A Note on Optimum Stratification of Populations for Estimating the Population Means

Abstract

The note gives the results of'a study carried out to find the optimum points of stratification (OPS) for estimating the population means of some standard distributions. The distributions considered here are the normal and the set of chi-square distributions. The OPS for the various gamma distributions can be easily derived from the known OPS of the corresponding chi-square distributions. The OPS depend on the type of allocation envisaged. I n this note attention has been confined to the proportional, equal, and optimum allocations.Tables of OPS are given for the distributions and allocations mentioned above. Some other interesting results follow :(i) Equalization of stratum totals as suggested by Ransen, Hurwitz and Madow (1953) does not lead to OPS for any of the populations considered.(ii) Equalization of Jdf (8)) *& gives an excellent approximabioii to the OPS for both equal and optimum allocations. (iii) The OPS for equal and optimum allocations almost coincide.To put in other words, if strata are defined by OPS optimum allocation differs only slightly from equal allocation. New rules are suggested for the family of distributions considered in this paper for all the three types of allocations. Practical Considerations in Stratifying PopulationsThe main purpose of stratifying a population is to reduce the variance of some estimator of its mean. It is known that even deliberately badly selected stratification will achieve the purpose with proper allocation (say proportional). With a similar allocation, the reduction in variance becomes more substantial if the strata are more homogeneous. Sometimes the homogeneity is achieved if strata are formed by combining together geographically contiguous areas. A more intimate knowledge of the population makes a greater reduction in variance possible by better stratification and allocation. For optimum allocation a knowledge of different strata variances is required. It is unlikely that one has this information for the character under study. Either estimates of the variances or the actual variances for a closely related variable are substituted for this purpose. Similarly for optimum stratification one needs the knowledge of the distribution of the variable under study. The non-availability of this information 1 Manuscript received August 8, 1962.

Obtaining Strata Boundaries under Proportional Allocation with Varying Cost of Every Unit

“…Also available methods present in the literature such as Aoyama (1954), Ekman (1959), Sethi (1963) etc. are mostly classical methods by which we obtain approximately optimum strata boundaries.…”

“…Dalenius and Gurney (1951) showed that if stratification is not done in well defined way in some cases would lead us to the loss in precision. Subsequently the attempts for determining OSB have been made by several authors, such as Mahalanobis (1953), Hansen and Hurwitz (1953), Sethi (1963), Aoyama (1954)), Ekman (1959), Rizvi et al (2000) etc. Des R. (1964) studied the performance of equal size stratification with equal allocation by comparing it with optimum stratification when same allocation of samples in different strata is used.…”

A Procedure for Computing Optimal Stratum Boundaries and Sample Sizes for Multivariate Surveys

“…Several other numerical and computational methods [5], [7]- [29] have been developed for determining OSB when a single main variable is under study.…”

Constructing Strata of Primary Sampling Units for the Residential Energy Consumption Survey

“…Dalenius (1950Dalenius ( , 1957 and others developed methods of approximating optimal stratum boundaries under Neyman or proportional allocation. Lavallée and Hidiroglou (1988) developed an algorithm for stratifying skewed distributions based on stratum boundary work by Sethi (1963). Stratification does not need to be optimal to be useful, however.…”