Optimal allocation in J-stage sampling with stratification from finite population

Summary The known results of optimal allocation in two-stage simple random sampling with stratification from finite population are generalized for J-stages. The optimalization is treated by minimizing the cost function for given variance as well as by minimizing the variance for given cost. It turns out that for both cases the optimal values differ only for the first-stage units and are the same for all the subsequent. The general formula for minimum variance of the unbiased estimator of the population mean per element is given.     

A comparison of approximately optimal stratification given proportional allocation with other methods of stratification and allocation

Summary In this article, we shall present an approximately optimal method for constructing stratum boundary points when the sample is allocated proportionally. The method is based on an equal partitioning of the cumulative off ^1/3, wheref is the distribution of the stratification variable. We show that in many practical situations this technique compares favourably with approximately optimal stratification and allocation methods previously suggested.

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Zusammenfassung In diesem Artikel stellen wir eine annähernd optimale Methode zur Festlegung von Stratabegrenzungspunkten dar, die für proportional angeordnete Samples gilt. Die Methode basiert auf einer gleichen Einteilung derf ^1/3-Kummulation, wobeif die Verteilung der Stratifikations-variablen darstellt. Wir zeigen, daß diese Technik in vielen praktischen Fällen gegenüber den bisher vorgeschlagenen Methoden zur optimalen Stratifikation und Zuordnung nicht schlecht abschneidet.

     

Optimum allocation in stratified sampling with replacement

In a stratified sample, when sampling is done with replacement in each stratum a better estimate of the population mean can be achieved by considering the distinct units only. An explicit expression for the variance for the mean, of a stratified sample based on the distinct units only, is obtained. Then the optimum allocation for the different stratum are obtained by minimizing this variance subject to (i) total sample size being fixed, or (ii) the expected number of distinct units being fixed. Neyman’s solutions are obtained as special cases. The solutions finally arrived at are algebraically complex, hence, numerical methods are applied. In all examples, the variance of the estimates obtained by this method are smaller than the variances obtained by Neyman’s allocation. A part of this work was supported by the Office of the Ordinance Research; U.S.A. Grant (DA-AROL(D)-31-124-G83) when the author was at University of California, Berkeley.     

Admissible unbiased variance estimation in finite population sampling under randomized response

LetP be the proportion of units in a finite population possessing a sensitive attribute. We prove the admissibility of (i) an unbiased estimator of the variance of a general homogeneous linear unbiased estimator ofP and (ii) an unbiased estimator of the population varianceP(1−P), based on an arbitrary but fixed sampling design, under the randomized response plans due to Warner (1965) and Eriksson (1973). Admissibility of an unbiased strategy for estimating the population variance is also established.     

Some aspects of random permutation models in finite population sampling theory

Summary We first consider Neyman's optimum allocation of sample size to strata in the light of available auxiliary information for which a suitable random permutation model is assumed. For a special case of this model the allocation of the sample size reduces to the same as when a certain superpopulation regression model is assumed. Motivated by this, more generally, we discuss some optimality results under random permutation models and compare them with the corresponding results when a superpopulation regression model is assumed.     

Estimation of finite population variance using random non-response in survey sampling

In this paper, an estimator of finite population variance proposed by Isaki (1983) is studied under the two different situations of random non-response suggested by Tracy and Osahan (1994). A distribution is proposed for the number of sampling units on which information could not be obtained due to random non-response. The estimators for the mean square errors of the proposed strategies are also suggested. This paper was written while both authors were members of the Dept. of Econometrics, Monash University, Clayton 3168, Australia. This paper was presented on SISC—1996, Sydney, Australia. The opinions and results discussed in this paper are of authors and not necessarily of their institutes.     

On generalized estimators in sampling from finite populations

Summary A comparative study of the generalized estimators proposed byKoop andMurthy is made.

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Zusammenfassung In der vorliegenden Arbeit vergleicht der Autor verschiedene vonKoop undMurthy beschriebene Estimatoren.

     

Estimation of a finite population total in varying probability sampling for multi-character surveys

The problems of estimating population total in multi-charter surveys are considered in a unified set up. Alternative estimators for Rao-Hartley-Cochran (1962), Midzuno-Sen (1952,53) and other varying probability sampling schemes are proposed when the measure of size is not well related to the study variables. Some of the proposed estimators are found superior to the existing alternatives. A numerical study is carried out to investigate the performances of the proposed alternatives.     

A chain ratio-type estimator in finite population double sampling using two auxiliary variables

Summary In this paper we construct a chain ratio-type estimator using two auxiliary variables. The performance of the constructed estimator relative to the simple mean, ratio-type estimate based on double sampling andChand's ratio-type estimator is investigated. A numerical illustration is given.     

Subjective Bayesian multivariate stratified sampling from finite populations

Summary In this paper we obtain optimum allocation formula in stratified sampling from a finite population withk characteristics under study using the superpopulation approach put forth byEricson [1969a]. Allocation at the second phase is also considered using information obtained from the first phase. Two different approaches are employed: a Bayesian posterior analysis and a Bayesian preposterior analysis. It is also shown that our allocation formulae for the second phase observations include the results ofMohd. Zubair Khan [1976] andDraper/Guttman [1968a] as special cases when the unknown characteristicX _ij possessed by the (i, j)-th element is scalar valued and the stratum sizes are large.     

Optimum allocation in multivariate stratified random sampling with overhead cost

The problem of allocating the sample numbers to the strata in multivariate stratified surveys, where, apart from the cost involved in enumerating the selected individuals in the sample, there is an overhead cost associated with each stratum, has been formulated as a non-linear programming problem. The variances of the posterior distributions of the means of various characters are put to restraints and the total cost is minimized. The main problem is broken into subproblems for each of which the objective function turns out to be convex. When the number of subproblems happens to be large an approach has been indicated for obtaining an approximate solution by solving only a small number of subproblems.     

Optimal allocation mechanisms with single-dimensional private information

We study revenue-maximizing allocation mechanisms for multiple heterogeneous objects when buyers care about the entire allocation, and not just about the ones they obtain. Buyers’ payoff depends on their cost parameter and, possibly, on their competitors’ costs. Costs are independently distributed across buyers, and both the buyers and the seller are risk-neutral. The formulation allows for complements, substitutes and externalities. We identify a number of novel characteristics of revenue-maximizing mechanisms: First, we find that revenue-maximizing reserve prices depend on the bids of other buyers. Second, we find that when non-participation payoffs are type-dependent, revenue-maximizing auctions may sell too often, or they may even be ex-post efficient.     

An optimum sampling design to base HT-method of estimating a finite population total

The problem of constructing a sampling design with the value of the sum of second order inclusion probabilities attaining its lower bound for non-integral values of the expected effective size of a sample in the design has been considered in this paper. If the values of the characteristic of interest on all the units in the population are non-negative the design is admissible (in the sense of variance) with respect to Horvitz-Thompson estimator in the class of designs with the same set of values of the first order inclusion probabilities of the units. Again such a design is best to use Horvitz-Thompson estimator of population total in the sense of smallest average variance of the estimator under a special superpopulatio model.     

Optimal estimates and optimal predictors of finite population characteristics in the presence of nonresponse

Summary When samples from a finite population are studied, for instance by interview, there will usually be some units from which no response is obtained. In this paper optimal predictors of finite population characteristics, when nonresponse is present, are studied. The predictors are studied under simple regression superpopulation models. The optimal predictors are connected to the classical weighted sample estimates which are shown to be maximum likelihood estimates, provided the probability function is fully described by the sampling design. The predictors are compared with respects to their efficiencies for some simple models and a possible explanation to the fact that the poststratification estimate which compensate for nonresponse does no better than the simple estimate, is pointed out.     

Unbiased estimators of mean, variance and sensitivity level for quantitative characteristics in finite population sampling

This paper considers the problem of procuring reliable information on sensitive quantitative characteristics without exposing respondents’ identities. A generalized optional randomized response procedure is proposed, which enables us to estimate some unknown population parameters unbiasedly. In particular, conditions for the assurance of unbiased estimations of mean, variance and sensitivity level are studied. Efficiency comparisons are also carried out to study the performance of the proposed procedure.     

Optimum allocation in Bayesian stratified two-phase sampling when there arem-attributes

The problem of optimally allocating a sample amongk-strata at the second phase of a two-phase sampling procedure when the sampling is for proportion has been discussed byNewbold [1971] and then by the author [Khan, 1972].This paper deals with optimum allocation of the sample amongk-strata at the second phase of a two-phase sampling procedure, when the sampling is form-attributes. The problem is to estimate the proportion of each attribute in the population. Here (m–1) dimensional Dirichlet distribution is taken as the prior distribution.     

On Bayesian optimum allocation in some two phase sampling problems

Summary For estimating certain parametric functions, we consider the problem of allocatingN _i, the size of the sample from theith population,i=1,2,...,k, at the second phase of sampling of a two phase sampling procedure, given that we taken _i observations from the population at the first phase. We consider that the observations from theith population follow the exponential distribution with mean _i,i=1,2,...,k, and the functions to be estimated are (i) (d_i/_i) and (ii) (d_ii). When the total cost of sampling at the second phase is c _iN_i and is fixed, allocations using the Bayes approach are obtained so that the estimation is as precise as is possible.