## Doctoral Theses

### Contributions to Stratified Sampling and Some Related Problems.

11-28-1963

11-28-1964

#### Institute Name (Publisher)

Indian Statistical Institute

Doctoral Thesis

#### Degree Name

Doctor of Philosophy

Mathematics

#### Department

Theoretical Statistics and Mathematics Unit (TSMU-Kolkata)

#### Abstract (Summary of the Work)

In this thesis I have presented some of my results on problems connected with stratified sampling. Most of the results have already been published in various journals or have been submitted for publication. The thesis includes some extension of the published results and a much greater amount of evidence to support some conjectures than could be accomodated in papers submitted to journals.A major portion of the thesis in contained in Part I. This part deals with the problem of optimum stratification and the use of simple rules to approximate optimum stratification. In this thesis attention has been confined to Normal and Gamma distributions. It is believed that the results can be used for a large variety of populations.Some of the important conclusions of this part are given be low.i) Stratification that is optimum for minimum variance allocation is almost optimum for equal allocation.ii) Equal and minimum variance allocations are almost identical for stratification that is optimum for equal allocation.iii) Equalizing strata totals in a bad rule of stratification for equal or minim um variance allocation.iv) Equalizing stratum ranges is a bad rule (for the populations considered) of stratification for equal or minimum variance allocation.v) Equalizing the product of stratus range and stratum weight is a good rule for some of the populations (very skew amongst the populations considered) but bad for others (the more symmetrical amongst the populations considered).vi) Equalizing oumulatives of the squire-root of the frequency function gives a close approximation to optimum stratification for both equal and minimum variance allocations.vii) the optimum points of stratification for proportional allocation for any Gamma distribution divides its aggregate into almost the some proportions into which the optimum points of stratification for proportional allocation for Normal distribution divide its total frequency.The main results of this part have already been published in the Australian Journal of statistics (1963).Part II deals with some other problems of stratified sampling as well as some problems whose scope is not confined to stratified sampling alone. The problems are listed below.i) Decision about the number of strata.ii) Optimum method of selecting a pair of unite from a stratum and consequent modifications of optimum stratification rules.iii) Solution of certain problems in programming and its application to the problem of optimum allocation of sample size to strata in multipurpose surveys.iv) Extension of results derived for equal probability sampling to varying probability sampling.v) The problem of obtaining unbiased ratio estimates. Important conclusions from this part are listed below.i) The decision about the number of strata depends not only on the correlation between the stratification variable and estimation variable but also on the distribution of the stratification variable.ii) The best method of selecting a pair of units from a population is to arrange the units in a monotonic sequence of the values of the estimation variable and then draw a pair of units equally distant from the two ends.

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