Improved Separate Ratio Exponential Estimator for Population Mean Using Auxiliary Information

STATISTICS IN TRANSITION-new series, October 2011 Vol. 12, No. 2, pp. 401—412 IMPROVED SEPARATE RATIO EXPONENTIAL ESTIMATOR FOR POPULATION MEAN USING AUXILIARY INFORMATION Rohini Yadav1, Lakshmi N. Upadhyaya1, Housila P. Singh2 and S.Chatterjee1 ABSTRACT This paper advocates the improved separate ratio exponential estimator for population mean Y of the study variable y using the information based on auxiliary variable x in stratified random sampling. The bias and mean squared error (MSE) of the suggested estimator have been obtained upto the first degree of approximation. The theoretical and numerical comparisons are carried out to show the efficiency of the suggested estimator over sample mean estimator, usual separate ratio and separate product estimator. Key words: Study variable, auxiliary variable, stratified random sampling, separate ratio estimator, separate product estimator, bias and mean squared error. 1. Introduction In sampling theory the use of the proper auxiliary information always increases the precision of an estimator. Stratification is one of the design tools which yield increased precision. Stratified sampling entails first dividing the whole population of N units into non-overlapping subpopulations of N1 , N2 , ..., NL units, respectively, called strata that together comprise the entire population, so that N1 +N2 + ... + NL =N and then drawing an independent samples of size n1 , n 2 , ..., n L from each stratum. If the sample in each stratum is a simple random sample, the whole procedure is described as stratified random sampling. We can stratify the population in such a manner that (i) within each 1 Department of Applied Mathematics, Indian School of Mines, Dhanbad-826004, India. E-mail: , , , 2 School of Studies in Statistics, Vikram University, Ujjain-456 010, India. E-mail: 402 R.Yadav, L. N. Upadhyaya, H. P. Singh, S.Chatterjee: Improved... stratum there is as uniformity as possible and (ii) among various strata the difference are as great as possible. To obtain the full benefits from stratification, the values of the N h must be known. We use this technique because when we divide heterogeneous population into relatively more homogenous sub population, it reduces heterogeneity and hence increases precision of the estimator. This technique is also preferred because of its administrative convenient in carrying out the survey. The ratio estimate of the population mean Y can be made in two ways. One is to make a separate ratio estimate of the total of each stratum and add these totals. An alternative estimate is derived from a single combined ratio. Many authors Kadilar and Cingi (2003), Singh and Vishwakarma (2006), Singh and Vishwakarma (2010), Koyuncu and Kadilar (2010) etc. have suggested the estimators of population parameters in stratified random sampling. Let the population of size N is equally divided into L strata with N h elements L in the h th stratum such that N=  N . Let n be the size of the sample drawn h h h=1 from h th stratum of size N h by using simple random sampling without L replacement (SRSWOR) such that sample size n=  n . Let y and x be the study h h=1 and the auxiliary variables, respectively, assuming values y hi and x hi for the i th unit in h th stratum. Let Wh =  N h /N  be the stratum weight, f h =  n h /N h  be the sampling Nh Nh   Y = 1/N y , X = 1/N and     h  hi h h  x hi   h i=1 i=1   nh nh   y = 1/n y , x = 1/n  h  h   hi h  h   x hi  be the population means and sample i=1 i=1   fraction, means of the study variate y and the auxiliary variate x respectively. Our purpose L is to estimate the population mean Y=  W Y =Y of the study variable y. h h st h=1 When the population mean X h of the h th stratum of the auxiliary variable x is known then the usual separate ratio, product and regression estimators for population mean Y are respectively given as y  y rs =  Wh  h X h  h=1  xh  L (1.1) L  xh  y ps =  Wh  y h  h=1  Xh  (1.2)  L  ylrs = Wh  y h +b h X h -x h    h=1  where bh = s yxh /s 2xh (1.3)  is the sample regression coefficient of y on x of the nh  h th stratum, s yxh = 1/  n h -1  y hi -y h  x -x  is the sample covariance h hi i=1  nh s = 1/  n h -1  y hi -y h 2 yh between y and x,  2 is the sample mean i=1 2    x -x  is the sample mean  square/variance of y and s = 1/  n h -1 2 xh nh hi h i=1 square/variance of x in the h th stratum respectively. We know that   L Var yst =  Wh2 γ hS2yh (1.4) h=1 2 Nh    y -Y  is the population mean square/variance  where S = 1/  N h -1 2 yh hi h i=1 of the study variate y. The mean squared error of the estimators yrs , yps and ylrs are respectively given by   L MSE yrs = Wh2 γ h S2yh +R h2S2xh -2R hSyxh  (1.5) h=1   L MSE yps = Wh2 γ h S2yh +R h2S2xh +2R hSyxh  (1.6) h=1   Var ylrs = Wh2 γ hS2yh 1-ρ h2  L (1.7) h=1  1 1  γh =    n h Nh  , where and ρhyx = Shyx /SyhSxh  .  R h = Y h /X h  404 R.Yadav, L. N. Upadhyaya, H. P. Singh, S.Chatterjee: Improved... In this paper, we suggested an improved separate ratio exponential estimator of the population mean Y of the study variable y using the supplementary information of the auxiliary variable x. The bias and mean squared error have been obtained upto the first degree of approximation. 2. An improved separate ratio exponential estimator Motivated by Upadhyaya et al (2011), we suggested a separate ratio exponential estimator t RS of the population mean Y of the study variable y is defined as a L   X h -x h a t RS = Wh y h exp   h=1  X h +  a h -1 x h  (2.1) a  To obtain the bias and mean square error (MSE) of the estimator t RS at (2.1), we write yh =Yh 1+e0h  such that and x h =Xh 1+e1h  E  e0h  =E  e1h  =0 and ignoring the finite population correction (fpc) term, we have 2 E  e0h = 1 1 2 γ S2 , E  e1h = 2 γ hS2xh ,  2 h yh Yh Xh E  e0h e1h  = 1 γ hSyxh Yh X h (2.2) Expressing (2.1) in terms of e’s, we have  e   a -1  -1  t RS = Wh Yh 1+e0h  exp - 1h 1+  h  e1h    a h   a h    h=1   -1 -2 2  e   a -1    L   a h -1   e1h 1h h  = Wh Yh 1+e0h  11+   e1h  + 2 1+   e1h  -...  a h   a h   2a h   a h    h=1   2 2  e  L  a -1  2   a -1    a -1   e    a -1  = Wh Yh 1+e0h  1- 1h 1-  h  e1h +  h  e1h -... + 1h2 1-2  h  e1h +3  h   ah  h=1  ah   ah    ah   2a h    ah    a  L  e  a -1 2 + 1 e2 +... = Wh Yh 1+e0h  1- 1h + h 2 e1h  1h ah 2a h2 h=1  ah  L L  e e e e2  1  = Wh Yh 1+e0h - 1h - 0h 1h + 1h2  a h -  +... ah ah ah  2  h=1  Neglecting the terms of e’s having power greater than two, we have  L  e e2  1 e e  a t RS -Y = Wh Yh e0h - 1h + 1h2  a h -  - 0h 1h  ah ah  2  ah  h=1   (2.3) Taking expectation on both s (...truncated)