Sequential Test of Fuzzy Hypotheses

In testing statistical hypotheses, as in other statistical problems, we may be confronted with fuzzy concepts. This paper deals with the problem of testing hypotheses, when the hypotheses are fuzzy and the data are crisp. We first give new definitions for notion of mass (density) probability function with fuzzy parameter, probability of type I and type II errors and then state and prove the sequential probability ratio test, on the basis of these new errors, for testing fuzzy hypotheses. Numerical examples are also provided to illustrate the approach.


Introduction
Statistical analysis, in traditional form, is based on crispness of data, random variable, point estimation, hypotheses, parameter and so on.As there are many different situations in which the above mentioned concepts are imprecise.On the other hand, the theory of fuzzy sets is a well known tool for formulation and analysis of imprecise and subjective concepts.Therefore the sequential probability ratio test with fuzzy hypotheses can be important.The problem of statistical inference in fuzzy environments are developed in different approaches.
Delgado et al. [1] consider the problem of fuzzy hypotheses testing with crisp data.Arnold [2,3] presents an approach to test fuzzily formulated hypotheses, in which he considered fuzzy constraints on the type I and II errors.Holena [4] considers a fuzzy generalization of a sophisticated approach to exploratory data analysis, the general unary hypotheses automaton.Holena [5] presents a principally different approach and motivates by the observational logic and its success in automated knowledge discovery.Neyman-pearson lemma for fuzzy hypotheses testing and Neyman-pearson lemma for fuzzy hypotheses testing with vague data is given by Taheri et al. and Torabi et al. [6,7].Filzmoser and Viertl [8] present an approach for statistical testing at the basis of fuzzy values by introducing the fuzzy p-value.Some methods of statistical inference with fuzzy data, are reviewed by Viertl [9].Buckley [10,11] studies the problems of statistical inference in fuzzy environment.Thompson and Geyer [12] proposed the Fuzzy p-values in latent variable problems.Taheri and Arefi [13] exhibit an approach for testing fuzzy hypotheses based on fuzzy test statistics.Parchami et al. [14] consider the problem of testing hypotheses, when the hypotheses are fuzzy and the data are crisp.they first introduce the notion of fuzzy p-value, by applying the extension principle and then present an approach for testing fuzzy hypotheses by comparing a fuzzy p-value and a fuzzy significance level, based on a comparison of two fuzzy sets.
In present work, we first define a new approach for obtaining the probability (density) function, when the random variable is crisp and the parameter of interest is imprecise (fuzzy).Also, the type I and type II errors are introduced based on fuzzy hypotheses.Then, the sequential probability ratio test (SPRT) is defined and extended based on such hypotheses.
We organize the matter in the following way: In section 2 we describe some basic concepts of fuzzy hypotheses, density (Mass) probability function with fuzzy parameter and necessary definitions.In section 3 we come up sequential probability ratio test based on fuzzy hypotheses.In section 4 the previous definitions and the sequential probability ratio test will be illustrated by examples.

Preliminaries
In this section we describe fuzzy hypotheses, density (Mass) probability function with fuzzy parameter and  necessary definitions.
Let be a probability space, a random variable (RV) , where X is the probability measure induced by with respect to P  and is called the probability density function of X with respect to  .In a statistical context, the measure  is usually a "counting measure" or a "Lebesgue measure", hence is , respectively.

Canonical Fuzzy Numbers
Let , and   is called a normal fuzzy set if there exists and its membership function be strictly increasing on the interval  and strictly decreasing on the interval  , then   is called a canonical fuzzy number (Klir and Yuan, [16]).
The fuzzy canonical numbers (such as triangular or trapezoidal fuzzy numbers) are very realistic in fuzzy set theory, so we use this numbers for our goal.

Fuzzy Hypotheses
We define some models, as fuzzy sets of real numbers, for modeling the extended versions of the simple, the one-sided, and the two-sided ordinary (crisp) hypotheses to the fuzzy ones.
Testing statistical hypothesis is a main branch of statistical inference.Typically, a statistical hypothesis is an assertion about the probability distribution of one or more random variable(s).Traditionally, all statisticians assume the hypothesis for which we wish provide a test are well-defined.This limitation, sometimes, force the statistician to make decision procedure in an unrealistic manner.This is because in realistic problems, we may come across non-precise (fuzzy) hypothesis.For example, suppose that  is the proportion of a population which have a disease.We take a random sample of elements and study the sample for having some idea about  .In crisp hypothesis testing, one uses the hypotheses of the form: , and so on.However, we would sometimes like to test more realistic hypotheses.In this example, more realistic expressions about  would be considered as: "small", "very small", "large", "approximately 0.2", "essentially larger" and so on.Therefore, more realistic formulation of the hypotheses might be 0 : H  is not small.We call such expressions as fuzzy hypotheses.
We define some models, as fuzzy sets of real numbers, for modeling the extended versions of the simple, the one-sided, and the two-sided crisp hypotheses to the fuzzy ones (Akbari and Rezaei,[17]).
Definition 2.1 Let 0  be a real number and known.
x R f x  0 be the "support" or "sample" space of X and We call the new density  f x    as the fuzzy probability density (mass) function (FPDF) of X (Akbari and Rezaei [18]).We note that, (substitute the summation by integral in discrete cases).
Let  : = , be a random sample, with observed we state the following definitions: and the probability of type II error of  

Sequential Probability Ratio Test
Consider testing a null fuzzy hypothesis against a alternative fuzzy hypothesis.In other words, suppose a sample can be drawn from one of two FPDFs and it is desired to test that the sample came from one distribution against the possibility that is came from the other.If denotes the random variables, we want to test 1 2 , , o test was of the following form: e simple likelihood-rati The sequential test that we propose to consider em L ploys the likelihood-ratios sequentially.Define , , .
Similarly, the acceptance region can be defined as =1 = .
, where Copyright © 2011 SciRes.OJS j k When we considered the simple likelihood-ratio test fo , , , < , = 1, 2,..., 1, , , , r fixed sample size n , we determined k so that the test would have preass ned size ig  .We know want to determine 0 k and 1 k so that the quential probability ratio test will have preassigned se  and  for its respective sizes of type I and type II e ors.Note that rr , where, as before, is a shortened notation for For fixed  and  , the above equations are two equations in t two unknown 0 k and 1 k .A solution of these two equations would give t e sequential probability ratio test having the desired preassigned error sizes , , X X  and consequently is itself a RV.Denote it by eally, we would like to know the distribution of N or at least the expectation of N .One way of assessing the performance of the sequential probability ratio test would be to evaluate the expected sample size that is required under each hypothesis.The following lemma, given without proof (Lehmann,[20]), state that the sequential probability ratio test with crisp hypotheses is an optimal test if performance is measured using expected sample size.We can similarly prove this lemma with fuzzy hypotheses based on introduced FDPF.The procedure used in e p rforming a se ntial que probability ratio test is to continue sampling as long as and stop sampling as soon as , an e va n by the following: continue sam ling as as ratio test, and let


, and stop sampling as soon as . As before, let N be a he sa he sequential probability If the sequential probability ratio test leads to rejection an o e , s equation (Casella and Berger, [19])

Numerical Examples
In this section, we illustrate the proposed approach for me distributions and use the ability of package "Maple ( Behboodian, [6]) Let so 6" [21] for this examples.
Exam le 4. 1 Taheri and p X be a continues r.v. with PDF we want to test We can interpret 0   and 1   as the value of " 1 3 near to " and " for .We can interpret the canonical parameters as having values that are "near to ".Let

 0 1 ,
and  .As might be anticipated, the actual determ 0 k and 1 k from above equations can be a major comput onal pr ect.We note that the sample s ination of ize of a sequential pr ati oj obability ratio test is a random variable.The procedure says to continue sampling until n  first falls outside the interval  k k .The actual sam e size then depend on which rved; it is a function of the random variable 1 2

1 k 1 k
has error sizes  and  ; then 0 probability ratio test having the desired preassigned error sizes  and  ; however, since it is difficult to to find the k and corresponding to such a sequential probability ratio test, instead one can use that sequential probability ratio test defined by 0   and   of before equation and be assured that the the sum f the error sizes o   and   is less than or equal to the sum of the desire rror siz d e es  and  .