Almost unbiased ratio and product-type estimators in systematic sampling

Q¨ UESTII ´ O,vol. 22, 3, p. 403-416, 1998 ALMOST UNBIASED RATIO AND PRODUCT-TYPE ESTIMATORS IN SYSTEMATIC SAMPLING R. SINGH  Prestige Institute of Management, Mandsaur H.P. SINGH  Vikram University In this paper we have suggested almost unbiased ratio-type and producttype estimators for estimating the population mean Y of the study variate y using information on an auxiliary variate x in systematic sampling. The variance expressions of the suggested estimators have been obtained and comparedwith usual unbiasedestimator y  , Swain’s(1964)ratio estimator y  Rand Shukla’s product estimator y  P. It has been shown that the proposed estimators are more efficient than usual unbiased estimator y  , ratio estimator y  Rand product estimator y  P. An empirical study is carried out to demonstrate the superiority of the constructed estimators over the estimators y  , y  Rand y  P. Keywords: Almost unbiased ratio and product-type estimators, Auxiliary information, Bias, Variance, Systematic sampling. AMS Classification: 62 DO5 * Prestige Institute of Management, Mandsaur, M.P., India. **School of Studies in Statistics, Vikram University, Ujjain-456010, M.P., India. – Received February 1997. – Accepted March 1998. 403 1. INTRODUCTION Systematic sampling has got the nice feature of selecting the whole sample with just one random start. Apart from its simplicity, which is of considerable importance, this procedure in many situations provides estimators more efficient than simple random sampling and/or stratified random sampling for certain types of population [Cochran (1946, 77), Gautschi (1957), Hajeck (1959)]. Suppose the population consists of Nunits U = ( U1 ; U2 ; : : : ; UN ) numbered from 1 to Nin some order. Unless mention otherwise, we assume N = nk, where nand kare positive integers. Thus there will be ksamples (clusters) each of size n. We select one sample at random out of ksamples and observe the study variate yand auxiliary variate xfor each and every unit selected in the sample. Let ( yij ; xij;i = 1 ; 2 ;::: ; k; j = 1 ; 2 ;::: ; n ) denote the value of jth unit in the ith sample. The systematic sample means y  = ( 1 = n ) n ∑ j = 1yij ; x  = ( 1 = n ) n ∑ j = 1xij ; ( i = 1 ; 2 ;::: ; k ) are unbiased estimators of the population means ( Y ; X ) of ( y ; x ) respectively. It is assumed that the population mean Xof the auxiliary variate xis known. Thus the classical ratio and product estimators for Ybased on a systematic sample ( yij ; xij; i = 1 ; 2 ;::: ; k;j = 1 ; 2 ;::: ; n ) of size n, are respectively defined by y  R = ( y  = x  ) X(1.1) and y  P = y  ( x  = X ) (1.2) which are respectively due to Swain (1964) and Shukla (1971). The biases and variances of y  Rand y  Pto the first degree of approximation are, respectively, given by B ( y  R ) = ( N ? 1 ) nN Y f 1 + ( n ? 1 ) ρx g ( 1 ? kρ  ) C2 x (1.3) B ( y  P ) = ( N ? 1 ) nN Y f 1 + ( n ? 1 ) ρx g kρ  C2 x (1.4) Var ( y  R ) = ( N ? 1 ) nN Y2 f 1 + ( n ? 1 ) ρx g  ρ  2C2 y + ( 1 ? 2Kρ  ) C2 x  (1.5) Var ( y  P ) = ( N ? 1 ) nN Y2 f 1 + ( n ? 1 ) ρx g  ρ  2C2 y + ( 1 ? 2Kρ  ) C2 x  ; (1.6) and the variance of usual unbiased estimator y  is given by Var ( y  ) = ( N ? 1 ) nN f 1 + ( n ? 1 ) ρy g S2 y ; (1.7) 404 where ρx = E ( xij ? X )( xij ? X ) E ( xij ? X ) 2 ; ρy = E ( yij ? Y )( yij ? Y ) E ( yij ? Y ) 2 are intraclass correlation between a pair of units within the systematic sample for the auxiliary variate xand study variate y respectively; ρ = E ( yij ? Y )( xij ? X ) q  E ( yij ? Y ) 2E ( xij ? X ) 2  is the correlation coefficient between yand x; ρ  = [ f 1 + ( n ? 1 ) ρy g = f 1 + ( n ? 1 ) ρx g ] ; K = ρ ( Cy = Cx ) and ( Cy ; Cx ) are the coefficients of variation of the variates ( y ; x ) respectively. It is obvious from (1.3) and (1.4) that the estimator y  Rand y  Pare biased. In some situations, bias is disadvantegeous. To keep this in view Kushwaha and Singh (1989) suggested a class of almost unbiased ratio and product-type estimators for population mean Yusing Jack-knife technique introduced by Quenouille (1956). However, it has been observed that their estimators attained the minimum variance equals to the approximate variance of usual linear regression estimator in systematic sampling, for the well known optimal choice K  = ρ  K. Later Banarasi et al (1993) suggested a ratio, product and difference estimator in systematic sampling. Banarasi et al (1993) estimator attains the minimum variance equals to the approximate variance of usual linear regression estimator for the known value of K. However the optimum estimator in the class of estimators suggested by Banarasi et al (1993) is biased. In this paper an effort has been made to propose almost unbiased ratio and product-type estimators which depend only on the well known optimum choice K  = ρ  K. The value of Kcan be made known quite accurately either due to past experience or by pilot sample surveys. Various authors including Murthy (1967, p. 325), Reddy (1978) and Srivenkataramana and Tracy (1983) have advocated about the assessment of the value of K  . 2. THE CLASS OF RATIO-TYPE ESTIMATORS Let the correlation between yand xbe positive. Suppose d1 = y  ,d2 = y  ( X = x ) and d3 = y  ( X = x  ) such that di 2 D ( i = 1 ; 2 ; 3 ) , where Ddenotes the set of all possible ratio-type estimators for estimating population mean Y. By definition, the class D will consist of all drof the form dr = 3 ∑ i = 1 widi 2 D(2.1) 405 where 3 ∑ i = 1wi = 1 for wi 2 R(2.2) and wi ( i = 1 ; 2 ; 3 ) denotes the constants used for reducing the bias in the class of estimators and Rstands for the set of real numbers. Such procedure of estimation of mean has also been discussed by Singh and Singh (1993). 3. VARIANCE To obtain the variance of drto the first degree of approximation, we write e0 = ( y  ? Y ) = Y ; e1 = ( x  ? X ) = X such that E ( e0 ) = E ( e1 ) = 0 and E ( e2 0 ) = ( N ? 1 ) nN f 1 + ( n ? 1 ) ρy g C2 y E ( e2 1 ) = ( N ? 1 ) nN f 1 + ( n ? 1 ) ρx g C2 x E ( e0e1 ) = ( N ? 1 ) nN f 1 + ( n ? 1 ) ρy g 1 = 2 f 1 + ( n ? 1 ) ρx g 1 = 2ρCyCx : Expressing (2.1) in terms of e’s with (2.2), we have dr = Y + Y  e0 ? ( w2 + 2w3 ) e1 + O ( e2 )  : (3.1) Let us choose w2 + 2w3 = w(say, another constant)(3.2) Then, to the first degree of approximation, the variance of dris given by Var ( dr ) = ( N ? 1 ) nN Y2 f 1 + ( n ? 1 ) ρx g  ρ  2C2 y + w ( w ? 2ρ  K ) C2 x  (3.3) which is minimized for w = ρ  K = K  (say)(3.4) 406 Substitution of (3.4) in (3.3) yields the minimum variance of dras min. Var ( dr ) = ( N ? 1 ) nN f 1 + ( n ? 1 ) ρy g ( 1 ? ρ2 ) S2 y (3.5) where S2 y = N ( N ? 1 ) E ( yij ? Y ) 2. We have thus proved the following theorem. Theorem 3.1. Up to terms of order n ? 1, Var ( dr )  ( N ? 1 = nN f 1 + ( n ? 1 ) ρy g ( 1 ? ρ2 ) S2 y with equality holding if w = ρ  K = K  : 4. BIAS REDUCTION OF ORDER O ( n ? 1 O ( n ? 1 O ( n ? 1) Equation (3.5) shows that the class of estimators attain the minimum variance equal to that of the usual linear regression estimator y1rin systematic sampling, defined as y1r = y  + ˆ β  ( X ? x  ) (4.1) where ˆ β  is the systematic sample regression coefficient of yon x. From (3.2) and (3.4), we have w2 + 2w3 = ρ  K(4.2) We note from (2.2) and (4.2) that there are three unknown quantities to be determined from only two equations. It is, therefore, not possible to obtain unique values for the constants wi’s, ( i = 1 ; 2 ; 3 ) to be used for bias reduction. To get the unique values for these constants wi’s, ( i = 1 ; 2 ; 3 ) , we shall impose the additional linear restriction as 3 ∑ i = 1 B ( di ) = 0(4.3) where B ( di ) stands for the bias in the i ? th ( i = 1 ; 2 ; 3 ) estimator of the population mean Y. 407 Equations (2.2), (4.2) and (4.3) may be expressed as 2 6 4 1 1 1 0 1 2 B ( d1 ) B ( d2 ) B ( d3 ) 3 7 5 2 6 4 w1 w2 w3 3 7 5 = 2 6 4 1 ρ  K 0 3 7 5 (4.4) It is well known in systematic sampling that B ( d1 ) = B ( y  ) = 0(4.5) and to the first degree of approximation B ( d3 ) = ( N ? 1 ) nN Y f 1 + ( n ? 1 ) ρx g ( 3 ? 2ρ  K ) : (4.6) The bias of d2is given at (1.3). Thus using (1.3), (4.5) and (4.6) in (4.4), we get the values of w1,w2and w3as w1 = ( 1 ? kρ  ) 2 w2 = ( 3 ? 2kρ  ) Kρ  w3 = ( Kρ  ? 1 ) Kρ  9 > > > = > > > ; (4.7) Substitution of (4.7) in (2.1) yields an almost unbiased ratio-type estimator for Yas dru =  ( 1 ? Kρ  ) 2y  + ( 3 ? 2Kρ  ) Kρ  y  ( X = x  ) ? ( 1 ? Kρ  ) Kρ  y  ( X = x  ) 2  (4.8) with the variance Var ( dru ) = ( N ? 1 ) Nn f 1 + ( n ? 1 ) ρy g ( 1 ? ρ2 ) S2 y : (4.9) We have thus proved the following theorem. Theorem 4.1. The estimator dru at (4.8) is an «optimum almost unbiased ratio-type» in the class of ratio-type estimators drat (2.1), with the variance given at (4.9). 408 It is to be noted that the estimator dru at (4.8) is unbiased upto terms of order n ? 1. Same process may be repeated by considering B ( di ) , ( i = 1 ; 2 ; 3 ) to terms of order O ( n ? 2), to get the unbiased ratio-type estimator to terms of order O ( n ? 2) and so on. In many situations of practical importance ρy ' ρxand known, for instance, see Murthy (1967) and Srivenkataramana and Tracy (1983). Thus ρ  ! 1 and K ! K and hence the values of wi’s, ( i = 1 ; 2 ; 3 ) are given by w1 = ( 1 ? K ) 2 w2 = ( 3 ? 2K ) K w3 = K ( K ? 1 ) 9 > > = > > ; (4.10) Putting (4.10) in (2.1) we get the almost unbiased ratio-type estimator for Yas d  ru =  ( 1 ? K ) 2y  + K ( 3 ? 2K ) y  ( X = x  ) + K ( K ? 1 ) y  ( X = x  ) 2  (4.11) with the variance Var ( d  ru ) = ( N ? 1 ) Nn f 1 + ( n ? 1 ) ρy g ( 1 ? ρ2 ) S2 y : (4.12) It is further remarked that the estimator d  ru in (4.11) can also be obtained from the estimator dru with ρ  ' 1. The value of Kcan easily be guessed quite accurately either through pilot survey or experienced gathered in due course of time, for instance, see Murthy (1967, p. 325) and Reddy (1978). From (1.5), (1.7) and (4.9) we have Var ( y  ) ? Var ( dru ) = ( N ? 1 ) nN f 1 + ( n ? 1 ) ρy g ρ2S2 y > 0(4.13) and Var ( y  R ) ? Var ( dru ) = ( N ? 1 ) nN Y2 f 1 + ( n ? 1 ) ρx g C2 x ( 1 ? Kρ  ) 2 : (4.14) > 0 unless Kρ  = 1 : We have thus established the following theorems. Theorem 4.2. The inequality Var ( dru ) < Var ( y  ) always holds good. 409 Theorem 4.3. The inequality Var ( dru ) < Var ( y  R ) is always true except when Kρ  = 1 : Remark 4.1. It is customary in systematic sampling to arrange the population units such that ρy,ρxare small, preferably negative, since positive ρy,ρxinflate sampling variance. Thus it may be possible to assess ρy,ρxquite accurately for the arrangement used. This together with assessed K = ρ ( Cy = Cx ) leads to the optimal choice of K  .  5. A CLASS OF PRODUCT-TYPE ESTIMATORS Suppose d  1 = y  ,d  2 = y  ( x  = X ) and d  3 = y  ( x  = X ) 2such that d  i 2 D  for i = 1 ; 2 ; 3, where D  denotes the sets of all possible product-type estimators for estimating the population mean Y. By definition, the set D  will consist of all d  pof the form d  p = 3 ∑ i = 1w  id  i 2 D  (5.1) for 3 ∑ i = 1 w  i = 1 and d  i 2 R(5.2) where wi’s, ( i = 1 ; 2 ; 3 ) denote the constants used for bias reduction. As in section 3, the following theorem can easily be proved. Theorem 5.1. Up to terms of order n ? 1, Var ( d  p ) > ( N ? 1 ) nN f 1 + ( n ? 1 ) ρy g ( 1 ? ρ2 ) S2 y with equality holding if w = ? ρ  K = ? K  ; where w  = ( w  2 + 2w  3 ) . 410 Proceeding exactly in the same way as in section 2,3 and 4, we get the values of wi’s, ( i = 1 ; 2 ; 3 ) as w  1 = [ 1 + ρ  K ( 1 + ρ  K )] w  2 = ? ρ  K ( 1 + 2ρ  K ) w  3 = ρ  2K2 : 9 > > = > > ; (5.3) Use of these wi’s, ( i = 1 ; 2 ; 3 ) removes the bias of d  pupto terms of order n ? 1at (5.1). Thus the substitution of wi’s, ( i = 1 ; 2 ; 3 ) in (5.1) yields an almost unbiased product-type estimator d  pu =  f 1 + ρ  K ( 1 + ρ  K ) g y  ? ρ  K ( 1 + 2Kρ  ) y  ( x  = X ) + ρ  2K2y  ( x  = X ) 2  (5.4) with the variance Var ( d  pu ) = ( N ? 1 ) Nn f 1 + ( n ? 1 ) ρy g ( 1 ? ρ2 ) S2 y : (5.5) In case ρy ' ρx ) ρ  = 1, the expressions in (5.3) reduce to: w  1 = [ 1 + K ( 1 + K )] w  2 = ? K ( 1 + 2K ) w  3 = K2 9 > > = > > ; (5.6) and hence the estimator d  pu at (5.4) takes the form d  pu =  f 1 + K ( 1 + K ) g y  ? K ( 1 + 2K ) y  ( x  = X ) + K2y  ( x  = X ) 2  (5.7) with the variance given by Var ( d  pu ) = ( N ? 1 ) Nn f 1 + ( n ? 1 ) ρy g ( 1 ? ρ2 ) S2 y : (5.8) We have thus proved the following theorem. Theorem 5.2. The estimator d  pu at (5.4) is an «optimum almost unbiased producttype»in the class of product-type estimators d  pat (5.1), with the variance given at (5.5). 411