (2018) 7(1): 1808-1815 Proposed calibration estimators of population mean in two-stage stratified random sampling In this case, the population mean of the auxiliary variable can easily be obtained, i.e We have described details of development of an estimator of population L t t y L tij mean    ,     t  t using t 1 i 1 j 1  t 1 simple random sampling without replacement (SRSWOR) independently in each stratum  L Nt t 1 i 1 X ti   (13) Where X ti is the value of the auxiliary variable corresponding to ith fsu in the tth stratum Let Wt ' be calibrated weight.The calibration The estimator of  is given by estimator of Y is therefore, given by L y s  Wt y ts (10) L y sc  Wt ' yts (14) t 1 t 1 L W Such that t 1 t  The weight Wt is self design weight and it is given by t ,  L Where Wt ' is calibrated weight obtained by minimizing a distance measure  W L t 1 and     t  Wt ' t  qtWt , where q t is positive quantity unrelated to Wt , subject to calibration constraint t 1 The variance of y s is obtained as L W x ' t 1 L  1   1   V  y s   Wt   S tby    S twy  (11) n  nt  mt  t   t 1 t t    An unbiased estimator of V  y s  is obtained as  1   1   Vˆ  y s   Wt   stby    stwy  (12)   n   t  mt  t   t 1  t  The weight Wt can be calibrated if the information of an auxiliary variable x related to the study variate y is available in order to improve the efficiency of the estimator y s The information of an auxiliary variable x related to y may be available at fsu’s level in twostage stratified random sampling ts   (15) Where x ts is an estimator of  t , developed similarly as yts For the ready reference, xts is given by xts  L t nt nt  xti , Where xti  i 1 mt mt x j 1 tij , (16) The following function  Wt ' ,     L t 1 W t  Wt qtWt '     2  Wt ' xts  X (17) Is minimized with respect to Wt ' , where  is Langrangian multiplier This yields Wt ' as 1811 Int.J.Curr.Microbiol.App.Sci (2018) 7(1): 1808-1815  X   Wt xts  Wt '  Wt 1  qt xts  (18)   Wt qt xts Variance and variance estimator proposed calibration estimators The developed calibration estimator is given by The approximate variance of y sc1 has been derived following the procedure given by Sarndal et al., (2003, chapter 4&8), and is given by L L y sc  Wt yts  t 1 W x yts qt L    X  Wt xts  (19) Wt xts2 qt  t 1  t ts t 1 L  of  Wt V ( y sc1 )   2 Vtpsu  Vtssu (22) t 1 M t N t L t 1 Which is a combined regression estimator in stratified random sampling y sc is a class of Nt Nt Vtpsu    Iij Di D j i 1 j 1 estimators depending upon the value of qt Where For qt  , we get an estimator as ,  Ii  Ij  Iij   nt N t  nt  N t N t  1  Ii  , nt , Nt L L L y SC1   Wt y ts  t 1  Wt xts y ts t 1 L W x t 1 t ts L    X   Wt x ts  (20) t 1   , we get another estimator as x ts L t 1 = t 1 L t t Where Vi  Ui  kl / i ytij ytij '  j / i  j '/ i , mt M t  mt  and  j / i  mt M t M t2 M t  1 L    X   Wt xts    Wt xts  t 1 t 1 L  d i d j   N2 W  N N  n   Vˆ  y SC1    t  t t t st  st Vˆi   2t nt  nt t 1 N t M t     nt ts Wt xts t t The estimator of variance of y SC1 following Sarndal et al., (2003), is obtained as  Wt yts L W y W X t 1  kl / i   L y SC   Wt y ts  t t t 1 L Vtssu  U Vi  Ii Which is a combined regression estimator in two-stage stratified random sampling For q t  Di  Yti  Yt  BX ti  X t  and B  W Y X Where Vˆi  X (21)  M t  mt M t mt2 mt  1 di  yti  yts  Bˆ xti  xts  , t 1 Which is a combined ratio estimator in twostage stratified random sampling L Bˆ  Wt xts yts t 1 1812 L W x t 1 t ts  si  st  Vˆi   (23) ytij ytij ' , and Int.J.Curr.Microbiol.App.Sci (2018) 7(1): 1808-1815 Table.1 The estimate of Y based on y s , y sc1 and y sc along with their estimate of variance Estimator Estimate %RB 6.92 Estimate of variance 13.15 ys y sc1 y sc 26.77 13.54 29.12 1.25 8.05 9.74 29.16 1.39 10.75 11.24 NB: Actual population mean of Y  29.36 The approximate variance of y sc2 following Sarndal et al., (2003), and Singh et al., (1998), is obtained as X V  y sc     xs S Et2     Wt 1  f t  S Et nt t 1  L B   Wt Yti t 1 (24) Nt  Eti N t  t 1 , Where Eti  Yti  Yt  BX ti  X t  , f t  nt N t and L W X t t 1 ti The approximate consistent and unbiased as of X Vˆ  y sc2     xs Vy sc2  estimator s et2     is Wt 1  f t  set nt t 1 L  obtained (25) n  e ti n t  i 1 , t Where eti  yti  yts  Bxti  xts  , ˆ  B L W y t 1 t L ts W x t 1 an estimate of random sampling t ts xs  given in Appendix-C of Sarndal et al., (2003) has been used There are 50 fsu’s of varying size The variable under study  y  is population of 1985 and an auxiliary variable x  is the population of 1975 The 50 fsu’s are stratified into strata considering the value of x in ascending order The stratum I consists of 13 fsu’s, stratum II consists of 14 fsu’s, stratum III consists of 12 fsu’s, stratum IV consists of 11 fsu’s respectively L PSE The values of y and x in sub samples were used to compute the population mean of Y L W x t The samples of size fsu’s were drawn by SRSWOR independently from strata to 4, respectively This process has been repeated 300 times independently That means, we obtained 300 samples of size fsu’s from each stratum Sub samples of size ssu’s are drawn by SRSWOR from each sample of fsu’s in each stratum The values of y and x in sub samples were used to compute the population mean In this process, we get 300 Y estimates of t from 300 sub samples in each stratum ts t 1 , and is X in two-stage stratified Simulation A limited simulation study has been carried out with real data The population MU284 Y In this process, we get 300 estimates of t from 300 sub-samples in each stratum The averages of these 300 estimates from each stratum are used to get the estimate of Y ˆ Y Mathematically, let  ti be the estimate of t from ith stratum We compute 1813 Int.J.Curr.Microbiol.App.Sci (2018) 7(1): 1808-1815 ˆt  300 ˆ  ti 300 t 1 (26) been found that the calibration estimator y sc1 is relatively less biased than estimators y sc The average of 300 sub-samples in the tth stratum So, we get simulated estimate of Y as follows ˆ  L  W ˆ t t 1 t (27) The percent relative bias (%RB) of the estimate has been computed as follows % RB  ˆ     100 (28) Similarly, the approximate variances of the usual estimator y s , y sc1 and y sc2 are computed The percent standard error (PSE) of the estimate has been computed as follows: PSE   SE ˆ  100 ˆ (29) The simulation studies results are presented in the Table It has been found from the results of the Table that the regression type calibration estimator y sc1 and ratio type calibration estimator y sc have performed better than the usual estimator y s in two-stage stratified random sampling However, estimator y sc1 has been found best in comparison to estimator y sc as it has minimum PSE of 9.74 as against 11.24 for y sc It may be noted that the calibration estimator y sc1 is equivalent to combined weighted regression estimator and y sc is the usual combined ratio 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Sandeep Kumar 2018 Calibration Approach Based Estimators of Finite Population Mean in Two - Stage Stratified Random Sampling Int.J.Curr.Microbiol.App.Sci 7(01): 1808-1815 doi: https://doi.org/10.20546/ijcmas.2018.701.219 1815 ... estimator of population mean of the study variate y in two- stage stratified random sampling has been developed in section-2 The calibration estimators of the population mean in two- stage stratified random. .. 7(1): 180 8-1 815 Proposed calibration estimators of population mean in two- stage stratified random sampling In this case, the population mean of the auxiliary variable can easily be obtained, i.e... N t ) in the tth stratum i An attempt has been made in the present paper to develop calibration estimator of finite population mean in two- stage stratified random sampling Following the calibration