ON EFFICIENCY OF CLUSTER SAMPLING ON SAMPLING ON TWO OCCASIONS. Bijoy Kumar Pradhan 1. INTRODUCTION

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1 TATITICA, anno LXIV, n., 004 ON EFFICIENC OF CLUTER AMPLING ON AMPLING ON TWO OCCAION Bijo Kumar Pradhan. INTRODUCTION In sample surves cluster or area sampling is widel practised because of its low cost and time saving device to conduct large scale and complicated surves. Its use becomes more desirable when a list of elements is not available or units of the population are widel scattered and it is required to take repeated observations on the selected units. However, the cluster sampling is more efficient than the comparable simple random sampling onl when the intra-class correlation within the same cluster is negative and less than. ( M As, the relative efficienc of the cluster sampling is controlled b both the size of the cluster and the intra-class correlation coefficient, it decreases if the size of the cluster increases substantiall (ukhatme and ukhatme, 970. In practice the intra-class correlation is usuall positive and further increase in size of the cluster leads to substantial decrease in the relative efficienc. Zarkovich and Krane (965 have shown that the correlation between two characters in cluster sampling with clusters as sampling units increases with increase in cluster size and correlation coefficient between the clusters as units is expected to be larger than correlation coefficient in element sampling. In the following sampling on two occasions is considered to estimate population mean on second occasion when the sampling units are clusters and the observations on the first occasion are regarded as ancillar information for the observations on the second or current occasion.. NOTATION AND PRELIMINARIE Consider a simple random sample of n clusters drawn from a population which consists of N clusters of M units each. Let x and be the characters under stud in the first and second occasion respectivel. In the second occasion m of the n clusters selected on the first occasion are retained at random and the remaining un-m of the clusters are replaced b a fresh selection. x and are supposed to be correlated when the are observed on the same unit repeatedl.

2 84 B. Kumar Pradhan Let X ij Value of the chatacter under stud in the first occasion, for the j th unit of the i th cluster (i,,..., N and j,,..,m ij Value of the chatacter under stud in the second occasion, for the j th unit of the i th cluster (i,,..., N and j,,..,m Define, i. M X i. X i j and M j and second occasion respectivel. M i. i j as means of the i th cluster on the first M j N ii. X N X i. and i. as cluster population mean of x and respectivel. N i N i iii. X iv. X x ij and random sample of units. x ij and random sample of units. ij as sample means based on a simple ij as sample means based on a simple v. X x ij and random sample of units. ij as sample means based on a simple 3. A GENERALIZED ETIMATOR IN CLUTER AMPLING ON TWO UCCEIVE OCCAION Consider a generalized estimator t of the population mean on second or current occasion as t a x + b x + c + d (3. where a, b, c and d are suitable constants. We have, E(t (a + b X NM + (c + d NM (3. In order that t is an unbiased estimator of NM, we have (a+b0 and (c+d (3.3

3 On efficienc of cluster sampling on sampling on two occasions 85 Hence, the estimator (3. takes the form t a ( x x + c + (- c (3.4 The variance of estimator t is V(t a V ( x + a V ( x + c V ( + ( c V ( a(- c Cov (, x (3.5 other covariance terms being zero. Minimising the variance of t with respect to a and c when N is sufficientl large, the optimum values of a and c are / M ( b a. x M x b and c b (3.6 where x u, and x are intra-class correlation coefficients defined b n i, j<k i, j<k ( ij NM ( ik NM, (M -(NM - ( X ij X NM ( X ik X NM, (M (NM x ( ij NM i j x i NM ( X ij X NM j NM,, and the simple correlation coefficient between cluster means on both occasions is defined b b N i ( ( X X i NM i NM N N ( i NM ( Xi X NM i i /

4 86 B. Kumar Pradhan Using the optimum values of a and c given b (3.6, the estimator t reduces to / M ( t x x b x M x b b b (3.7 with variance b V opt ( t ( M b (3.8 Now, the optimum value of µ is given b further minimising to µ, which gives V opt(t with respect b (3.9 Thus, M V opt ( opt ( t b M n ( EFFICIENC OF CLUTER AMPLING ON TWO OCCAION If the samples on both occasions are drawn using RWOR, the variance of the optimum estimator t neglecting the finite population correction factor is given b V opt ( t Mn, (4. where is the simple correlation coefficient between values of units on first and second occasion. The relative efficienc of t compared to t is V ( t opt opt ( opt t M ( b V ( ( (4. t would be more efficient than t if

5 On efficienc of cluster sampling on sampling on two occasions 87 M M b i.e. - where M - b (4.3 Further, in order that t would be more efficient than t if M b b (4.4 which gives the upper limit of M. As b is expected to be greater than (Zarkovich and Krane, 965 is likel to be greater than unit and as such, even if is positive, the cluster sampling on both occasions provides more efficient estimate than the simple random sampling on both occasions. Tables: -3 have been computed below to show the relative efficienc of cluster sampling in sampling on two occasions compared to simple random sampling of elements for some specified values of,, and M. b TABLE Relative efficienc of cluster sampling over simple random sampling (M, 0.0 / b (M4, 0.0 / b (M5, 0.0 / b

6 88 B. Kumar Pradhan TABLE Relative efficienc of cluster sampling over simple random sampling (M, 0.05 / b (M4, 0.05 / b (M5, 0.05 / b TABLE 3 Relative efficienc of cluster sampling over simple random sampling (M, 0. / b (M4, 0. / b (M5, 0. / b

7 On efficienc of cluster sampling on sampling on two occasions 89 COMMENT: (a For fixed (intra-class correlation coefficient and, the efficienc increases with large increase in b( b (correlation coefficient between cluster means. (b For fixed b and, the efficienc decreases with increase in. Note: The same results are obtained if optimum estimator is formed b a weighted combination of double sampling regression estimator using variate values of the entire clusters in the first occasion and of the clusters relating to matched part in the second occasion and simple estimator using unmatched clusters in the second occasion. (Appendix - A. Department of tatistics Utkal Universit, Bhubaneswar BIJO KUMAR PRADHAN APPENDIX-A For estimating NM as NM, let us consider the following regression estimate of + b (x x t where b is the sample regression coefficient. Hence, V( t V ( + b V ( x + b V (x b COV (x,, other covariance terms being zero. Therefore, V( t ( + M ( b + (+ M b The estimate of given b NM from the unmatched portion in the second occasion is t with variance

8 90 B. Kumar Pradhan V( t ( M for large N. t in- The weighted estimate of NM, sa, t, is given b weighting versel proportional to their population variance. t and Hence t wt + w t w + w where w V( t and w V( t Therefore, the variance of t is given b b V( t ( M b u where. n The optimum value of µ which minimizes Hence, b V ( t is V ( ( opt t M b. ACKNOWLEDGEMENT Author is thankful to the referee for man useful suggestion for improving the manuscript. REFERENCE M.H. HANEN, W.N. HURWITZ, W.G. MADOW (953, ample urve Methods and Theor, Wile, New ork. P.V. UKHATME, B.V. UKHATME (970, ampling theor of urves with Applications, Food and Agriculture Organisation, Rome, econd Edition... ZARKOVICH, J. KRANE (965, ome efficient was of cluster sampling. Proceedings of 35th session of International tatistical Institute, Belgrade.

9 On efficienc of cluster sampling on sampling on two occasions 9 RIAUNTO ull efficienza del campionamento a grappolo nel campionamento a due occasioni In questo lavoro si mostra come anche quando il coefficiente di correlazione intraclasse tra le unità dello stesso cluster è positivo, il campionamento a grappolo su due occorrenze (occasions è più efficiente che il campionamento casuale semplice su due occorrenze (occasions se si vuole stimare la media di popolazione di un carattere studiato sulla (seconda occorrenza in esame. UMMAR On efficienc of cluster sampling on sampling on two occasions In this paper it has been shown that even if the intra-class correlation coeffcient among the units in the same cluster is positive, under certain condition, the cluster sampling on two occasions is likel to be more efficient than the simple random sampling on two occasions to estimate the population mean of the character under stud on current (second occasion.