Date of Submission


Date of Award


Institute Name (Publisher)

Indian Statistical Institute

Document Type

Doctoral Thesis

Degree Name

Doctor of Philosophy

Subject Name



Applied Statistics Unit (ASU-Kolkata)


Mukhopadhyay, Parimal (ASU-Kolkata; ISI)

Abstract (Summary of the Work)

The problem of drawing inference concerning the parameters of a finite population of identifiable units has been increasingly engaging the attention of statisticians. The central problem here is to devise a suitable method of selecting a sample from the population and to employ an appropriate estimator to estimate the finite population total or mean. A consider- able progress in this field of study has been made and many authors have contributed towards the development of the theory in this aspect of the problem of statistical inferenceNumerous papers have been written covering the first aspect of the problem, namely, method of selecting an appropriate sample from a given universe. It has been demonstrated that the unequal probability sampling provides more efficient estimator of population total than that obtained from an equal probability sampling.Hansen and Hurwitz (1943) were the first to indicate the utility of the method of selection with varying probability. They gave a method of se- lecting a single unit with probability proportional to size which can be easily extended to select more than one unit if the selection be made with replacement - the probability proprotional to size with replacement (pp- swr) scheme. Madow (1949) proposed the use of systematic sampling with unequal probabilities to avoid the possibility of units being selected more than once. Midsuno (1950), Narain (1951), among others, considered the problem of sampling with varying probabilities without replacement (wor). Closely following these authors Horvitz-Thompson (1952),Sen (1953), Yates and Grundy (1953) studied more general methods of sampling wor and with varying probabilities. The variance of the Horvitz-Thompson estima- tor (HTE) of population total is uniquely determined by the first order and second order inclusion-probabilities of units in a sample for a chosen design and reduces to zero if the variate'values are exactly proportional to the cor- responding inclusion-probabilities. As the values of the varlable of interest are unknown it seems reasonable to choose an auxiliary variable (usually known as size-measure) which is believed to be closely related to the main variable and attempt has been made to develop fixed-size sampling desings with inclusion-probabilities proportional to size-measures.such desings are called IPPS (inclusion-probability proportional to size) designs or rps de- signs (Hanurav (1967)). It was desired to construct aps designs or designs which are approximately #ps such that the variance of the HTE have vari- ance less than that of the customary Hansen-Hurwitz (1943) estimator of population total in the ppswr sampling scheme. Apart from the estimator suggested by Horvitz and Thompson themselves an alternative expression for variance of HTE was derived independently by Sen (1953) and Yates and Grundy (1953) which is valid only if the number of units in the sam- ple is fixed. The unbiased estimator of the variance of HTE proved to be disadvantageous since it is not always zero when the variance is zero. An alternative conditioanally unbiased estimator was suggested by Sen (1953) and by Yates and Grundy (1953) which possesses the particular property of being zero when the variance itself is zero. Both the estimators can assume negative values. However under some selection procedures it was demonstrated by Sen (1963), Raj (1956a), Rao and Singh (1963), Lanke (1974), Asok and Sukhatme (1974), among others, that the first estimator could take negative values for some of the pairs whereas the later one takes postive values for all the sample pairs.


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Creative Commons Attribution 4.0 International License
This work is licensed under a Creative Commons Attribution 4.0 International License.


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