Sampling with Unequal Probabilities and Without Replacement

Muhammad Qaiser Shahbaz (
Statistics, NCBA&E
May, 2003


After describing the basic theory of survey sampling with reference to equal and unequal probability sampling, some selected selection procedures have been discussed, which can be used with Horvitz and Thompson estimator. Some of the popular estimators of population total (other than the Horvitz–Thompson estimator) have been discussed. The Model based sampling inference has been presented along with the famous model based estimators.

Some approximate formulae for variance of the Horvitz–Thompson estimator that use only the first order inclusion probabilities have been obtained. Some special cases of these approximations have also been given.

Three new selection procedures for use with Horvitz–Thompson estimator have been developed. These selection procedures are applicable for a sample of size two and are strictly without replacement. Some fundamental results related to inclusion probabilities and joint inclusion probabilities have been verified for these newly developed selection procedures. Empirical study for these new selection procedures have been carried out in order to see their performance for various types of populations. The regression analysis has been carried out in order to see the effect of coefficient of variation and correlation coefficient on the variance of these estimators. It has been found that these two coefficients have significant effect on the variance of Horvitz–Thompson estimator under the newly developed selection procedures.

A general procedure has been developed by introducing a constant in the revised probabilities of selection that helps in developing a number of other selection procedures. It has been found that the Yates–Grundy draw-by-draw (1953) and the Brewer (1963a) procedures are special cases of the general selection procedure. Empirical study has been conducted to obtain a suitable value of the constant for various sorts of populations.

A series of modified Murthy estimators has been developed by using the general Murthy (1957) estimator. These estimators have been developed by using various selection procedures in the general Murthy (1957) estimator. It has been found that the estimator used by Durbin (1953) for his rejective procedure is a special case of Murthy (1957) estimator under the Durbin (1967) draw-by-draw procedure. The unbiasedness of the new estimators has been verified and their design-based variances have been obtained. Empirical study has been carried out in order to see the performance of the new estimators.

The model based study of the modified Murthy estimator under the Durbin (1953) draw-by-draw procedure has been conducted and it is found that this estimator achieves the Godambe–Joshi (1965) lower bound to the variance of any estimator in unequal probability sampling.