Chi-square goodness-of-fit test for vague data Przemys law Grzegorzewski Systems Research Institute Polish Academy of Sciences Newelska 6, 01-447 Warsaw, Poland and Faculty of Math. and Inform. Sci., Warsaw Institute of Technology Plac Politechniki 1 00-665 Warsaw, Poland pgrzeg@ibspan.waw.pl Abstract Testing goodness-of-fit plays a central role in data analysis. This problem seems to be much more complicated in the presence of vague data. In this paper we suggest how to generalize the well-known chisquare goodness-of-fit test for situations with fuzzy data. Keywords: Chi-square test, fuzzy sets, goodness-of-fit tests, vague data. 1 Introduction Most of statistical procedures are based on fairly specific assumptions regarding the underlying population distribution, like normality, exponentiality, etc. Therefore it might be desirable to check whether these assumptions are reasonable. It is a crucial point because if the assumptions regarding the distribution are not justified then the results of the statistical inference might be not appropriate. Statistical procedures for testing hypotheses about the underlying distribution are called goodness-of-fit tests. The above mentioned problems become much more complicated when the data are not necessarily precise. Many statistical tests for vague data have been proposed recently (for the review see, e.g., [7], [8]). But they were generally constructed either under specific assumptions on the underlying distribution ([4]) or as distribution-free tests (e.g. [3], [5], [6]). Anna Jȩdrej Faculty of Math. and Inform. Sci., Warsaw Institute of Technology Plac Politechniki 1 00-665 Warsaw, Poland In this paper we consider the problem of testing goodness-of-fit with vague data. We suggest how to modify a classical chi-square test for such data. Since the chi-square test is a general one and it could be applied both for the discrete and continuous distributions, it is a good candidate for this generalization. 2 Chi-square goodness-of-fit test Suppose that a random sample X 1,..., X n is drawn from a population with unknown cumulative distribution function F. We wish to test the null hypothesis H 0 : F (x) = F 0 (x) for all x, (1) that the population c.d.f. is F 0 (which is completely specified), against H 1 : F (x) F 0 (x) for some x. (2) To verify H 0 a goodness-of-fit test should be used. One of the most popular goodness-of-fit test is the chi-square test. To apply this test data must first be grouped into categories and then the observed frequencies for these categories are compared with the frequencies expected under the null hypothesis. In the case of a discrete distribution these categories appear in a natural way and are relevant to the distribution under study. When the distribution F 0 is continuous we have to arrange classes which are counterparts of above mentioned categories. Let us group our data into k mutually exclusive class intervals (ξ 0, ξ 1 ],..., (ξ k 1, ξ k ], where ξ 0 < ξ 1 <,..., < ξ k R {, + }.
Let n i be the observed frequency in the class interval Ξ i = (ξ i 1, ξ i ], where i = 1,..., k. Of course, we have n 1 +... + n k = n. If the null hypothesis holds then the probability p i that a given observation belongs to the class interval Ξ i = (ξ i 1, ξ i ] is given by p i = P (ξ i 1 < X ξ i H 0 ) = F 0 (ξ i ) F 0 (ξ i 1 ). (3) Thus np i is the expected frequency for the ith class under H 0. The idea of the test is to compare observed and expected frequencies and if the differences between n i and np i are large then it is an indication that H 0 is not correct. The test statistic is T = k (n i np i ) 2. (4) np i For large samples T is approximately chisquare distributed with k 1 degrees of freedom. We reject H 0 in favor of H 1 if T χ 2 1 α,k 1, where χ2 1 α,k 1 is the quantile of order 1 α from the chi-square distribution with k 1 degrees of freedom. 3 Vague data It may happen that a sample used for making decision consists of observations that are not necessarily crisp but may be vague as well. In order to describe the vagueness of data we use the notion of a fuzzy number, introduced by Dubois and Prade [2]. We say that a fuzzy subset A of the real line R, with the membership function µ A : R [0, 1], is a fuzzy number if and only if a) A is normal (i.e. there exists an element x 0 such that µ A (x 0 ) = 1), b) A is fuzzy convex (i.e. µ A (λx 1 + (1 λ)x 2 ) µ A (x 1 ) µ A (x 2 ), x 1, x 2 R, λ [0, 1]), c) µ A is upper semicontinuous, d) suppa is bounded, where suppa = cl({x R : µ A (x) > 0}) and cl is the closure operator. A useful notion for dealing with a fuzzy number is a set of its α cuts. The α cut of a fuzzy number A is a nonfuzzy set defined as A α = {x R : µ A (x) α}. (5) A family {A α : α (0, 1]} is a set representation of the fuzzy number A. According to the definition of a fuzzy number it is easily seen that every α cut of a fuzzy number is a closed interval. Hence we have A α = [A L α, A U α ], where A L α = inf{x R : µ A (x) α}, (6) A U α = sup{x R : µ A (x) α}. (7) A space of all fuzzy numbers will be denoted by FN(R). A notion of fuzzy random variable was introduced by Kwakernaak [12], [13]. Other definitions of fuzzy random variables are due to Kruse [10] or to Puri and Ralescu [14]. Our definition is similar to those of Kwakernaak and Kruse. Suppose that a random experiment is described as usual by a probability space (Ω, A, P ), where Ω is a set of all possible outcomes of the experiment, A is a σ algebra of subsets of Ω (the set of all possible events) and P is a probability measure. Then mapping X : Ω FN(R) is called a fuzzy random variable if {X(α, ω) : α (0, 1]} is a set representation of X(ω) for all ω Ω and for each α (0, 1] both Xα L = Xα L (ω) = inf X α (ω) and Xα U = Xα U (ω) = sup X α (ω) are usual real-valued random variables on (Ω, A, P ) (see [11]). A fuzzy random variable X is considered as a perception of an unknown usual random variable V : Ω R, called an original of X (if only vague data are available, it is of course impossible to show which of the possible originals is the true one). Similarly n dimensional fuzzy random sample X 1,..., X n may be treated as a fuzzy perception of the usual random sample V 1,..., V n (where V 1,..., V n are independent and identically distributed crisp random variables). For more information we refer the reader to [11]. A random variable is completely characterized by its probability distribution P θ. In statistical reasoning we assume that a probability
distribution under study belongs to a family of distributions P = {P θ : θ Θ}. Then very often we identify the distribution with its parameter θ and restrict statistical inference to that parameter. However, if we deal with a fuzzy random variable we cannot observe parameter θ directly but only its vague image. Using this reasoning together with Zadeh s extension principle Kruse and Meyer [11] introduced the notion of fuzzy parameter of fuzzy random variable θ which may be considered as a fuzzy perception of the unknown parameter θ. It is defined as a fuzzy subset of the parameter space Θ with membership function µ θ e : Θ [0, 1]. Of course, if our data are crisp, i.e. X = V, we get θ = θ. 4 Chi-square test for vague data Suppose µ X1,..., µ Xn denote membership functions of fuzzy numbers which are observations of a fuzzy random sample X 1,..., X n. Suppose our sample comes from the unknown distribution F, and our goal is to test the null hypothesis H 0 : F = F e θ, (8) against H 1 : F F e θ, (9) where the distribution Fθ e is completely specified by a fuzzy parameter θ described by its membership function µ θ e. In this section we propose how to apply the chi-square test for fuzzy data. As it was shown in Sec. 2, data firstly have to be grouped into class intervals. However, a fuzzy observation, contrary to the crisp one, may belong to more than one class. Therefore, a natural question immediately arises: How to compute observed frequencies for classes intervals using fuzzy data? Let Ξ 1,..., Ξ k, where Ξ i = (ξ i 1, ξ i ] and ξ 0 < ξ 1 <,..., < ξ k R {, + }, denote, as before, mutually exclusive class intervals. Let w(x j ) = µ Xj (x)dx (10) R denote the width of a fuzzy observation X j (see [1]). Moreover, let w(ξ i X j ) = ξ i ξ i 1 µ Xj (x)dx (11) denote the width of the intersection of the fuzzy observation X j with the class interval Ξ i = (ξ i 1, ξ i ]. Hence for each i = 1,..., k we can compute coefficients ñ i = µ Xj (x)dx n ξ i 1 µ Xj (x)dx j=1 ξ i R (12) which are counterparts of frequencies corresponding to classes Ξ i. One can check that ñ 1 +... + ñ k = n, similarly as the sum of frequencies is equal to the sample size. However, since the coefficients ñ i can assume arbitrary nonnegative value (not necessarily integer) we call them rather quotas than frequencies. As we remember, the main idea of the chisquare test is to compare the observed frequencies for class intervals with the frequencies expected under the null hypothesis. Thus, we have to solve another problem: How to determine the frequencies expected under the null hypothesis that the actual distribution is F Λ(θ)? Suppose fθ e denotes the density function corresponding to Fθ e, which may be considered as a fuzzy perception of the density f θ. In other words, a density f θ belongs to fθ e with the same grade µ θ e (θ) as θ belongs to the fuzzy parameter θ.for each i = 1,..., k we can determine following values π i = ξ i ξ i 1 Θ Let C be a constant such that C = µ e θ (θ)f θ(x)dθdx. (13) k π i. (14) Thus, if the null hypothesis holds, the probability that a fuzzy observation belongs to the
class interval Ξ i = (ξ i 1, ξ i ] is given by the following formula p i = π i C. (15) One can easily check that p i [0, 1] for each i = 1,..., k and k p i = 1. Hence n p i denotes the expected frequency for the ith class under H 0. So now we are able to define a test statistic for testing H 0 against H 1 with fuzzy data. It is given by T = k (ñ i n p i ) 2. (16) n p i One can see that it looks like (4). In fact we have the same structure and the only difference is that n i and p i in (16) are computed in a different way than in (4). Thus it is not surprising that for large samples - as simulations show - statistic (16) is approximately chi-square distributed with k 1 degrees of freedom. Therefore, we reject H 0 in favor of H 1 if T χ 2 1 α,k 1, where χ2 1 α,k 1 is the quantile of order 1 α from the chi-square distribution with k 1 degrees of freedom. We have examined the suggested chi-square goodness-of-fit test for fuzzy data using Monte-Carlo simulations (see [9]). Our simulation study showed that our test works properly but it is a little bit liberal. Till now, we have considered the chi-square goodness-of-fit test for a simple null hypothesis. However, the more typical situation in practise is that the null hypothesis is composite, i.e. it states the form of the distribution but not all the relevant parameters. If the null hypothesis is composite then we have to estimate the expected frequencies from the data (generally using the method of maximum likelihood). Then we can apply the same test statistic (4) but now its asymptotic distribution is the chi-square with k r 1 degrees of freedom, where r is the number of parameters which had to be estimated in order to estimate expected frequencies. Taking into account all these remarks our goodness-of-fit test for fuzzy data could be also applied for complex hypotheses. In such case the method of maximum likelihood and Zadeh s extension principle would be useful in estimating expected frequencies. Then the final decision would be based on the relationship between test statistic (16) and the appropriate quantile of the chi-square distribution with k r 1 degrees of freedom. 5 Conclusions In the present paper we have considered the problem of goodness-of-fit testing in fuzzy environment. We have proposed a generalization of the classical chi-square goodness-of-fit test for fuzzy data. Unfortunately, as in the classical case with crisp data, the chi-square test requires large samples. Since not always such big samples are available other goodnessof-fit techniques for vague data are still required. References [1] S. Chanas, On the interval approximation of a fuzzy number, Fuzzy Sets and Systems 122 (2001), 353 356. [2] D. Dubois, H. Prade, Operations on fuzzy numbers, Int. J. Syst. Sci. 9 (1978), 613 626. [3] P. Grzegorzewski, Statistical inference about the median from vague data, Control and Cybernetics 27 (1998), 447 464. [4] P. Grzegorzewski, Testing statistical hypotheses with vague data, Fuzzy Sets and Systems 112 (2000), 501 510. [5] P. Grzegorzewski, Distribution-free tests for vague data, In: Soft Methodology and Random Information Systems, Lopez-Diaz M., Gil M.A., Grzegorzewski P., Hryniewicz O., Lawry J. (Eds.), Springer, Heidelberg, 2004, pp. 495-502. [6] P. Grzegorzewski, Two-sample median test for vague data, In: Proceedings of the 4th Conference of the European Society for Fuzzy Logic and Technology -
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