Testing Fuzzy Hypotheses Using Fuzzy Data Based on Fuzzy Test Statistic

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1 Joural of Ucertai Systems Vol.5, No.1, pp.45-61, 2011 Olie at: Testig Fuzzy Hypotheses Usig Fuzzy Data Based o Fuzzy Test Statistic Mohse Arefi 1,, S. Mahmoud Taheri 2,3 1 Departmet of Statistics, Faculty of Scieces, Uiversity of Birjad, Birjad, Ira 2 Departmet of Mathematical Scieces, Isfaha Uiversity of Techology, Isfaha , Ira 3 Departmet of Statistics, School of Mathematical Scieces, Ferdowsi Uiversity of Mashhad, Mashhad, Ira Received 9 July 2009; Revised 16 May 2010 Abstract This paper deals with the problem of testig hypothesis whe both the hypotheses ad the available data are fuzzy. First, four differet kids of fuzzy hypotheses are defied. The, a procedure is developed for costructig the fuzzy poit estimatio based o fuzzy data. Also, the cocept of fuzzy test statistic is defied based o the α-cuts of the fuzzy ull hypothesis ad the α-cuts of the costructed fuzzy poit estimatio. Fially, by itroducig a credit level, we propose a method to evaluate the fuzzy hypotheses of iterest. The proposed method is employed to test the fuzzy hypotheses for the mea of a ormal distributio, ad the variace of a ormal distributio. A practical example i lifetime testig is provided, to show the applicability of the proposed method i applied studies. c 2011 World Academic Press, UK. All rights reserved. Keywords: credit level, fuzzy data, fuzzy hypothesis, fuzzy umber, fuzzy test statistic, lifetime testig, testig hypothesis 1 Itroductio ad Motivatio I classical approaches to testig statistical hypotheses, it is assumed that both the uderlyig hypotheses ad the available data are crisp. For example, if the differece betwee two populatio meas is to be tested, the ordiary ull hypothesis stipulates that the differece betwee two populatio meas is precisely equal to zero. I additio, it is usually assumed that the collected observatios are precise. However, we would sometimes like to test if two meas are early equal or ot. O the other had, sometimes the available observatios are ot precise. For istace, i ecoomic studies we may wish to test if the meas of icomes of households of two iterested populatios are approximately equal or ot. I such a case, the hypothesis of exact equality of meas seems to be urealistic. As a example of imprecise data, cosider the problem of lifetime testig. I lifetime aalysis, the data available are usually reported as imprecise data. For istace, measurig the lifetime of a battery may ot yield a exact result. A battery may work perfectly over a certai period but be losig i power for some time, ad fially go dead completely at a certai time. I this case, the data may be reported as imprecise quatities such as: about 1000 (h), approximately 1400 (h), about betwee 1000 (h) ad 1200 (h), essetially less tha 1200 (h), ad so o. The classical procedures for testig hypotheses are ot appropriate for dealig with such imprecise cases. After the iceptio of the otio of fuzzy sets by Zadeh 32, there have bee attempts to aalyze the problem of testig hypotheses for these situatios usig fuzzy set theory. See Taheri 21 for a review of some related works. I the preset work, we cosider the fuzzy hypotheses istead of crisp oes, ad itroduce a procedure to test such hypotheses based o a fuzzy test statistic whe the available data are fuzzy. Our proposed procedure is a extesio of Taheri ad Arefi s approach to testig statistical hypotheses 22, i which the data are assumed to be crisp. But, here we cosider testig fuzzy hypotheses whe the available data are fuzzy, too. I this extesio, we costruct a fuzzy test statistic usig the α-cuts of the fuzzy ull hypothesis ad the α-cuts of the fuzzy poit estimatio. The, we itroduce a procedure for testig the fuzzy hypotheses of iterest. Correspodig author. Arefi@Birjad.ac.ir (M. Arefi); Tel.: (+98) , Fax: (+98)

2 46 M. Arefi ad S.M. Taheri: Testig Fuzzy Hypotheses Usig Fuzzy Data Based o Fuzzy Test Statistic Testig statistical hypotheses uder imprecise (fuzzy) costraits were ivestigated by some authors. The problem of testig hypotheses with fuzzy data was cosidered by Casals ad Gil 5, Casals et al. 6, Filzmoser ad Viertl 9, Grzegorzewski 12, ad Wu 31. This topic, usig fuzzy radom variables, was studied by Körer 16 ad Moteegro et al. 19. Testig fuzzy hypotheses was discussed by Arold 1, 2, Taheri ad Arefi 22, Taheri ad Behboodia 23, 24, ad Wataabe ad Imaizumi 29. Testig fuzzy hypotheses with fuzzy data was ivestigated by Grzegorzewski 13, Kruse ad Meyer 17, Taheri ad Behboodia 25, ad Torabi et al. 27. For some other recet works o testig hypothesis i fuzzy eviromet, see Buckley 3, 4, Deoeux ad Masso 7, Hryiewicz 14, Thompso ad Geyer 26, ad Viertl 28. Also, the problem of poit estimatio i the fuzzy eviromet has bee ivestigated by some authors, e.g. Gil et al. 11 ad Wu 30. This paper is orgaized as follows: I Sectio 2, we recall some prelimiary cocepts about fuzzy umbers ad iterval arithmetic. I Sectio 3, we itroduce differet kids of fuzzy hypotheses. A ew method to test fuzzy hypotheses based o fuzzy data is give i Sectio 4. I Sectio 5, we apply our method to test fuzzy hypotheses for the mea ad for the variace of a ormal distributio. A practical example i lifetime testig is provided i Sectio 6. I Sectio 7, we compare our method with some other works. A brief coclusio is provided i Sectio 8. 2 Prelimiary Cocepts I this sectio, we recall some prelimiary cocepts about fuzzy umbers ad iterval arithmetic. For details, the reader ca refer to stadard texts, e.g. Klir ad Yua 15. A fuzzy set à of the uiverse X is defied by a membership fuctio à : X 0, 1. A α-cut of Ã, writte as Ãα, is defied as Ãα = x Ã(x) α}, for 0 < α 1. A fuzzy umber M is a fuzzy set of the real umbers satisfyig: i) M(x) = 1 for some x, ii) Mα is a closed bouded iterval for 0 < α 1. A triagular fuzzy umber T 1 = (a 1, a 2, a 3 ) T is defied by three umbers a 1 < a 2 < a 3 as T 1 (x) = x a1 a 2 a 1 a 1 x < a 2, a 3 x a 3 a 2 a 2 x < a 3. Also, the membership fuctios of fuzzy sets T 2 = (a 1, a 2 ) EL ad T 3 = (a 2, a 3 ) ES for a 1 < a 2 < a 3 are defied as x a1 T 2 (x) = a 2 a 1 a 1 x < a 2, 1 x < a2, T 1 a 2 x, 3 (x) = a 3 x a 3 a 2 a 2 x < a 3. Let I = a, b ad J = c, d be two closed itervals. The, based o the iterval arithmetic, we have I + J = a + c, b + d, I J = a d, b c, I.J = α 1, β 1, α 1 = miac, ad, bc, bd}, β 1 = maxac, ad, bc, bd}, I J = α 2, β 2, α 2 = mi a c, a d, b c, b d }, β 2 = max a c, a d, b c, b d }, where, zero does ot belog to J = c, d i the last case. 3 Fuzzy Hypotheses I this sectio, we recall some models as fuzzy sets of real umbers for modellig the extesios of simple, oe-sided, ad two-sided ordiary (crisp) hypotheses to fuzzy oes (see also 22). Defiitio 1 Let θ 0 be a kow real umber. i) Ay hypothesis of the form (H : θ is approximately θ 0 ) is called a fuzzy simple hypothesis.

3 Joural of Ucertai Systems, Vol.5, No.1, pp.45-61, Figure 1: Three forms of fuzzy hypotheses i Example 1 ii) Ay hypothesis of the form (H : θ is ot approximately θ 0 ) or, equivaletly, (θ is away from θ 0 ) is called a fuzzy two-sided hypothesis. iii) Ay hypothesis of the form (H : θ is essetially smaller tha θ 0 ) is called a fuzzy left oe-sided hypothesis. iv) Ay hypothesis of the form (H 1 : θ is essetially larger tha θ 0 ) is called a fuzzy right oe-sided hypothesis. I the ext sectio, we ivestigate some methods to test the followig forms of hypotheses: H0 : θ is approximately θ a) 0, H 0 : θ is H 0, H 1 : θ is ot approximately θ 0, H 1 : θ is H 1, H0 : θ is approximately θ b) 0, H 1L : θ is essetially larger tha θ 0, H0 : θ is approximately θ c) 0, H 1S : θ is essetially smaller tha θ 0, For the above hypotheses, we suppose that where a 1 a 1 ad a 3 a 3. H 0 : θ is H 0, H 1 : θ is H 1L, H 0 : θ is H 0, H 1 : θ is H 1S. H 0 = (a 1, θ 0, a 3 ) T, H1 = H c 0, H1L = (a 1, θ 0 ) EL, H1S = (θ 0, a 3) ES, Note 1 It should be metioed that case (a) is a atural geeralizatio of a crisp simple hypothesis versus a two-sided hypothesis of the form: H 0 : θ = θ 0 v.s. H 1 : θ θ 0. Moreover, case (b) is a atural geeralizatio of the crisp simple hypothesis H 0 : θ = θ 0 v.s. oe-sided hypothesis H 1 : θ > θ 0. A similar argumet holds for case (c). Example 1 I the above fuzzy hypotheses, suppose that a 1 = 1, a 1 = 2, θ 0 = 3, a 3 = 5, ad a 3 = 6, the we have some fuzzy hypotheses i which the related membership fuctios are show i Fig. 1: (a) (c). 4 Testig Fuzzy Hypotheses Based o Fuzzy Data Let X 1, X 2,..., X be a radom sample from a probability desity fuctio (or probability mass fuctio) f(x; θ), where the parameter θ is ukow. Suppose that the available data of the radom sample are observed as the fuzzy umbers X 1, X 2,..., X rather tha the crisp data x 1, x 2,..., x. We ca obtai a fuzzy poit estimatio for θ as follows. Defiitio 2 Let θ = u(x 1, x 2,..., x ) be a poit estimatio for θ. By substitutig the α-cut of the fuzzy umbers X i, i = 1,..., for x i, i = 1,..., ito θ, the α-cut of the fuzzy poit estimatio θ is obtaied as follows θ α := u(x 1, x 2,..., x ); x i X } i α, i = 1, 2,...,. I the followig, we itroduce a procedure for testig a fuzzy simple hypothesis agaist a fuzzy two-sided hypothesis ad a fuzzy oe-sided hypothesis, respectively.

4 48 M. Arefi ad S.M. Taheri: Testig Fuzzy Hypotheses Usig Fuzzy Data Based o Fuzzy Test Statistic 4.1 Testig Fuzzy Simple Hypothesis agaist Fuzzy Two-sided Hypothesis Suppose that we are iterested i testig the followig fuzzy hypotheses H0 : θ is approximately θ 0, H 0 : θ is H 0, H 1 : θ is ot approximately θ 0, H 1 : θ is H 1, where H 0 = (a 1, θ 0, a 3 ) T is a triagular fuzzy umber ad its α-cuts are θ 0 α = a 1 +(θ 0 a 1 )α, a 3 (a 3 θ 0 )α. I the crisp case, the decisio rule for testig a crisp ull hypothesis H 0 : θ = θ 0 agaist a crisp alterative H 0 : θ θ 0, at the sigificace level β, is Q0 Q 1 β/2 or Q 0 Q β/2 RH 0 (Rejectio of H 0 ), Q β/2 < Q 0 < Q 1 β/2 AH 0 (Acceptace of H 0 ), where Q 0 is the value of the crisp test statistic (uder H 0 ), ad Q β/2 ad Q 1 β/2 are the β/2 ad 1 β/2 quatiles of the crisp test statistic. Now, we itroduce a approach for testig the above fuzzy hypotheses based o the fuzzy data X 1, X 2,..., X. i) First, we obtai a fuzzy poit estimatio θ usig Defiitio 2. ii) By substitutig α-cuts of the fuzzy poit estimatio θ for the poit estimatio θ, ad the α-cuts of H 0 for θ 0 i the crisp test statistic (Q 0 ) ad by usig the iterval arithmetic, we obtai the α-cuts of the so-called fuzzy test statistic Z. Subsequetly, the obtaied fuzzy test statistic is used to provide a approach for testig the fuzzy hypotheses based o the followig quadruplet procedure 22 (see Fig. 2). a) We calculate the total area uder the graph of Z, deoted by A T. b) We obtai the area uder the graph of Z, but to the right of the vertical lie through Q 1 β/2 ad to the left of the vertical lie through Q β/2, deoted by A R. c) We choose a value for the credit level φ from (0, 1. d) Fially, we decide to reject or accept H 0 i the followig way AR /A T φ RH 0, A R /A T < φ AH 0. Remark 1: Note that, i the uderlyig curret eviromet, we come across two kids of ucertaity. The first kid of ucertaity (probabilistic oe) is related to the radomess of data ad, i testig hypothesis, it is cotrolled by the sigificace level β (or the cofidece level 1 β). But, the secod kid of ucertaity (possibilistic oe) is due to the impreciseess (fuzziess) of the data as well as due to the impreciseess of the hypotheses of iterest. This kid of ucertaity may be cotrolled by credit level φ i makig the decisio whether to accept or reject H 0. It is obvious that the selected value of φ is more or less subjective so that, by icreasig the credit level φ (as by icreasig the cofidece level 1 β), we guard agaist the rejectio of H 0. Remark 2: As a special case, suppose that we wat to test the followig crisp hypothesis based o the fuzzy data X 1, X 2,..., X : H0 : θ = θ 0, H 1 : θ θ 0. The α-cuts of the fuzzy test statistic Z are obtaied by the proposed approach, by substitutig α-cuts of the fuzzy poit estimatio θ for the poit estimatio θ i the crisp test statistic (Q 0 ). The, we ca test these hypotheses based o the quadruplet procedure provided.

5 Joural of Ucertai Systems, Vol.5, No.1, pp.45-61, Figure 2: The related quatities i testig a fuzzy simple hypothesis versus a fuzzy two-sided hypothesis 4.2 Testig Fuzzy Simple Hypothesis agaist Fuzzy Right Oe-sided Hypothesis Suppose that we wish to test the followig fuzzy hypotheses H0 : θ is approximately θ 0, H 1L : θ is essetially larger tha θ 0, H 0 : θ is H 0, H 1 : θ is H 1L, where H 0 = (a 1, θ 0, a 3 ) T is a triagular fuzzy umber for which θ 0 α = a 1 + (θ 0 a 1 )α, a 3 (a 3 θ 0 )α. Let θ = u(x 1, x 2,..., x ) be a poit estimatio for θ. I the ordiary case, the decisio rule for testig a crisp ull hypothesis H 0 : θ = θ 0 agaist a crisp alterative H 0 : θ > θ 0, at the sigificace level β, is of the form Q0 Q 1 β RH 0, Q 0 < Q 1 β AH 0, where Q 0 is the value of the crisp test statistic (uder H 0 ), ad Q 1 β is the (1 β)-quatile of the distributio of the crisp test statistic. Now, we itroduce a approach for testig the above fuzzy hypotheses based o the fuzzy data X 1, X 2,..., X. i) First, usig Defiitio 2, we obtai a fuzzy poit estimatio θ. ii) By substitutig the α-cuts of the fuzzy poit estimatio θ for the poit estimatio θ, ad the α-cuts of H 0 for θ 0 i the crisp test statistic (Q 0 ) ad by usig the iterval arithmetic, we obtai the α-cuts of the so-called fuzzy test statistic Z. Now, we use the fuzzy test statistic thus obtaied to provide a approach for testig fuzzy right oe-sided hypotheses based o the followig quadruplet procedure 22 (see Fig. 3). a) We calculate the total area uder the graph of Z, deoted by A T. b) We obtai the area uder the graph of Z, but to the right of the vertical lie through Q 1 β, deoted by A R. c) We choose a value for the credit level φ from (0, 1. d) Fially, we decide whether to reject or accept H 0 i the followig way AR/A T φ RH 0, A R/A T < φ AH 0. Remark 3: As a special case, suppose that we wat to test the followig crisp hypothesis based o the fuzzy data X 1, X 2,..., X : H0 : θ = θ 0, H 1 : θ > θ 0.

6 50 M. Arefi ad S.M. Taheri: Testig Fuzzy Hypotheses Usig Fuzzy Data Based o Fuzzy Test Statistic Figure 3: The related quatities i testig a fuzzy simple hypothesis versus a fuzzy right oe-sided hypothesis Figure 4: The related quatities i testig a fuzzy simple hypothesis versus a fuzzy left oe-sided hypothesis The α-cuts of the fuzzy test statistic Z are obtaied usig the above approach, but oly by substitutig the α-cuts of the fuzzy poit estimatio θ for the poit estimatio θ i the crisp test statistic (Q 0 ). Fially, we ca test these hypotheses based o the proposed quadruplet procedure. Remark 4: We ca also apply the above procedure for testig a fuzzy simple hypothesis agaist a fuzzy left oe-sided hypothesis based o fuzzy data (see Fig. 4). Remark 5: It is obvious that if i testig crisp hypotheses the value of the observed test statistic is close to the related quatile, the the classical methods for makig the decisio whether to accept or reject the ull hypothesis are very sesitive. I such cases, we propose to use the followig alterative methods: I1) Testig crisp hypotheses with fuzzy test statistic based o Buckley s approach 3. I2) Substitutig the fuzzy data for the crisp data ad testig the crisp hypotheses based o the method suggested i Remarks 2 ad 3. I3) Substitutig the fuzzy hypotheses for the crisp hypotheses ad testig them based o Taheri ad Arefi s approach 22. I4) Substitutig the fuzzy hypotheses for the crisp hypotheses ad usig the fuzzy data istead of the crisp data to test such hypotheses based o the method proposed i Subsectios 4.1 ad 4.2. Let us ow illustrate the above differet cases through the followig umerical example. Example 2 Assume that, based o a radom sample of size = 100 from a populatio with the distributio N(θ, σ 2 = 9), we wat to test some hypotheses about the mea θ at the sigificace level β = The observed value of the test statistic is z 0 = x θ0 σ/.

7 Joural of Ucertai Systems, Vol.5, No.1, pp.45-61, A1) Let the mea of the radom sample be x = We wat to test the followig hypotheses H0 : θ = 2, H 1 : θ > 2. Here, z 0 = < z 1 β = Hece, we accept the ull hypothesis H 0. A2) I case A1, let the mea of the radom sample be x = Here, z 0 = > z 1 β = ad, therefore, we reject the ull hypothesis H 0. A3) I case A2, let the mea of the radom sample be x = , but we wat to test the followig hypotheses H0 : θ = , H 1 : θ > Here, z 0 = < z 1 β = Hece, we accept the ull hypothesis H 0. A4) Cosider the case A1. Let the mea of the radom sample be x = We wat to test the followig hypotheses H0 : θ = , H 1 : θ > Here, z 0 = > z 1 β = Hece, we reject the ull hypothesis H 0. Cosider the above cases. The pairs (A1, A2) ad (A3, A4) have differet results with respect to acceptig or rejectig the ull hypothesis H 0 with a small chage i the sample mea. The pair (A2, A3) has differet results with respect to acceptig or rejectig the ull hypothesis H 0 with a small chage i the hypotheses. The pair (A1, A4) have differet results for acceptig or rejectig the ull hypothesis H 0 with a slight chage i the sample mea ad the hypotheses. For testig the pairs (A1, A2) ad (A3, A4), we ca use Buckley s approach ad the method proposed i Remarks 2 ad 3. For the pair (A2, A3) (ad also for the pairs (A1, A2) ad (A3, A4)), we ca defie the fuzzy hypotheses i a suitable maer, ad the test such hypotheses based o Taheri ad Arefi s approach. For the pair (A1, A4) (ad also for the pairs (A1, A2), (A2, A3), ad (A3, A4)), we ca defie the fuzzy hypotheses i a suitable maer, ad the test such hypotheses with fuzzy data based o the method proposed i Subsectios 4.1 ad Testig Fuzzy Hypotheses i the Normal Distributio 5.1 Testig Fuzzy Hypotheses for the Mea Suppose that we have take a radom sample of size from a N(θ, σ 2 ) (σ 2 kow) ad we have further observed the fuzzy umbers X 1, X 2,..., X. Now, we wat to test the followig fuzzy hypotheses at the sigificace level β: H0 : θ is approximately θ 0, H 0 : θ is H 0, H 1L : θ is essetially larger tha θ 0, H 1 : θ is H 1L. The usual poit estimatio for θ is θ = x. By substitutig the α-cuts of X i, i = 1,...,, ( X i α = X L i, X U i ) for x i i the poit estimatio, the α-cuts of the fuzzy poit estimatio X will be obtaied as Xα = 1 X i L, X i U 1 = X i L, 1 X i U = X L, X U. i=1 i=1 Uder the crisp ull hypothesis H 0 : θ = θ 0, the value of the crisp test statistic is z 0 = θ θ 0 σ/. By substitutig the α-cuts of the fuzzy poit estimatio X for θ ad the α-cuts of H0 for θ 0 i z 0, ad usig the iterval arithmetic, the α-cuts of the fuzzy test statistic are obtaied to be Xα Zα = H 0 α σ/ = X L a 3 + (a 3 θ 0 )α σ/ X U a 1 (θ 0 a 1 )α, σ/. i=1

8 52 M. Arefi ad S.M. Taheri: Testig Fuzzy Hypotheses Usig Fuzzy Data Based o Fuzzy Test Statistic Figure 5: Fuzzy hypotheses i Example 3 For example, let X i = (x i r i, x i, x i +r i ) T =: (x i, r i ) T, i = 1,...,, be the symmetric triagular fuzzy umbers with X i α = x i (1 α)r i, x i +(1 α)r i. The, the fuzzy poit estimatio is obtaied as X = (x r, x, x+r) T obeyig Xα = x (1 α)r, x + (1 α)r. Hece, the α-cuts of the fussy test statistic are obtaied as Zα = Xα H 0α σ/ = = x (1 α)r a3+(a 3 θ 0)α σ/, x+(1 α)r a1 (θ0 a1)α σ/ z 0 (1 α)(r + a 3 θ 0 ) σ, z 0 + (1 α)(r + θ 0 a 1 ) σ, where x = 1 i=1 x i ad r = 1 i=1 r i. Now, usig the above fuzzy test statistic, we ca apply the quadruplet procedure (proposed i Subsectio 4.2) for testig the fuzzy hypotheses of iterest. Example 3 Assume that, based o a radom sample of size = 50 from a populatio N(θ, σ 2 = 9), we observe the fuzzy data i Table 1. Table 1: The fuzzy data from a ormal populatio i Example 3 (x i, r i ) T (x i, r i ) T (x i, r i ) T (x i, r i ) T (x i, r i ) T (x i, r i ) T (1.8, 0.2) T (2.8, 0.3) T ( 3.4, 0.4) T ( 2.1, 0.2) T (2.1, 0.2) T ( 0.4, 0.2) T (2.9, 0.4) T ( 4.6, 0.3) T (2.4, 0.1) T (1.0, 0.2) T (1.4, 0.1) T (0.9, 0.2) T (0.9, 0.2) T (6.8, 1.4) T (1.9, 0.3) T (0.8, 0.1) T (3.0, 0.6) T ( 1.2, 0.2) T (3.7, 0.7) T (3.8, 0.4) T (6.0, 0.8) T (5.0, 1.0) T (0.3, 0.1) T ( 2.8, 0.4) T (1.2, 0.2) T (4.4, 0.4) T (3.1, 0.4) T (3.0, 0.6) T (1.6, 0.2) T (1.6, 0.3) T (1.7, 0.3) T (6.0, 1.2) T (7.3, 1.5) T (6.9, 1.2) T (6.9, 1.0) T (0.9, 0.2) T (0.5, 0.1) T (1.3, 0.3) T (5.1, 1.0) T (6.2, 1.1) T (1.8, 0.2) T (5.7, 1.0) T (3.4, 0.5) T (3.4, 0.4) T (1.3, 0.2) T (5.8, 1.1) T (4.9, 1.0) T ( 0.7, 0.1) T ( 1.3, 0.2) T (5.8, 1.0) T A) Suppose that we wat to test the followig fuzzy hypotheses at the sigificace level β = 0.05 H0 : θ is approximately 2, H 0 : θ is H 0, H 1L : θ is essetially larger tha 2, H 1 : θ is H 1L, where H 0 = (1.75, 2, 2.25) T ad H 1L = (1.80, 2) EL (see Fig. 5). The fuzzy poit estimatio is X = (1.924, 2.416, 2.908) T ad the fuzzy test statistic is obtaied as Z = ( , , ) T = ( , , ) T with the followig α-cuts Zα = ( (1 α)), ( (1 α)). 3 3

9 Joural of Ucertai Systems, Vol.5, No.1, pp.45-61, Figure 6: The related quatities i Example 3 for the fuzzy hypotheses Figure 7: The related quatities i Example 3 for the crisp hypotheses Based o the fuzzy test statistic, we obtai A T = = ad A R = Sice A R /A T = , we reject H 0 for every credit level φ (0, (see Fig. 6). B) Cosider the above fuzzy data. Now, suppose that we wat to test the followig crisp hypotheses (which is equivalet to the case a 1 = θ 0 = a 3 i the above fuzzy hypotheses): H0 : θ = 2, H 1 : θ > 2. (1) Based o Remark 3, the α-cuts of fuzzy test statistic are calculated as Zα = Xα θ 0 σ/ = = = x (1 α)r θ0 σ/, x+(1 α)r θ0 σ/ z 0 (1 α) r σ, z 0 + (1 α) r σ ( (1 α)) 50 3, ( (1 α)) Hece, the fuzzy test statistic is Z = ( , , ) T = ( , , ) T, ad we obtai A T = = , A R = , ad A R /A T = The ull hypothesis i (1) is, therefore, rejected for every credit level φ (0, (see Fig. 7). Sice r (r + θ 0 a 1 ) ad r (r + a 3 θ 0 ), it is cocluded that i this case, uder the fuzzy hypotheses, we may reject H 0 with a higher credit level tha we would the crisp hypotheses case.

10 54 M. Arefi ad S.M. Taheri: Testig Fuzzy Hypotheses Usig Fuzzy Data Based o Fuzzy Test Statistic Figure 8: Z ad AR i Example 4 Example 4 Cosider the fuzzy data i Example 3. Suppose that we wish to test the followig fuzzy hypotheses at the sigificace level β = 0.05: H0 : θ is approximately 2, H 1 : θ is away from 2, H 0 : θ is H 0, H 1 : θ is H 1, where H 0 = (1.5, 2, 2.5) T ad H 1 = H c 0. The α-cuts of the fuzzy test statistic are calculated as Zα = = z 0 (1 α)(r + a 3 θ 0 ) σ, z 0 + (1 α)(r + θ 0 a 1 ) σ ( α) 50 3, ( α) Hece, the fuzzy test statistic is Z = ( , , ) T = ( , , ) T, ad we obtai A T = = ad A R = A R1 + A R2 = = Sice A R /A T = , we reject H 0 for every credit level φ (0, (see Fig. 8). 5.2 Testig Fuzzy Hypotheses for the Variace Assume that we have take a radom sample of size from a populatio N(µ, θ) (µ is ukow) ad that we have observed the fuzzy umbers X 1, X 2,..., X. Suppose further that we wat to test the followig fuzzy hypotheses at the sigificace level β: H0 : θ is approximately θ 0, H 1L : θ is essetially larger tha θ 0. The usual poit estimatio for θ is θ = s 2 = 1 1 i=1 (x i x) 2. By substitutig the α-cuts of X i, i = 1,...,, ( X i α = X L i, X U i ) for x i i the poit estimatio θ, ad usig the iterval arithmetic, the α-cuts of the fuzzy poit estimatio S 2 are obtaied as follows ( S 2 1 α = 1 i=1 X i L, X i U X L, X ) ( U. X i L, X i U X L, X ) U = S 2L α, S 2U α,

11 Joural of Ucertai Systems, Vol.5, No.1, pp.45-61, where Xα = X L, X U = 1 i=1 X L i, X U i = S 2L α = max 0, mi S 2U α = max 1 1 i=1 ( X L i X U ) 2, 1 1 i=1 ( X L i X U ) 2, 1 X L i=1 i, 1 X U i=1 i, 1 1 i=1 ( X L i X U )( X U i X L ), 1 1 i=1 ( X L i X U )( X U i X L ), 1 1 i=1 ( X i U X L ) 2, 1 1 i=1 ( X i U X L ) 2. Uder the crisp ull hypothesis H 0 : θ = θ 0, the crisp test statistic ( 1)S2 θ 0 is distributed accordig to χ 2 ( 1), with Q 0 = ( 1)θ θ 0 = ( 1)s2 θ 0 as its observed value. By substitutig the α-cuts of the fuzzy poit estimatio S 2 for θ = s 2 ad the α-cuts of H 0 = (a 1, θ 0, a 3 ) T (i this case a 1 > 0) for θ 0 i Q 0, ad usig the iterval arithmetic, the α-cuts of the fuzzy test statistic are obtaied to be Zα = ( 1) S 2 α H 0 α ( 1) S 2L α, S 2U α = a 1 + α(θ 0 a 1 ), a 3 α(a 3 θ 0 ) = ( 1) S 2L α a 3 α(a 3 θ 0 ), ( 1) S 2 U α. a 1 + α(θ 0 a 1 ) Now, based o the above fuzzy test statistic, we ca test the hypotheses of iterest by employig the quadruplet procedure proposed i Subsectio 4.2. Example 5 Suppose that, based o a radom sample of size = 20 from a populatio N(µ, θ), we observe the symmetric triagular fuzzy umbers i Table 2 as fuzzy observatios. Table 2: The fuzzy data from a ormal populatio i Example 5 (r 1i, x i, r 2i) T (r 1i, x i, r 2i) T (r 1i, x i, r 2i) T (r 1i, x i, r 2i) T (r 1i, x i, r 2i) T (1.03, 1.29, 1.55) T (1.21, 1.51, 1.81) T (2.10, 2.63, 3.16) T (2.82, 3.53, 4.24) T (1.80, 2.25, 2.70) T (0.18, 0.23, 0.28) T (2.18, 2.72, 3.26) T (1.10, 1.37, 1.64) T (2.46, 3.08, 3.70) T (0.43, 0.54, 0.65) T (0.56, 0.70, 0.84) T (1.69, 2.11, 2.53) T (1.30, 1.62, 1.94) T (2.44, 3.05, 3.66) T (1.29, 1.61, 1.93) T (2.83, 3.54, 4.25) T (2.63, 3.29, 3.95) T (3.13, 3.91, 4.69) T (2.08, 2.60, 3.12) T (3.88, 4.85, 5.82) T Here, the α-cuts of the fuzzy poit estimatio are obtaied as follows (see Fig. 9) S 2 α = S 2L α, S 2U α, where S 2L α = α α(1 α) (1 α) 2 0 α , α α(1 α) (1 α) < α 1, ad S 2U α = α α(1 α) (1 α) 2. Now, suppose that we wat to test the followig hypotheses at the sigificace level β = 0.05: H0 : θ is approximately 2, H 0 : θ is H 0, H 1L : θ is essetially larger tha 2, H 1 : θ is H 1L, where H 0 = (1.5, 2, 2.5) T ad H 1L = (1.75, 2) EL. The α-cuts of the fuzzy test statistic are

12 56 M. Arefi ad S.M. Taheri: Testig Fuzzy Hypotheses Usig Fuzzy Data Based o Fuzzy Test Statistic Figure 9: The fuzzy poit estimatio i Example 5 Figure 10: Z ad AR i Example 5 for the fuzzy hypotheses where Zα = ( 1) S 2L α a 3 α(a 3 θ 0 ), ( 1) S 2 U α = 19 S 2L α a 1 + α(θ 0 a 1 ) α, 19 S 2 U α = α Z L α = α α(1 α) (1 α) α 0 α , α α(1 α) (1 α) α < α 1, ZL α, Z U α, ad Z U α = α α(1 α) (1 α) α Usig the trapezoidal rule 10, we obtai A T = ad A R = Sice A R /A T = , we reject H 0 for every credit level φ (0, (see Fig. 10). Now, suppose that, based o the fuzzy data i Example 5, we wat to test the followig crisp hypotheses istead of the above fuzzy oes (which is equivalet to the case a 1 = θ 0 = a 3 i the above fuzzy hypotheses) H0 : θ = 2, (2) H 1 : θ > 2. Based o Remark 3, the α-cuts of the fuzzy test statistic are calculated as follows Zα = ( 1) S 2L U α ( 1) S 2 α, = 9.5 S θ 2L α, 9.5 S 2U α = 9.5 S 2 α. 0 θ 0 Usig the trapezoidal rule 10, we obtai A R = 0. Hece A R /A T = 0, ad we accept the ull hypothesis i (2) for every credit level (see Fig. 11). Sice θ 0 a 3 α(a 3 θ 0 ) ad a 1 + α(θ 0 a 1 ) θ 0 for 0 < α 1, it

13 Joural of Ucertai Systems, Vol.5, No.1, pp.45-61, Figure 11: Z ad AR i Example 5 for the crisp hypothesis is cocluded that, i this case, for the fuzzy hypotheses, we may reject H 0 with a higher credit level tha we did i the crisp hypotheses case. 6 Testig Fuzzy Hypotheses for Mea of a Expoetial Distributio Assume that we have take a radom sample of size from a expoetial distributio Exp(θ) with the followig desity f(x) = 1 θ e x/θ, x > 0, θ > 0. Suppose that we observe the fuzzy umbers X 1, X 2,..., X. We wat to test the followig fuzzy hypotheses H0 : θ is approximately θ 0, H 0 : θ is H 0, H 1L : θ is essetially larger tha θ 0, H 1 : θ is H 1L. The usual poit estimatio for θ is θ = x. By substitutig the α-cuts of X i, i = 1,...,, ( X i α = X L i, X U i ) for x i i the poit estimatio, we obtai the fuzzy poit estimatio X with the α-cuts Xα = X L, X U = 1 X i=1 i L, 1 X i=1 i U. Uder the crisp ull hypothesis (H 0 : θ = θ 0 ), the crisp test statistic 2X θ 0 is distributed accordig to χ 2 (2), with Q 0 = 2θ θ 0 = 2x θ 0 as its observed value. By substitutig the α-cuts of the fuzzy poit estimatio X for θ ad the α-cuts of H 0 for θ 0 i Q 0, ad usig the iterval arithmetic, the α-cuts of the fuzzy test statistic are obtaied as follows Zα = 2 Xα H 0 α = 2 X L a 3 (a 3 θ 0 )α, 2 X U. a 1 + (θ 0 a 1 )α Now, based o the above fuzzy test statistic, we ca test the hypotheses of iterest usig the quadruplet procedure proposed i Subsectio 4.2. Example 6 Lifetime testig: The followig data (the ceters of fuzzy umbers, x i ) show the lifetimes (i 1000 km) of frot disk brake pads o a radomly selected set of 40 cars (same model) that were moitored by a dealer etwork (see, 18, pp. 337). But, i practice measurig the lifetime of a disk may ot yield a exact result. A disk may work perfectly over a certai period but be brakig for some time, ad fially be uusable at a certai time. So, such data may be reported as imprecise quatities. Assume that the lifetimes of frot disk brake pads are reported as fuzzy umbers i Table 3. I fact, imprecisio is formulated by fuzzy umbers X i = (x i, s i ) R, with s i = 0.05x i, i = 1, 2,..., 40, as follows X i (t) = 1 t x i s i x i t x i + s i, 0 otherwise.

14 58 M. Arefi ad S.M. Taheri: Testig Fuzzy Hypotheses Usig Fuzzy Data Based o Fuzzy Test Statistic Figure 12: Z ad AR i Example 6 Table 3: The fuzzy data of the lifetimes (i 1000 km) of frot disk brake pads i Example 6 X i = (x i, s i) R Xi = (x i, s i) R Xi = (x i, s i) R Xi = (x i, s i) R Xi = (x i, s i) R (86.2, 4.3) R (45.1, 2.3) R (52.1, 2.6) R (54.2, 2.7) R (59.0, 3.0) R (38.4, 1.9) R (41.0, 2.1) R (56.4, 2.8) R (81.3, 4.1) R (62.4, 3.1) R (45.5, 2.3) R (36.7, 1.8) R (42.2, 2.1) R (51.6, 2.6) R (34.4, 1.7) R (22.7, 1.1) R (22.6, 1.1) R (40.0, 2.0) R (38.8, 1.9) R (50.2, 2.5) R (48.8, 2.4) R (81.7, 4.1) R (61.5, 3.1) R (53.6, 2.7) R (50.7, 2.5) R (42.8, 2.1) R (102.5, 5.1) R (42.7, 2.1) R (80.6, 4.0) R (64.5, 3.2) R (73.1, 3.7) R (28.4, 1.4) R (46.9, 2.3) R (45.9, 2.3) R (33.8, 1.7) R (59.8, 3.0) R (31.7, 1.6) R (33.9, 1.7) R (50.6, 2.5) R (56.7, 2.8) R Suppose that the lifetime of the frot disk brake pad has a expoetial distributio with a ukow mea θ. The fuzzy poit estimatio for the parameter θ based o the fuzzy data is X = (x, s) R = ( , ) R with α-cuts Xα = , (1 α) Suppose that we wat to test the followig fuzzy hypotheses at the sigificace level β = 0.10: H0 : θ is approximately 45, H 0 : θ is H 0, H 1L : θ is essetially larger tha 45, H 1 : θ is H 1L, where H 0 = (40, 45, 50) T ad H 1L = (43, 45) EL. The α-cuts of the fuzzy test statistic are calculated as follows α Zα =,. 50 5α α Also, the fuzzy test statistic is obtaied as follows 10z z < z , z 4102 Z(z) = z 45 < z , 0 otherwise. Hece, A T = , A R = , ad A R /A T = Therefore, H 0 is rejected for every φ (0, (see Fig. 12).

15 Joural of Ucertai Systems, Vol.5, No.1, pp.45-61, A Compariso Study I this sectio, we compare our method with two well kow methods proposed for testig statistical hypotheses i the fuzzy eviromet. 7.1 Compariso with Buckley s Approach Buckley 3 studied the problem of testig crisp hypotheses based o the fuzzy test statistic. He first cosidered the cofidece itervals for the parameter of iterest as the α-cuts of a fuzzy poit estimatio. The, the fuzzy test statistic could be defied based o the α-cuts of the fuzzy poit estimatio. Fially, the statistical hypotheses could be evaluated usig a credit level. Our proposed approach has the followig two advatages over Buckley s. 1) While Buckley cosiders the problem of testig based o crisp data ad crisp hypotheses, we assume that both the hypotheses ad the data are fuzzy. Our method is, therefore, more coveiet i real world studies. 2) We itroduce a fuzzy poit estimatio based o the α-cuts of fuzzy data for obtaiig the fuzzy test statistic, whereas Buckley uses a set of cofidece itervals as the α-cuts of a fuzzy poit estimatio. The fuzzy poit estimatio for α = 1 is reduced to the usual crisp poit estimatio i our method, but Buckley s sometimes does ot yield a usual crisp poit estimatio. For istace, the cofidece iterval for σ 2 of a ormal distributio is as follows ( 1)s 2 χ 2 (1 α/2, 1), ( 1)s2 χ 2. (α/2, 1) For α = 1, usig Buckley s method, the poit estimatio of σ 2 would be obtaied as ( 1)s2, which is χ 2 (0.5, 1) ot a usual poit estimatio. O the other had, based o our proposed method described i Subsectio 5.2, we obtai s 2 = 1 1 i=1 (x i x) 2 as the poit estimatio of σ 2, which is exactly the usual crisp poit estimator of σ 2 (see also 8). 7.2 Compariso with Wu s Approach Wu 31 proposed a approach for testig the fuzzy mea of a ormal distributio based o fuzzy data. He used the followig otatios for testig the fuzzy hypothesis H 0 : µ = µ 0 agaist H 1 : µ µ 0 (where is a orderig betwee two fuzzy umbers) x L α = 1 ( x i ) L α core( µ 0 ), i=1 x U α = 1 ( x i ) U α core( µ 0 ), i=1 where ( x i ) L α = ift x i (t) α} ad ( x i ) U α = supt x i (t) α}, ad core( µ 0 ) is the ceter of the fuzzy umber (e.g. core( µ 0 ) = µ 0 if µ 0 (µ 0 ) = 1). The, he proposed to accept H 0 i the α-cut sese if x L σ α < z 1 β ad σ x U α < z 1 β, ad to accept H 1 i the α-cut sese if x L α z 1 β ad x U α z 1 β. The, he itroduced degrees of optimism ad pessimism ad also a degree of belief to evaluate the hypotheses of iterest. Fially, by trasferrig the basic problem to a optimizatio problem, ad by solvig the problem, oe could decide whether to accept or reject the hypotheses. He itroduced a similar approach for testig a simple fuzzy hypothesis agaist a two-sided fuzzy hypothesis. Some of the advatages of our method over Wu s method are as follows: 1) We use the α-cuts of the fuzzy ull hypothesis for calculatig the fuzzy test statistic while, Wu used the ceter of the fuzzy ull hypothesis (core( µ 0 )). Now, cosider differet fuzzy hypotheses with differet spreads ad similar ceters. Based o our proposed approach, oe obtais differet fuzzy test statistics (ad so, differet results i testig the hypotheses), whereas, by applyig Wu s approach, oe obtais similar results for testig such differet hypotheses. σ σ

16 60 M. Arefi ad S.M. Taheri: Testig Fuzzy Hypotheses Usig Fuzzy Data Based o Fuzzy Test Statistic 2) For testig fuzzy hypotheses, we obtai a fuzzy test statistic based o all the α-cuts of the fuzzy data, but Wu s approach is based oly o ( x i ) L α = ift x i (t) α} ad ( x i ) U α = supt x i (t) α} of the fuzzy data. 3) I additio to the sigificat level β, we use oe additioal criterio (the credit level) to evaluate the fuzzy hypotheses. However, Wu suggested three additioal criteria which may cofuse the decisio maker i evaluatig the hypotheses of iterest i practical problems. 8 Coclusio We exteded a approach to the problem of testig fuzzy hypotheses whe the available data are fuzzy, too. I the proposed approach, the fuzzy hypotheses are tested based o two criteria: a sigificace level (comig derived from the radomess of data) ad a credit level (stemmig from the fuzzy viewpoit). This approach is especially suitable for testig crisp/fuzzy hypotheses whe the observed value of the test statistic is close to the quatile of the distributio of the test statistic. The advatage of the proposed approach is that it is a atural aalogue of the usual approach to the problem of testig statistical hypotheses, sice decisio makig is essetially based o the so called fuzzy test statistic. The proposed approach is geeral ad ca be applied for testig fuzzy hypotheses with ay type of fuzzy data. We illustrated the proposed approach through some umerical examples. I additio, the applicability of the approach was explaied by a practical example of lifetime testig. Extesios of the proposed method to test the parameters of regressio models ca be cosidered for future work. I additio, studyig the problem of testig hypothesis withi the framework of graular computig 20, 33 is a potetial subject for further research. Ackowledgemets The authors would like to thak Professor R. Viertl for readig the mauscript ad for his valuable suggestios ad commets. The secod author is partially supported by Fuzzy Systems ad Applicatios Ceter of Excellece at Shahid Bahoar Uiversity of Kerma, Ira. Refereces 1 Arold, B.F., A approach to fuzzy hypothesis testig, Metrika, vol.44, pp , Arold, B.F., Testig fuzzy hypothesis with crisp data, Fuzzy Sets ad Systems, vol.94, pp , Buckley, J.J., Fuzzy statistics: hypothesis testig, Soft Computig, vol.9, pp , Buckley, J.J., Fuzzy Statistics, Spriger, Heidelberg, Casals, M.R., ad M.A. Gil, A ote o the operativeess of Neyma-Pearso tests with fuzzy iformatio, Fuzzy Sets ad Systems, vol.30, pp , Casals, M.R., M.A. Gil, ad P. Gil, The fuzzy decisio problem: a approach to the problem of testig statistical hypotheses with fuzzy iformatio, Europea Joural of Operatioal Research, vol.27, pp , Deoeux, T., M.H. Masso, ad P.A. Hébert, Noparametric rak-based statistics ad sigificace tests for fuzzy data, Fuzzy Sets ad Systems, vol.153, pp.1 28, Falsafai, A., ad S.M. Taheri, O Buckley s approach to fuzzy estimatio, Soft Computig, (to appear), DOI: /s Filzmoser, P., ad R. Viertl, Testig hypotheses with fuzzy data: the fuzzy P-value, Metrika, vol.59, pp.21 29, Fiey, R.L., ad G.B. Thomas, Calculus, 2d Editio, Addiso Wesley, New York, Gil, M.A., N. Corral, ad P. Gil, The fuzzy decisio problem: a approach to the poit estimatio problem with fuzzy iformatio, Europea Joural of Operatioal Research, vol.22, pp.26 34, Grzegorzewski, P., Testig statistical hypotheses with vague data, Fuzzy Sets ad Systems, vol.112, pp , 2000.

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