# Sampling and Estimation Under 'Deep Stratification'.

December 1988

## Date of Award

Winter 12-12-1989

## Institute Name (Publisher)

Indian Statistical Institute

## Document Type

Master's Dissertation

## Degree Name

Master of Technology

Computer Science

## Department

Applied Statistics Unit (ASU-Kolkata)

## Supervisor

Roy, Bimal Kumar (ASU-Kolkata; ISI)

## Abstract (Summary of the Work)

1. Introduction to Deep StratificationIf the study variables are all related to single supplementary variable (x) for which information is available for all population units then stratification and allocation may be done in an optimum fashion using the data on x. But all the characteristics of interest may not be related to one supplementary variable but to two or more auxiliary variables. In such a situation, the unite may first be grouped into primary strata with respect to the most important of the stratification variables and then within each of the primary strata so formed, secondary or sub-strata may be constructed according to another supplementary variable, and so on. This procedure is known as multiple stratification or deep strati- fication.However, our problem is associated to two-way strati- fication of the population units. Suppose a population consists of N units and these N units are grouped into p x q cells of a matrix of order p by q. So each cell constitutes one stratum. p denotes the number of levels of first stratification variable and g denotes the number of levels of second stratification variable.Let bij= 1(1)p, j = 1(1)g denote the number of population units in the (i, j) th cell. Let Uijk k = 1(1)Bij denote the k-th population unit of the (i, j) th cell. Our problem is to draw a sample of n units from the population considering the following restrictions -(i) MINR(1) < total number of sample units in the i-th row â‰¤ MAXR(i), for all i = 1(1)p(ii) MINC (j) S total number of sample units in the j-th column â‰¤MAXC(j) for all j = 1(1)qwhere MINR(1), MAXR(i), MINC(j), MAXC(j), n are given.So our problem is to select a matrix A = ((aij)), i = 1(1)pij = 1(1)q following the restrictions :- (A)MINC(i)â‰¤ qÆ©i=Î› 2ijâ‰¤MAXC(i), for all i = 1(1)p(B)MINC(j)â‰¤ pÆ©i=Î› 2ijâ‰¤MAXC(j), for all j = 1(1)q(C)2ijâ‰¤bijvi=1(1)p, j = 1(1)qand (D) pÆ©i=Î› pÆ©j=Î› 2ij=n (given).We mey get a large number of solution matrices A satisfying the above criteria. Suppose PNO denotes the possible number of solution matrices. Select one matrix out of PNO matrices randomly.For a selected matrix A = ((aij)), if aij > odraw a sample of aij units from bij population units of the (i, j) th cell with SRSWOR sampling scheme, i = 1(1)p, j = 1(1) q.Suppose P, is the probability that one particular unit of the cell numbered i, will be included in the sample and is Fij the probability that one particular unit of the cell numbered i and another particular unit of the cell numbered j will be included in the sample. The (i,j) th cell will be numbered as e, where 8 will be obtained by the formulas = q(i - 1) + j. %3D.

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ISI-DISS-1988-