Test of one-sided hypotheses on the expected value

Test of one-sided hypotheses on the expected value of a fuzzy random variable? ¶ Montenegro, M., Colubi, A., Casals, M.R., and Gil-Alvarez, M.A. Dpto. de Estad¶³stica, I.O. y D.M., Universidad de Oviedo, 33071 Oviedo, Spain

Abstract. In this paper we present a procedure to test one-sided haypotheses about the population expected value of a fuzzy random variable. This procedure is based on a parameterized ranking function making the hypotheses being equivalent to classical ones for the population mean of a real-valued random variable.

1

Introduction

In previous papers (Montenegro et al., 2001, 2002) we have analyzed the problem of testing \two-sided" hypotheses on the population (fuzzy) expected value of a fuzzy random variable. The techniques to test such a type of hypotheses have been based on an operational generalized metric on the space of fuzzy numbers with compact support. However, these techniques cannot be applied to test one-sided hypotheses on this expected value. In fact, one-sided hypotheses do not make a wellde¯ned sense in case of fuzzy random variables since to rank fuzzy numbers we have to specify an ordering/preordering among them. For this purpose we can consider a suitable ranking function (like the parameterized one introduced by Campos and Gonz¶ alez, 1989). L¶ opez-D¶³az and Gil (1998) have proved this function is easy to compute, and when it is applied to the expected value of a fuzzy random variable, we obtain the classical expected value of a real-valued random variable. On the basis of the last result, we can reduce the problem of testing one-sided hypotheses on the population mean of a fuzzy random variable to the problem of testing the mean of a real-valued random variable. In this paper we ¯rst present some possible procedures to test the onesided hypotheses on the population mean of a fuzzy random variable. We illustrate later one of these procedures with an example. Finally, we will make some remarks to compare the approach in this paper with the one in previous ones (Montenegro et al., 2001, 2002). ?

The research in this paper has been partially supported by MCYT Grants DGE-PB98-1534 and BFM2001-3494. Their ¯nancial support is gratefully acknowledged.

2

Preliminaries

Let Kc (IR) denote the class of nonempty compact intervals. Let Fc (IR) be © e: the space of fuzzy numbers with compact support, that is, Fc (IR) = A ª e e e IR ! [0; 1] j A® 2 Kc (IR) for all ® 2 [0; 1]g, where A® = fx 2 IR j A(x) ¸ ®g e0 = clfx 2 IR j A(x) e for ® 2 (0; 1] and A > 0g. Given a probability space (­; A; P ), a mapping X : ­ ! Fc (IR) is said to be a fuzzy random variable (FRV for short) associated with this space in Puri and sense (1986) i® the ®-level mapping de¯ned so that ¡ Ralescu's ¢ X® (!) = X (!) ® for all ! 2 ­, is a random compact convex set whatever ® 2 [0; 1] may be. A fuzzy random variable X : ­ ! Fc (IR) is said to be integrably bounded if, and only if, kX0 k 2 L1 (­; A; P ) (with kX0 k(¢) = supx2X0 (¢) jxj). The exe ) such pected value of an integrably bounded FRV X is the fuzzy number E(X ¡ ¢ e ) = Aumann integral (1965) of X® (and, because of the convexity that E(X ® ¡ ¢ £ ¤ e ) = E(inf X® ); E(sup X® ) ) of values of X , this is equivalent to say that E(X ® for all ® 2 [0; 1]). eB e 2 Fc (IR) In accordance with Campos and Gonz¶ alez (1989), given A; e is said to be greater than or equal to B e in the ¸-average and ¸ 2 [0; 1], A e if, and only if, V ¸ (A) e º¸ B) e ¸ V ¸ (B), e where sense (and it is denoted by A ¡ ¢ R ¸ e e e e e º¸ A, e e V (A) = [0;1] ¸ sup A® +(1¡¸) inf A® d®. In case A º¸ B but not B e e ¸ B. we will write A The parameterized ranking function V ¸ leads to a reasonable ranking and it is quite convenient for computational purposes. In particular, when we deal with the expected value of a FRV, we obtain (see L¶ opez-D¶³az and Gil, 1998) e ) exists we have that that whenever E(X ¡ ¢ e )) = E V ¸ ± X : V ¸ (E(X

3

Stating the problem of testing one-sided hypotheses on the expected value of a FRV

Given a random sample of n independent observations, X1 ; : : : ; Xn , from X , the aim of this paper is testing (for a ¯xed subjective choice of the parameter e versus the alternative e ) º¸ A ¸) the null \one-sided" hypothesis H0 : E(X e e 2 Fc (IR). e \one-sided" hypothesis H1 : A ¸ E(X ) for a given A On the basis of the de¯nition of the ranking relation and the result by L¶ opez-D¶³az and Gil, the preceding¡ problem ¢ is equivalent to that of testing e versus the one-sided the one-sided hypothesis H0 : E V ¸ ± X ¸ V ¸ (A) ¡ ¸ ¢ ¸ e hypothesis H1 : E V ± X < V (A); which is a classical problem of testing on the mean of the real-valued random variable V ¸ ± X based on the random sample V ¸ ± X1 ; : : : ; V ¸ ± Xn .

4

Solving the problem of testing one-sided hypotheses on the expected value of a FRV

The problem above can be solved by using classical procedures. In this respect, we can ¯nd in the Classical Inference literature di®erent methods. The methods based on the normality of the involved FRVs are now applicable. Thus, if X is a normal FRV ¡with variance ¾ 2 in Puri and Ralescu's ¢ e sense (1985), that is X® (!) = X(!)© E(X ) ® for all ! 2 ­; ® 2 [0; 1], where X is a real-valued random variable having normal distribution N (0; ¾ 2 ), then V ¸ ± X is a real-valued random variable having normal distribution e )); ¾ 2 ). Consequently, whatever the speci¯ed ¸ 2 [0; 1] and A e2 N (V ¸ (E(X Fc (IR) maybe, we have that Theorem 1. Given a random sample of n independent observations, X1 ; : : : ; Xn , from a fuzzy random variable X having normal distribution with variance e ) º¸ ¾ 2 , to test at the signi¯cance level ® 2 [0; 1] the null hypothesis H0 : E(X e against the alternative H1 : A e ), the hypothesis H0 should be e ¸ E(X A rejected whenever e V ¸ ± X ¡ V ¸ (A) > tn¡1;® ; i2 Pn h ¸ ¸ =n(n ¡ 1) i=1 V ± Xi ¡ V ± X

r

where tn¡1;® is the 100(1 ¡ ®) fractile of Student's t-distribution with n ¡ 1 degrees of freedom, and V ¸ ± X denotes the sample fuzzy Pn mean of the n associated real-valued random variables (i.e., V ¸ ± X = i=1 V ¸ ± Xi =n). However, the assumption of normality for X is not too realistic. Only a few practical situations could be properly modeled by this type of FRVs. In fact, real-life FRVs are commonly simple fuzzy random variables (that is, they take on a ¯nite number of di®erent values), whence the associated real-valued random variable V ¸ ± X will be also simple. To test the considered hypotheses for this type of variables we can make use, for instance, of asymptotic techniques. Thus, consider a probability space (­; A; P ). Let X be a fuzzy random variable associated with it, so that on ­ the fuzzy random variable takes on r di®erent values, x e1 ; : : : ; x er . For each n 2 N , consider n independent fuzzy random variables having identical distribution that X on ­. Let fn = (f1 ; : : : ; fr¡1 ) 2 [0; 1]r¡1 , with fl = relative frequency of x el (l 2 f1; : : : ; r ¡ 1g) in the performance of the n fuzzy random variables. Let V ¸ ± X denote the sample fuzzy mean of the n associated real-valued random variables. Then, Theorem 2. Given a random sample of n independent observations, X1 ; : : : ; Xn , from a simple fuzzy random variable X , to test at the signi¯cance level e against the alternative e ) º¸ A ® 2 [0; 1] the null hypothesis H0 : E(X

e ), the hypothesis H0 should be (asymptotically) rejected whene ¸ E(X H1 : A h i e > °® ; where °® is the 100(1 ¡ ®) fractile of the ever 2n V ¸ ± X ¡ V ¸ (A) b 1 Â2 + : : : + ¸ b k Â2 , linear combination of chi-square independent variables ¸ 1;1

1;k

b1 ; : : : ; ¸ bk (k ∙ r ¡ 1) being the nonnull eigenvalues of the matrix with ³h ¸ i´ ³h i´ e B; where H V ¸ ± X ¡ V ¸ (A) e B t H V ¸ ± X ¡ V ¸ (A) is the Hessian matrix 0 h h i i1 e e @ 2 V ¸ ± X ¡ V ¸ (A) @ 2 V ¸ ± X ¡ V ¸ (A) C B ¢¢¢ C B @fn1 @fn1 @fn1 @fn(r¡1) C B C B .. .. .. C; B . . . B h h i iC C B B @ 2 V ¸ ± X ¡ V ¸ (A) e e C @ 2 V ¸ ± X ¡ V ¸ (A) A @ ¢¢¢ @fn(r¡1) @fn1 @fn(r¡1) @fn(r¡1)

³ ´¡1 and B is an (r ¡ 1) £ (r ¡ 1) matrix such that B t B = IXF (fn ) , where ¡ F ¢¡1 £ IX (f¤n ) is the inverse of the sample Fisher information matrix fnl (±lm ¡ fnm ) lm .

5

Illustrative example

The conclusions in Theorem 2 are now illustrated by means of a real-life example, in which data were supplied by members of the Departamento de Medio Ambiente of the Consejer¶³a de Agricultura in the Principado de Asturias in Spain. Example. Consider the population of days of a given year, and consider a random sample of 50 days in which visibility (variable X ) has been observed. x2 ), MEDIUM Variable X takes on the values PERFECT (~ x1 ), GOOD (~ (~ x3 ), POOR (~ x4 ), and BAD (~ x5 ). Experts in the measurement of these values have described them in terms of the fuzzy sets (meaning fuzzy percentages) and based on S- and ¦-curves, and triangular and trapezoidal fuzzy numbers, whose support is strictly contained in [0; 100] as follows (see Figure 1): x~1 = Tra(90; 95; 100; 100); x~2 = Tri(70; 90; 100); 8 S(40; 50) > > < 1 x ~3 = 1 ¡ S(70; 80) > > : 0

in [40; 50] in [50; 70] in [70; 80] otherwise,

8 S(20; 30) > > < 1 x ~4 = 1 ¡ S(40; 50) > > : 0

in [20; 30] in [30; 40] in [40; 50] otherwise,

x~5 = S(0; 20):

1

: : PERFECT ---GOOD

= =

= MEDIUM ¢ ¢ ¢¢ = POOR = BAD

20

40

60

80

100

Fig. 1. Values of the variable visibility

For the considered sample the observed fuzzy data have been collected in Table 1. values of X x ~1 x ~2 x ~3 x ~4 x ~5 absolute frequencies 4 21 12 8 2 Table 1. Data of variable visibility from days in the sample

e against the alternative H1 : e ) º:5 U To test the null hypothesis H0 : E(X e e e U Â:5 E(X ), where U ) denotes the value RATHER GOOD ON THE AVERAGE, which can be assumed to be modeled by the ¦-curve ¦(60; 70; 80; 90), on the basis of Theorem 2 the hypotesis H0 should be rejected at the significance level ® = :05 (actually, the p-value of the test is given by .0399).

6

Additional and concluding remarks

An interesting open problem in connection with the study in this paper is that of discussing the e®ects of choosing the value of parameter ¸ on the power of tests in Theorems 1 and 2.

In this respect, we have developed a broad introductory analysis. In this way, it should be pointed out that although the test in Montenegro et al. (2001, 2002) cannot be applied to deal with one-sided hypotheses, the ideas in this paper can be used also to test two-sided hypotheses. For purposes of comparing these ideas with those in the previous papers, we have performed some simulation studies allowing us to conclude that for a proper choice of ¸ (the choice depending on the shape of variable values), the power function of the asymptotic test by Montenegro et al. (2001, 2002) could be slightly improved by employing the test based on V ¸ . In this way, Example I. Consider a FRV X taking on 5 di®erent values in a population. Assume these values are asymmetric triangular ones, whose centers and widths have been obtained by a randomization proccess, and the 5 values are e the population expected value of X . equally likely. Let U 10000 samples of size 300 have been simulated, and 20 di®erent null hye © :02(i ¡ 1) e ) = U ei = U pothesis have been considered, namely, [i]: E(X (i = 1; : : : ; 20). Test 1 denotes the test in Montenegro et al. (2001, 2002) (with the metric using Lebesgue measure on [0; 1] for both, the ®-levels and the convex linear combinations). Test 2 involves the statistic in Theorem 2 applied to test the null hye )) = V0:1 (U ei ), Test 3 involves the same statistic applied to pothesis V0:1 (E(X e )) = V0:5 (U ei ), and Test 4 involves the same test the null hypothesis V0:5 (E(X e )) = V0:9 (U ei ). statistic applied to test the null hypothesis V0:9 (E(X The following table gathers the percentage of rejections at level ® = :05 (i.e., when the real percentage of rejections for the null hypothesis being true equals 5%). Null Test Test Test Test

hyp. 1 2 3 4

[1] 5.2 5.2 5.2 5.3

[2] 6.6 6.3 6.6 6.8

[3] 9.2 8.9 9.2 9.2

[4] 13.2 12.8 13.2 13.3

[5] 18.6 17.7 18.6 19.1

[6] 26.2 25.3 26.2 27.1

[7] 33.5 32.1 33.6 34.5

[8] 44.7 43 44.7 45.7

[9] 53.3 51.4 53.3 53.9

[10] 62.7 60.8 62.7 63.5

[11] 71.6 69.6 71.6 72.5

[12] 78.7 77.0 78.7 79.3

[13] 85.0 83.6 85.0 85.3

[14] 89.7 88.7 89.8 90.3

[15] 93.1 92.7 93.1 93.3

[16] 96.4 95.7 96.4 96.7

[17] 97.6 97.1 97.6 97.7

[18] 98.6 98.2 98.5 98.6

[19] 99.3 99.2 99.3 99.4

[20] 99.6 99.5 99.6 99.6

Example II. Consider a FRV Y taking on 10 di®erent values in a population. Assume these values are S-, Z- and ¦-curves, which have been obtained by f the a randomization proccess, and the 10 values are equally likely. Let W population expected value of Y. 10000 samples of size 500 have been simulated, and 20 di®erent null hyf © :01(i ¡ 1) e fi = W pothesis have been considered, namely, [i]: E(Y) = W (i = 1; : : : ; 20). Tests 1 to 4 have meanings similar to those in Example I. The following table gathers the percentage of rejections at level ® = :05 (i.e., when the real percentage of rejections for the null hypothesis being true equals 5%).

Null Test Test Test Test

hyp. 1 2 3 4

[1] 5.5 5.1 4.8 5.0

[2] 5.8 5.9 6.5 9.3

[3] 7.9 6.9 10.5 24.2

[4] 12.0 9.4 17.3 46.9

[5] 17.7 12.5 26.2 69.4

[6] 25.3 16.5 37.2 86.6

[7] 35.4 21.4 50.4 95.4

[8] 47.2 28.1 62.9 98.9

[9] 58.7 33.5 73.3 99.7

[10] 70.0 39.8 82.8 100

[11] 80.1 46.9 89.3 100

[12] 88.4 55.4 94.2 100

[13] 93.6 60.9 96.9 100

[14] 97.0 68.4 99.4 100

[15] 98.8 74.5 99.8 100

[16] 99.5 80.5 99.9 100

[17] 99.9 84.5 100 100

[18] 100 89.0 100 100

[19] 100 92.1 100 100

[20] 100 94.5 100 100

On the basis of these simulations, we realise there are not general conclusions to get. Thus, in Example I Tests 1 and 3 show a similar behavior, whereas Tests 2 and 4 are worse on the average. On the other hand, in Example II Test 4 is clearly the most powerful one. Figure 2 includes the graphical representation of the power function of Tests 1 (continuous line) and 3 in Example I, whereas Figure 3 includes the graphical representation of the power function of Tests 1 (continuous line) and 4 in Example II.

Fig. 2. Figure 2

Fig. 3. Figure 3

Anyway, it should be emphasized that Tests 2 to 3 could be valuable to e ) = A e in case of rejecting it, but in case of test the null hypothesis E(X e is not equivalent to E(X e ) = A, e whence e ) »¸ A acceptance we know that E(X the improvement in the power function would not mean a real advantage.

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