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Authors requiring further information regarding Elsevier’s archiving and manuscript policies are encouraged to visit: http://www.elsevier.com/copyright http://www.elsevier.com/copyright Author's personal copy Bootstrap testing fuzzy hypotheses and observations on fuzzy statistic Mohammad Ghasem Akbari a,*, Abdolhamid Rezaei b a Department of Statistics, Faculty of Sciences, University of Birjand, South Khorasan, Iran b Department of Statistics, School of Mathematical Sciences, Ferdowsi University of Mashhad, Mashhad, Iran a r t i c l e i n f o Keywords: Fuzzy canonical number Yao–Wu signed distance Fuzzy data Fuzzy hypotheses Fuzzy confidence Testing hypotheses Bootstrap method a b s t r a c t The bootstrap is a simple and straightforward method for calculating approximated biases, standard deviations, confidence intervals, testing statistical hypotheses, and so forth, in almost any nonparametric estimation problem. A new approach for bootstrap testing fuzzy hypotheses based on fuzzy test statistic is introduced. In this paper we describe bootstrap method that is designed directly for hypothesis testing for fuzzy data based on Yao–Wu signed distance. � 2010 Elsevier Ltd. All rights reserved. 1. 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. The point estimation ap- proaches frequently are used in statistical inference. 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 testing hypotheses with fuzzy data can be important. The problem of testing statistical hypotheses when the hypotheses are crisp or fuzzy have been studied by a few authors. Arnold (1996, 1998) presented an approach to test fuzzily for- mulated hypotheses, in which he considered fuzzy constraints on the type I and II errors. Taheri and Behboodian (1991) state and prove a Neyman–Pearson Lemma for testing fuzzy hypotheses. Their approach has been extended by Torabi, Behboodian, and Taheri (2006) for the cases when the data are fuzzy, too. For some other recent works in testing hypothesis using fuzzy approaches, see Buckley (2005, 2006), Desimpelaere and Marchant (2007), Fil- zmoser and Viertl (2004), Hyniewicz (2006), Thompson and Geyer (2007), Viertl (2006). The bootstrap using fuzzy data, is developed in different approaches. Montenegro, Colubi, Casals, and Gil (2004) have presented asymptotic one-sample procedure. Korner’s asymptotic develop- ment (2000) concerns general fuzzy random variables (taking on way-either finite or infinite-number of values in the space of com- pact convex fuzzy sets of a finite-dimensional Euclidean space). Gonzalez-Rodriguez, Montenegro, Colubi, and Gil (2006) have shown that the one-sample method of testing the mean of a fuzzy random variable can be extended to general ones (more precisely, to those whose range is not necessarily finite and whose values are fuzzy subsets of finite-dimensional Euclidean space). In this paper we construct a new method for bootstrap testing hypotheses in fuzzy environment which is completely different from those mentioned above. For this purpose we organize the matter in the following way: in Section 2 we describe some basic concepts of canonical fuzzy numbers, Yao and Wu (2000) signed distance and fuzzy hypothe- ses. In Section 3 we come up bootstrap testing fuzzy hypotheses based on fuzzy test statistic for mean based on Yao–Wu signed dis- tance. Section 4 provide a Bootstrap testing fuzzy hypotheses based on fuzzy test statistic for variance based on Yao-Wu signed distance. A brief conclusion is provided in Section 5. 2. Preliminaries In this section we study canonical fuzzy numbers, Yao–Wu signed distance and fuzzy hypotheses. 2.1. Canonical fuzzy numbers Let X be the universal space, then a fuzzy subset ~x of X is defined by its membership function l~x : X !½0; 1�. We denote by ~xa ¼ fx : l~xðxÞ P ag the a-cut set of ~x and ~x0 is the closure of the set fx : l~xðxÞ > 0g, and (1) ~x is called normal fuzzy set if there exist x 2 X such that l~xðxÞ¼ 1; (2) ~x is called convex fuzzy set if l~xðkx þð1 � kÞyÞ P min ðl~xðxÞ; l~xðyÞÞ for all k 2 ½0; 1�; 0957-4174/$ - see front matter � 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.eswa.2010.02.030 * Corresponding author. Tel./fax: +98 511 8828605. E-mail address: g_z_akbari@yahoo.com (M.G. Akbari). Expert Systems with Applications 37 (2010) 5782–5787 Contents lists available at ScienceDirect Expert Systems with Applications j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / e s w a Author's personal copy (3) the fuzzy set ~x is called a fuzzy number if ~x is normal convex fuzzy set and its a-cut sets are bounded 8a – 0; (4) ~x is called a closed fuzzy number if ~x is fuzzy number and its membership function l~x is upper semicontinuous; (5) ~x is called a bounded fuzzy number if ~x is a fuzzy number and its membership function l~x has compact support. If ~x is a closed and bounded fuzzy number with xLa ¼ inffx : x 2 ~xag and xUa ¼ supfx : x 2 ~xag and its membership function is strictly increasing on the interval xLa; x L 1 � � and strictly decreasing on the interval xU1 ; x U a � � for any a 2 ½0; 1�, then ~x is called canonical fuzzy number. Let ‘‘�” be a binary operation � or � between two canonical fuzzy numbers ~a and ~b. The membership function of ~a � ~b is de- fined by l~a�~bðzÞ¼ sup x�y¼z minfl~aðxÞ; l~bðyÞg 8z 2R for �¼� or � and �¼þ or -. In the following, let �int be a binary operation �int or �int between two closed intervals ~aa ¼ aLa; aUa � � and ~ba ¼ b L a; b U a h i . Then ~aa�int ~ba is defined by ~aa�int ~ba ¼fz 2R : z ¼ x � y; x 2 ~aa; y 2 ~bag: If ~a and ~b be two closed fuzzy numbers. Then ~a � ~b and ~a � ~b are also closed fuzzy numbers. Furthermore, we have ð~a � ~bÞa ¼ ~aa�int ~ba ¼ a L a þ b L a; a U a þ b U a h i ; ð~a � ~bÞa ¼ ~aa�int ~ba ¼ a L a � b U a ; a U a � b L a h i : 2.2. Yao–Wu signed distance Now we define a signed distance between fuzzy numbers which is used later. Several ranking methods have been proposed so far, by Cheng (1998), Modarres and Sadi-Nezhad (2001) and Nojavan and Gha- zanfari (2006). In this paper we use another ranking system for canonical fuzzy numbers which is very realistic and is defined by Yao and Wu as the following: Definition 2.1. For each a; b 2R, define the signed distance d� of a and b by d�ða; bÞ¼ a � b. Thus, we have the following way to define the rank of any two numbers on R. For each a; b 2R, d�ða; bÞ > 0 () d�ða; 0Þ > d�ðb; 0Þ() a > b; d�ða; bÞ < 0 () d�ða; 0Þ < d�ðb; 0Þ() a < b; d�ða; bÞ¼ 0 () d�ða; 0Þ¼ d�ðb; 0Þ() a ¼ b: Definition 2.2. For each ~a; ~b 2FðRÞ, define the signed distance of ~a and ~b as follows: dð~a; ~bÞ¼ Z 1 0 Mað~aÞ� Mað~bÞ � � da ¼ Z 1 0 d� Mað~aÞ; Mað~bÞ � � da; where Mað~aÞ and Mað~bÞ are equal to aLaþa U a 2 and bLaþb U a 2 , respectively, furthermore dð~a; ~bÞ means the distance of ~a to ~b. Definition 2.3 (Yao and Wu, 2000). For each ~a; ~b 2FðRÞ, define the ranking ; and � of ~a and ~b by dð~a; ~bÞ > 0 () dð~a; 0Þ > dð~b; 0Þ() ~a ~b; dð~a; ~bÞ < 0 () dð~a; 0Þ < dð~b; 0Þ() ~a ~b; dð~a; ~bÞ¼ 0 () dð~a; 0Þ¼ dð~b; 0Þ() ~a � ~b: 2.3. 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 to provide a test are well-defined. This limitation, some- times, forces 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 h is the proportion of a population which has a disease. We take a random sample of elements and study the sample for having some idea about h. In crisp hypothesis testing, one uses the hypotheses of the form: H0 : h ¼ 0:2 versus H1 : h – 0:2 or H0 : h 6 0:2 versus H0 : h > 0:2, and so on. However, we would sometimes like to test more realistic hypotheses. In this example, more realistic expressions about h would be considered as: small, very small, large, approximately 0.2, and so on. Therefore, more realistic formulation of the hypotheses might be H0 : h is small, versus H1 : h is not small. We call such expressions as fuzzy hypotheses. We define some models, as fuzzy sets of real numbers, for mod- eling the extended versions of the simple, the one-sided, and the two-sided crisp hypotheses to the fuzzy ones. Definition 2.4. Let h0 be a real number and known. (i) Any hypothesis of the form (H : h is approximately h0) is called to be a fuzzy simple hypothesis. (ii) Any hypothesis of the form (H : h is not approximately h0 ) is called to be a fuzzy two-sided hypothesis. (iii) Any hypothesis of the form (H : h is essentially smaller than h0 ) is called to be a fuzzy left one-sided hypothesis. (iv) Any hypothesis of the form (H1 : h is essentially larger than h0 ) is called to be a fuzzy right one-sided hypothesis. We denote the above definitions by (w) H0 : h is approximately h0 H1 : h is not approximately h0 � or H0 : h is eH 0 H1 : h is eH 1; � (ww) H0 : h is approximately h0 H1 : h is certainly larger than h0 � or H0 : h is eH 0L H1 : h is eH 1; � (www) H0 : h is approximately h0 H1 : h is certainly smaller than h0 � or H0 : h is eH 0R H1 : h is eH 1: � The above areas are shown in Figs. 1–3. 3. Bootstrap testing fuzzy hypotheses based on fuzzy test statistic for mean Suppose that we have canonical fuzzy random sample ~x ¼ð~x1; ~x2; . . . ; ~xnÞ. 3.1. Testing fuzzy simple hypothesis fuzzy against the fuzzy two-sided hypothesis We want to test the fuzzy null hypotheses eH 0 : the mean of observations ðhÞ is approximately h0 versuseH 1 : the mean of observations ðhÞ is not approximately h0 . M.G. Akbari, A. Rezaei / Expert Systems with Applications 37 (2010) 5782–5787 5783 Author's personal copy We generate B bootstrap fuzzy random sample ~x� 1 ; ~x� 2 ; . . . ; ~x� B (i.e., each ~x� b is a fuzzy sample of size n drawn randomly and replacement from ~x). We need a distribution that estimates the population of treat- ment times under H0 . Note first that the empirical distribution (i.e., putting probability 1n on each member of ~x) is not an approx- imate estimate for distribution because it does not obey H0. Some- how we need to obtain an estimate of the population that has mean ~h0. A simple way is to translate the empirical distribution so that it has the desired mean. In other words, we use as our esti- mated null distribution the empirical distribution on the values ~xci ¼ ~xi � ex �fh0; i ¼ 1; 2; . . . ; n; because 1n� n i¼1~xci ¼ fh0 . We compute t ~x� b c � � ¼ d ex�bc ;eh0� �cSe�bðexcÞ b ¼ 1; 2; . . . ; B; where (1) d is Yao–Wu signed distance. (2) ex�bc ¼ 1n�ni¼1~x�bci b ¼ 1; 2; . . . ; B. (3) cSe�bðexcÞ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1nðn�1ÞPni¼1 d2 ~x�bci ; ex�bc� � r b ¼ 1; 2; . . . ; B. The cth percentile of tð~x�bc Þ is estimated by the value t̂c such that # t ~x� b c � � 6 t̂c n o B ¼ c: Finally, the fuzzy bootstrap confidence interval at the level 1 � 2c using fuzzy data is ePa ¼ 1n X n i¼1 xi � t̂c sð~xÞffiffiffi n p ; 1 n Xn i¼1 xi � t̂1�c sð~xÞffiffiffi n p " # ; ( xi 2 ~xia; i ¼ 1; 2; . . . ; n ) ; where s2ð~xÞ ¼ 1 n�1 Pn i¼1 d 2ð~xi; ~�xÞ. If B � c is not an integer, the following procedure can be used. Assuming c 6 0:5, let k ¼ ½ðB þ 1Þc�, the largest integer 6 ðB þ 1Þc. Then we define the empirical c and 1 � c quantizes by the kth larg- est and ðB þ 1 � kÞth largest values of tð~x�bc Þ, respectively. Example 3.1. Suppose that we have taken a fuzzy random sample of size n ¼ 9 from a population and we observed the following triangular fuzzy data: and using of the ability of package ‘‘it MINITAB 13”, we show the percentiles and histogram of t ~x� b c � � in the following using 10,000 bootstrap samples. If B ¼ 10; 000, the estimate of the 5% point is the 500th largest value of the t ~x� b c � � s and the estimate of the 95% point is the 9500th largest value of the t ~x� b c � � s. The a-cuts of fuzzy bootstrap confidence interval (c ¼ 0:05 or 90%) using fuzzy data is ePa ¼ 19 X 9 i¼1 xi �1:781 20:36ffiffiffi 9 p ; 1 9 X9 i¼1 xi þ2:088 20:36ffiffiffi 9 p " # ; ( xi 2 ~xia; i ¼ 1;2; . . . ;9 ) ¼½46:153þ2:22a;76:167�1:556a�: We show the following the distribution of tð~x�bc Þ computed using 10000 bootstrap samples (Fig. 4). We obtain the a- cuts of the so-called fuzzy test statistics -10 0 10 0 100 200 300 400 500 600 700 800 900 1000 Fr eq ue nc y Fig. 4. bootstrap distribution of t ~x� b c � � . Fig. 2. The fuzzy hypotheses eH 0L versus eH 1 . Fig. 3. The fuzzy hypotheses eH 0R versus eH 1 . Fig. 1. The fuzzy hypotheses eH 0 versus eH 1 . 5784 M.G. Akbari, A. Rezaei / Expert Systems with Applications 37 (2010) 5782–5787 Author's personal copy eZ a ¼ ePa � ~h0asð~xÞffiffi n p ; and use the fuzzy test statistics to provide an approach for testing above fuzzy hypotheses, based on the following assumptions (see Fig. 5). ASSUMPTIONS 1. CT be the total area under eZ . 2. C1 and C2 be the areas according to Fig. 5. 3. CR ¼ C1 þ C2 DECISION RULE If CRCT 6 2c, then we accept H0 . If CRCT P 2c, then we reject H0 . Example 3.2. Consider Table 1. Now suppose that we want to test the following fuzzy hypotheses eH 0 : h isð57; 60; 62ÞeH 1 : h is notð57; 60; 62Þ: ( Here, eH 0 suggests that h is approximation 60, and eH 1 suggests that h is away from 60. We have eZ a ¼ ½�2:339 þ 0:622a; 2:899 � 0:671a�; CR ¼ C1þ C2 ¼ 0:061 þ 0:18 ¼ 0:241; CT ¼ 3:39. since CRCT ¼ 0:071 6 0:1, thus accept H0 (Table 2). 3.2. Testing fuzzy simple hypothesis fuzzy against the fuzzy left one- sided hypothesis We want to test the fuzzy null hypotheses H0 : h is approximately h0; H1 : h is certainly smaller than h0: � We obtain the a-cuts of the so-called fuzzy test statistics eZ a ¼ ePa �eh0as ðexÞffiffi n p ; and use the fuzzy test statistics to provide an approach for testing above fuzzy hypotheses, based on the following assumptions (see Fig. 6) ASSUMPTIONS 1. CT be the total area under eZ . 2. C1 be the area according to Fig. 6. 3. CR ¼ C1 DECISION RULE If CRCT 6 c, then we accept H0. If CRCT P c, then we reject H0. Example 3.3. Consider Table 1. Now suppose that we want to test the following fuzzy hypotheses eH 0 : h is approximatelyð:; 60; 62Þ;eH 1 : h is certainly smaller thanð:; 60; 62Þ: ( We have eZ a ¼ ½�2:339 þ 0:622a; 2:382 � 0:23a�; CR ¼ C1 ¼ 0:061; CT ¼ 2:79. since CRCT ¼ 0:022 6 0:05, thus accept H0 . 3.3. Testing fuzzy simple hypothesis fuzzy against the fuzzy right one- sided hypothesis We want to test the fuzzy null hypotheses H0 : h is approximately h0; H1 : h is certainly larger than h0: � We obtain the a-cuts of the so-called fuzzy test statistics eZ a ¼ ePa �eh0asð~xÞffiffi n p ; and use the fuzzy test statistics to provide an approach for testing above fuzzy hypotheses, based on the following assumptions (see Fig. 7) Fig. 5. eZ in testing fuzzy simple hypothesis versus fuzzy two-sided hypothesis. Table 1 Fuzzy random sample of size n ¼ 9 from a population. N Observation N Observation N Observation 1 (32, 35, 40) 4 (60, 63, 63) 7 (70, 73, 75) 2 (80, 82, 82) 5 (41, 45, 47) 8 (54, 56, 59) 3 (60, 60, 60) 6 (93, 95, 96) 9 (34, 35, 36) Table 2 Percentiles of the t distribution with 8 degree of freedom, the Nð0; 1Þ distribution and the bootstrap distribution of tð~x�bc Þ. Percentile 5% 10% 90% 95% t8 �1.86 �1.40 1.40 1.86 Normal �1.65 �1.28 1.28 1.65 Bootstrap �2.088 �1.537 1.319 1.781 Fig. 6. eZ in testing fuzzy simple hypothesis versus fuzzy left one-sided hypothesis. M.G. Akbari, A. Rezaei / Expert Systems with Applications 37 (2010) 5782–5787 5785 Author's personal copy ASSUMPTIONS 1. CT be the total area under eZ . 2. C2 be the area according to Fig. 7. 3. CR ¼ C2 DECISION RULE If CRCT 6 c, then we accept H0. If CRCT P c, then we reject H0. Example 3.4. Consider Table 1. Now suppose that we want to test the following fuzzy hypotheses eH 0 : h is approximatelyð28; 30; :Þ;eH 1 : h is certainly larger thanð28; 30; :Þ: ( We have eZ a ¼ ½2:38 þ 0:327a; 7:097 � 0:524a�; CR ¼ C2 ¼ 2:772; CT ¼ 16:9. since CRCT ¼ 0:164 P 0:05, thus reject H0 . 4. Bootstrap testing fuzzy hypotheses based on fuzzy test statistic for variance Suppose that we have canonical fuzzy random samples ~x ¼ð~x1; ~x2; . . . ; ~xnÞ. We generate B bootstrap fuzzy random sample ~x� 1 ; ~x� 2 ; . . . ; ~x� B and for each we compute (Fig. 8) v2� b ¼ ðn � 1Þs2�bð~xÞ s2ð~xÞ b ¼ 1; 2; . . . ; B; where (1) s2� b ð~xÞ ¼ 1 n�1 Pn i¼1d 2ð~x�bi ; ex�Þ. (2) d is Yao–Wu signed distance. (3) ex� ¼ 1n�ni¼1~x�bi . (4) s2 ðexÞ ¼ 1n�1 Pni¼1 d2ð~xi; exÞ. The cth percentile of v2�b is estimated by the value t̂c such that #fv2�b 6 t̂cg B ¼ c: Finally, the a-cuts of bootstrap confidence interval using fuzzy data is eP�a ¼ Pn i¼1ðxi � �xÞ 2 t̂1�c ; Pn i¼1ðxi � �xÞ 2 t̂c " # : xi 2 ~xia; i ¼ 1; 2; . . . ; n ( ) ; whenever its membership function is given by leP�ðyÞ¼ sup06a61 aIeP�aðyÞ: Example 4.1. Suppose that we have taken a fuzzy random sample of size n ¼ 12 from a population and we observed the following triangular fuzzy data: (Table 3). The last line of Table 4 shows the percentiles of v2� b for variance computed using 10000 bootstrap samples. The fuzzy bootstrap confidence interval (c ¼ 0:05 or 90%) using fuzzy data is eP�a ¼ P12 i¼1ðxi � �xÞ 2 15:27 ; P12 i¼1ðxi � �xÞ 2 4:523 " # : xi 2 ~xia i ¼ 1; 2; . . . ; n ( ) ; and we have for some a0’s a 0 0.1 0.2 Confidence interval [317.68, 1115.83] [318.1, 1112.66] [318.39, 1109.59] a 0.3 0.4 0.5 Confidence interval [318.8, 1106.63] [319.24, 1103.24] [319.71, 1101.02] a 0.6 0.7 0.8 Confidence interval [320.21, 1098.38] [320.74, 1095.84] [321.3, 1093.41] a 0.9 1 Confidence interval [321.9, 1091.08] [322.52, 1088.86] We obtain the a-cuts of the so-called fuzzy test statistics eZ�a ¼ðn � 1ÞeP�aeh0a ; and now we can similarly apply the previous section for analyzing of this section. 0 10 20 0 100 200 300 400 500 Fr eq ue nc y Fig. 8. Bootstrap distribution of v2�b . Table 3 Fuzzy random sample of size n ¼ 12 from a population. N Observation N Observation N Observation 1 (33, 35, 36) 5 (60, 63, 66) 9 (100, 103, 105) 2 (80, 82, 84) 6 (70, 70, 72) 10 (54, 56, 58) 3 (85, 87, 87) 7 (70, 73, 76) 11 (40, 40, 42) 4 (90, 90, 90) 8 (65, 70, 73) 12 (94, 96, 99) Fig. 7. eZ in testing fuzzy simple hypothesis versus fuzzy left one-sided hypothesis. 5786 M.G. Akbari, A. Rezaei / Expert Systems with Applications 37 (2010) 5782–5787 Author's personal copy 5. Conclusion The proposed procedure is based on a significance level. Exten- sion of the proposed method to test the parameters of linear mod- els, such as regression models, design of experiment is a potential area for the future work. We can use the Yao–Wu signed distance for estimation of coefficients of linear models and apply the pro- posed method in this paper. Acknowledgements Authors are grateful to the referees of the journal for their sug- gestions and would like to thank the Co-Editor-in-Chief Doctor Vijayakumar for the helpful comments and useful helps. 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Neyman–Pearson Lemma for fuzzy hypotheses testing withy vague data. Metrika, 64, 289–304. Viertl, R. (2006). Univariate statistical analysis with fuzzy data. Computational Statistics and Data Analysis, 51, 133–147. Yao, J. S., & Wu, K. (2000). Ranking fuzzy numbers based on decomposition principle and signed distance. Fuzzy Sets and Systems, 11, 275–288. Table 4 Percentiles of the v2 distribution with 7 and 11 degrees of freedom and the bootstrap distribution of v2�b . Percentile 0.005 0.01 0.025 0.05 0.95 0.975 0.99 0.995 v2;7 0.989 1.239 1.69 2.167 14.067 16.013 18.475 20.278 v2;11 2.603 3.053 3.816 4.575 19.675 21.92 24.725 26.757 Bootstrap 2.699 3.077 3.85 4.523 15.27 16.889 18.29 21.349 M.G. Akbari, A. Rezaei / Expert Systems with Applications 37 (2010) 5782–5787 5787