Estimating the expected value of fuzzy random variables in the stratified random sampling from finite populations
Introduction
Puri and Ralescu introduced in 1986 (see [10]) the concept of fuzzy random variables (FRVs) to extend random variables and vectors as well as many well-known results for the latter.
In a previous paper, Lubiano and Gil [7] have studied the estimation of the expected value of a FRV in simple random sampling and random sampling with replacement from finite populations. We make a explicit use of the results in Lubiano and Gil's paper in proving the ones in the present paper.
The study developed in this paper generalizes Lubiano and Gil's one by considering a more complex, and usually more precise, sampling technique, as it is the stratified random sampling. The discussion of different possible types of allocation is stressed, and conditions are established to guarantee the advantages of certain sampling procedures with respect to other ones. Moreover, the final part of this paper is dedicated to examining a way to quantify the degree of adequacy of a variate for stratification, and finally we present an inferential method to test the convenience of a given variate.
Section snippets
Preliminaries
Let be the class of the nonempty, compact and convex subsets of the Euclidean space (), and let be the class of the mappings (fuzzy subsets of ) such that for all α∈[0,1], where if α∈(0,1], and . is referred to as the α-level set of .
The space can be endowed with a linear structure, induced by the product by a scalar and the Minkowski addition given, respectively, byfor all
Estimation of the expected value of a fuzzy random variable in the stratified random sampling
Let us consider a finite population with N individuals or sampling units U1,…,UN, in such a way that they are grouped into L disjoint subpopulations called strata with sizes N1,…,NL, respectively. Let be an FRV defined upon , whose expected value we will try to estimate on the basis of a stratified random sample.
In the stratified random sampling, a random sample (simple or with replacement) of size n is drawn from the population, where nh of the units in this sample belong to the hth
The relative gain in precision due to the stratification
In this section, we are going to define and examine a way to quantify the degree of adequacy of a variate for stratification. We will consider either the random sampling with replacement in each stratum, or the random sampling without replacement from a population whose size is very big in comparison with the sample size. With these assumptions, the assumptions of independence and identity of distribution (i.i.d.) will be valid either exactly or approximately, respectively.
Before getting
Illustrative example
Some of the conclusions in the study developed in this paper are illustrated by means of the following example: Example Consider the population of roads of a certain country in which the variable VISIBILITY (denoted by ) can be observed. Variable takes on the values PERFECT (), GOOD (), MEDIUM (), POOR (), and BAD (). Experts in the measurement of these values have described them in terms of the fuzzy sets (meaning fuzzy percentages) and based on S-curves, and triangular and
Concluding remarks
The study in this paper could be developed for a more general measure of deviation/error, like the one introduced recently by Körner and Näther [5].
Acknowledgements
The research in this paper has been supported in part by DGESYC Grant No. PB98-1534, DGXII Grant No. PL97-0421, CICYT Grant No. AGF98-1087 and FEDER Grant No. 1FD97-0042. Their financial support is gratefully acknowledged.
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