666663 Journl of Uncertin Systems Vol.9, No., pp.6-74, 5 Online t: www.jus.org.uk Credibility Hypothesis Testing of Fuzzy Tringulr Distributions S. Smpth, B. Rmy Received April 3; Revised 4 April 4 Abstrct This pper introduces the notion of testing hypothesis bout prmeters involved in credibility distributions nd defines criterion clled membership rtio criterion for testing given null credibility hypothesis ginst n lterntive credibility hypothesis. The proposed membership rtio criterion hs been used to develop credibility tests for testing hypotheses bout credibility prmeters which pper in some specil types of tringulr credibility distributions. The tests derived re shown to hve higher credibility vlues under lterntive credibility hypothesis in clss of credibility tests hving given credibility level under null credibility hypothesis. Some illustrtive exmples re lso included to demonstrte the theory developed in this pper. 5 World Acdemic Press, UK. All rights reserved. Keywords: credibility hypothesis, credibility test, credibility rejection region, power of credibility test, best credibility rejection region, membership rtio criterion Introduction In rel life situtions, uncertinties of vrious types rise. Uncertinty cn be brodly clssified s Uncertinty cused by rndomness nd Uncertinty cused by Fuzziness. In the words of Liu [3], rndomness is bsic type of objective uncertinty wheres fuzziness is bsic type of subjective uncertinty. Probbility theory is very old brnch of mthemtics ment for rndom phenomen hving lot of pplictions in diversified fields of study. Credibility theory is reltively new brnch of mthemtics, which is similr to Probbility theory. It is specificlly ment for studying the behvior of fuzzy phenomen. Liu nd Liu [4] initited the first study on Credibility theory nd Liu [] crried out further refinements of the theory. Mny reserchers hve studied pplictions of Credibility theory in different fields of study. Peng nd Liu [5] developed methodology for modeling prllel mchine scheduling problems with fuzzy processing times. Zheng nd Liu [6] designed fuzzy optimiztion model for fuzzy vehicle routing problem by considering trvel times s fuzzy vrible. Ke nd Liu [] considered fuzzy project scheduling problem. Wng, Tng nd Zho [4] designed n inventory model without bckordering for the fuzzy vribles. Wng, Zho nd Tng [5] considered fuzzy progrmming model for vendor selection problem in supply chin. Smpth [6] nd Smpth nd Deep [7] hve pplied Credibility theory in designing cceptnce smpling plns for situtions involving both fuzziness nd rndomness. Smpth nd Vidynthn [] nd Smpth nd Klivni [8] hve used Credibility theory in Clustering of fuzzy dt. A thorough review of the existing literture on Credibility theory indictes severl developments in Credibility theory, which re nlogues of those vilble in Probbility theory. For exmple, fuzzy vribles, credibility distributions, expecttion, vrince, moments nd relted results corresponding to Credibility Theory re nlogues of rndom vribles, probbility distributions, moments etc defined under Probbility theory. Even though significnt mount of works hve been done in Credibility theory one cn feel the bsence of studies on the inferentil spects (prllel to Theory of Estimtion nd Testing of Hypotheses in Probbility distributions) of prmeters (constnts) involved in credibility distributions. Recently, reserchers hve turned their ttention in this direction for Uncertinty distributions. The problem of estimting prmeters in Uncertinty distributions hs been studied by Wng nd Peng [3] nd Wng, Go nd Guo []. Another significnt work is due to Wng, Go nd Guo [] in which the ide of testing Uncertinty hypothesis hs been introduced. Smpth nd Rmy [9] hve lso considered distnce bsed test procedure for testing uncertinty hypotheses. Corresponding uthor. Emil: smpth959@yhoo.com (S. Smpth).
Journl of Uncertin Systems, Vol.9, No., pp.6-74, 5 63 It is to be noted tht so fr no work hs been done in testing hypotheses bout prmeters (constnts) ppering in credibility distributions. Different credibility distributions ment for vrious prcticl situtions re vilble in the literture. For detiled discussion on them one cn refer to []. The choice of the credibility distribution nd prior knowledge bout the prmeters which pper in credibility distribution re bsolutely necessry for prctitioner. For exmple, if one identifies tht tringulr credibility distribution is pproprite for given sitution, credibility vlue of the fuzzy vrible ssuming vlues in given set nd other chrcteristics of interest like its expected vlue, vrince etc cn be rrived t only if vlues of the three prmeters bnd, c of the tringulr credibility distribution re fully known. When prctitioner is presented with different sets of vlues by domin experts bout the prmeters, the question of choosing n pproprite set for given sitution rises. To del with these types of problems, we consider tool which is similr to the Theory of Testing of Hypothesis in Probbility distributions bsed on the ide of Neymn-Person theory nd mke study on its properties. The pper is orgnized s follows. The second section of this pper gives brief description on Credibility theory nd the third section gives certin definitions needed in the process of developing tests for prmeters in credibility distributions. The fourth section is devoted for identifying optiml tests in the cse of prmeters involved in tringulr credibility distributions. Conclusions nd directions for future work re given in the fifth section of this pper. Credibility Theory In this section, we give brief introduction to Credibility Theory [, 4] s well s description bout the terminology used in Credibility Theory. Let be nonempty set, be lgebr over. Elements of re clled events. CrA indictes the credibility of occurrence of event A which is the number ssigned to ech event A tht stisfies following xioms. Axiom of Normlity: Cr ; Axiom of Monotonicity: CrA CrB whenever A B; Axiom of Self-Dulity: CrA CrA c for ny event A ; Axiom of Mximlity: Cr Ai supcr Ai for ny events A i with supcr Ai.5. i i i Credibility Mesure Spce: The set function Cr is clled credibility mesure if it stisfies the xioms of normlity, monotonicity, self-dulity nd mximlity. The triplet,,cr is known s credibility spce. Fuzzy Vrible: Fuzzy vrible is mesurble function from the credibility mesure spce,,cr to the set of rel numbers. Membership Function: The membership function of fuzzy vrible defined on the credibility mesure spce,,cr is defined s x Cr x ^, x R. Credibility Inversion Theorem: Let be fuzzy vrible defined on the credibility mesure spce,,cr with membership function. Then for ny set B of rel numbers, Cr B sup ( x ) sup ( x ). xb xb Credibility Distribution: Let be fuzzy vrible defined on the credibility mesure spce,,cr with membership function. Then the credibility distribution : R, is defined s x Cr x, for ll x R. By credibility inversion theorem, the credibility distribution is given by x sup ( y ) sup ( y ), for ll. yx yx x R 3 Testing Credibility Hypotheses There re mny credibility distributions vilble in the literture. Some populr nd widely studied distributions re equipossible, tringulr nd trpezoidl credibility distributions. The membership functions ssocited with them re given below.
64 S. Smpth nd B. Rmy: Credibility Hypothesis Testing of Fuzzy Tringulr Distributions Equipossible: An equipossible fuzzy vrible is determined by the pir b, of rel numbers with the membership function if x b ( x) otherwise. Tringulr: A tringulr fuzzy vrible is fully determined by the triplet bc,, of rel numbers with membership function x if x b b ( x) c x if b x c. c b Throughout this pper, we use the symbolic expression ~ Tri, b, c to denote the fct tht the fuzzy vrible hs fuzzy tringulr distribution defined on bc,,. Trpezoidl: A trpezoidl fuzzy vrible is defined on the qudruplet, b, c, d of rel numbers with membership function x if x b b ( x) if b x c d x if c x d. d c The credibility distributions identified by the bove membership functions contin certin unknown constnts. Equipossible membership function hs two prmeters, nmely, nd b, the tringulr membership function contins three prmeters, nmely,, bnd c nd the trpezoidl membership function hs four prmeters, bcnd,, d. In our further discussion we refer these constnts s credibility prmeters. In this present work, we consider test criterion for developing procedure to test whether given credibility distribution is the one ssocited with the fuzzy environment under study. To fcilitte the understnding of the proposed criterion we present below certin definitions. These definitions re the credibility versions of relted ides from sttisticl theory of testing of hypotheses. Support: Support of the credibility distribution of fuzzy vrible is nothing but the collection of vlues ssumed by the fuzzy vrible with nonzero credibility. For exmple, in the cse of ~ Tri,4,5, the support of fuzzy vrible is x x 5. Credibility Hypothesis: It is sttement bout the credibility distribution of fuzzy vrible. For exmple, H : ~ Tri,4,7 ~ Tri,4,7. is credibility hypothesis which sttes the fuzzy vrible Null Credibility Hypothesis: A credibility hypothesis tht is being tested for possible rejection is known s null credibility hypothesis. Alterntive Credibility Hypothesis: A credibility hypothesis, which will be ccepted in the event of rejecting null credibility hypothesis is known s lterntive credibility hypothesis. In this pper, the letters H nd K re used to denote the null nd lterntive credibility hypotheses respectively. The formultion of null nd lterntive credibility hypotheses is domin oriented subject which tkes into ccount the opinions expressed by subject experts. For exmple, consider the menstrul cycle length of young women. It is believed tht the menstrul cycle length is lunr cycle length. However, if one suspects the chnge in life style nd living conditions hve ffected the menstrul cycle length then one cn think of suitble credibility hypotheses testing problem. If gynecologist believes tht bsed on experience, the cycle length hs incresed nd identifies its Tri 9,3,33, then the null nd lterntive credibility hypotheses distribution s tringulr credibility distribution cn be tken s H : ~ Tri 6, 8,3 nd K Tri : ~ 9,3,33, respectively. Credibility Test: It is rule tht helps the prctitioner to either ccept or reject null credibility hypothesis if the experimentl vlues of the underlying fuzzy vribles re known.
Journl of Uncertin Systems, Vol.9, No., pp.6-74, 5 65 Credibility Rejection Region: The subset of the support of fuzzy vrible contining the points leding to the rejection of the null credibility hypothesis s identified by credibility test is known s Credibility Rejection Region nd is denoted by. C H is rejected. C Mthemticlly, the credibility rejection region is defined s Type I nd Type II Errors: Type I error is the ction of rejecting the null credibility hypothesis when it is true nd Type II error is the ction of ccepting the null credibility hypothesis when it is flse. Level of Significnce: The mximum credibility for committing Type I error by credibility test is known s level of significnce nd is denoted by. Cr C. Power of Credibility Test: The power of the credibility test with rejection region C is defined by Best Credibility Rejection Region: A credibility rejection region C is sid to be best credibility rejection region of level if it hs mximum power under lterntive credibility hypothesis thn tht of ny other credibility rejection region of sme level. Tht is, C the best credibility rejection region, if Cr C K Cr C K ny subset of the support of the fuzzy vrible such tht Cr C H Here Cr A H nd Cr A K. where C is indicte the credibility vlues of the event A corresponding to the credibility distributions defined under null nd lterntive credibility hypotheses respectively. For exmple, Cr A H Sup H( x) Sup H( x) A A where H is the membership function ssocited with the credibility distribution specified under null credibility hypothesis. Computtion of the bove credibility is bsed on credibility inversion theorem. It is to be noted tht credibility vlue of the fuzzy vrible lying in rejection region under the lterntive credibility hypothesis needs to be mximum, becuse it is ssocited with correct ction, nmely the ction of rejecting the null credibility hypothesis when the lterntive is true. Further such credibility vlue computed under the null credibility hypothesis should be smll, becuse the corresponding event is n incorrect ction. Best Credibility Test: The rule tht helps the prctitioner to identify best credibility rejection region is known s best credibility test. Membership Rtio criterion: It is pertinent to note tht the definitions stted bove re nothing but the credibility theory nlogues of the definitions vilble in Neymn-Person pproch relted to the clssicl theory of testing of sttisticl hypothesis. Hence, it is decided to develop method for determining best credibility test. Following the lines of Neymn-Person we suggest criterion for testing null credibility hypothesis ginst n lterntive credibility hypothesis using the membership functions corresponding to the credibility distributions mentioned under the hypotheses s given below. For testing the null credibility hypothesis, H : hs credibility distribution ginst the lterntive credibility hypothesis K : hs credibility distribution, the membership rtio criterion is Reject the null credibility hypothesis if the observed vlue of the fuzzy vrible C where C x ( x) ( x) k, nd re the membership functions corresponding to the credibility distributions nd ; k being constnt selected so tht Cr C H. The set of vlues stisfying the inequlity used in C s stted bove hve reltively higher vlues with respect to the membership function defined under the lterntive credibility hypothesis when compred to the function given under the null credibility hypothesis. Hence, the credibility of the event defined by such observtions will be reltively higher under lterntive credibility hypothesis when compred to the sme under null credibility hypothesis. Therefore, when the observed vlue of the fuzzy vrible is member of the set defined by the membership rtio criterion, the decision of rejecting the null credibility hypothesis nd ccepting the lterntive is more meningful. Cr C H. The choice of the constnt k is governed by the vlue of It is interesting to note tht the bove criterion hve certin chrcteristics tht re not in line with those possessed by Neymn-Person lemm. The test identified by the bove criterion need not be unique s observed in the lter prt of this work. Further, the bsence of dditive property of the credibility mesure mkes it difficult to give generl proof for estblishing the optiml property (Best Credibility Rejection Region) of the test obtined using the bove criterion. Hence, it becomes necessry to mke studies on the properties of the test derived using the bove criterion for specific distributions. Even though the membership rtio criterion cn be pplied to ny credibility distribution we confine ourselves to the tringulr credibility distributions. The following section of this pper studies
66 S. Smpth nd B. Rmy: Credibility Hypothesis Testing of Fuzzy Tringulr Distributions certin spects relted to the existence of best credibility test for testing hypotheses bout tringulr credibility distribution. 4 Test for Tringulr Credibility Distribution The credibility distribution function of the tringulr fuzzy vrible ~ Tri(, b, c) is if x x if x b ( b ) ( x) x c b if b x c ( c b) if x c. In this section, we consider the problem of developing best credibility rejection region for testing H : ~ Tri(, b, c) ginst the lterntive credibility hypothesis K : ~ Tri(, b, c) for specific choices of the prmeters. Before considering the question of developing best credibility test for fuzzy tringulr distributions, we list below certin importnt propositions ssocited with tringulr credibility distribution Tri(, b, c). These propositions re needed in studying the properties of the test obtined using membership rtio criterion. ~ Tri, b, c nd, x, ( b ), x will hve Proposition 4.: If then ll intervls of the form credibility. Proof: Consider the intervl x, x where x x b. The credibility of the event tht the fuzzy vrible belongs to the intervl is given by x Cr x, x ( ) ( ) ( ). Sup x Sup x x b x( x, x ) x( x, x ) b If Cr x, x then Solving for x we get x ( b ). Since the expression for Cr x, x x, ( b ), x the form x. b is free from x of the stted choice, we conclude tht ll intervls of will hve credibility. It my be noted tht, ( b ) is the longest intervl with credibility tht lies to the left of b. Proposition 4.: If ~ Tri, b, c nd, credibility greter thn. Proof: Let x ( b ) nd x b. Since x ( b ), we get Cr x x then ll intervls of the form b x ( ),, x bwill hve the Then the credibility of fuzzy vrible lies in the intervl, x Cr x x Sup x Sup x, ( ) ( ). x( x, x ) x( x, x ) b,. Since the expression for Cr x, x the form ( b ), x, x bwill hve credibility greter thn. Proposition 4.3: If ~ Tri, b, c nd, then ll intervls of the form c c b x credibility. Proof: For ny intervl x x where b x x c,, x x is x of the stted choice, we conclude tht ll intervls of ( ),, x c will hve the
Journl of Uncertin Systems, Vol.9, No., pp.6-74, 5 67 If Cr x, x c x Cr x x Sup x Sup x, ( ) ( ). x( x, x ) x( x, x ) cb then x c ( c b). is free from x of the stted choice, we conclude tht ll intervls of the form will hve credibility. Note tht c ( c b), c is the lrgest set in the region to the right of b which hs credibility. Proposition 4.4: All intervls of the form x, c ( c b), x b where, will hve the credibility greter thn. Proof is strightforwrd nd hence omitted. Since the expression for Cr x, x c ( c b), x, x c Proposition 4.5: All intervls of the form ( x, x), x (, b) nd x ( b, c) hve credibility vlue greter thn. Proof: The credibility of ssuming vlues in ( x, x ), where x (, b) nd x ( b, c) is Cr x, x ( ) ( ) Sup x Sup x x( x, x ) x( x, x ) mx x, x x c x mx, b c b ( ). Hence the proposition is proved. Consider the problem of testing the null credibility hypothesis H : ~ Tri(, b, c) ginst the lterntive credibility hypothesis K : ~ Tri(, b, c) where by using the membership rtio criterion defined in the previous section. The rejection region cn be identified by the membership rtio criterion on mking use of the nture of the membership rtio. The nture of the membership rtio depends on the reltive positions of the prmeters used in the tringulr credibility distributions defined under null nd lterntive credibility hypotheses. Tble gives ten possible cses tht rise when we tke into ccount the ordering of the prmeters involved in the credibility distributions. Tble: Possible ordering of prmeters Cse Ordering of the prmeters b c b c b c b c 3 b b c c 4 b b c c 5 b c b c 6 b b c c 7 b b c c 8 b b c c 9 b b c c b c b c It is interesting to observe tht in the cses,,,3,5 nd 6 the inequlities, b b nd c cre stisfied. Whenever the prmeters in the two fuzzy vribles Tri(, b, c) nd Tri(, b, c ) stisfy these conditions we cll one of them s right shift of the other. Formlly, we define the right shift of fuzzy tringulr vrible s follows.
68 S. Smpth nd B. Rmy: Credibility Hypothesis Testing of Fuzzy Tringulr Distributions Right Shift: The tringulr fuzzy vrible Tri(, b, c) is sid to be right shift of nother tringulr fuzzy vrible Tri(, b, c ) if, b b nd c c. Similr definition cn be given for left shift of tringulr fuzzy vrible s given below. Left Shift: The tringulr fuzzy vrible Tri(, b, c) is sid to be left shift of nother tringulr fuzzy vrible Tri(, b, c) if, b b nd c c. Tble gives the vlues of the rtio ( x) ( x) with respect to right shifted tringulr credibility distributions. The first column gives different situtions nd the second column gives the vlues of the rtio ( x) ( x). It is esy to verify tht in ll these five cses, the rtio is non decresing in x. Tble : Vlues of membership rtio for right shifted tringulr distributions Cse b c b c b c b c b b c c b c b c b b c c ( x) ( x) ( x) if x c ( x) if x c if x ( x) x c b if x c ( x) c x b if c x c if x x c b if x b ( x) c x b ( x) c x c b if b x c c x c b if c x c if x x b if x b ( x) x b ( x) x c b if b x c c x b if c x c if x x b if x b x b ( x) x c b if b x b ( x) c x b c x c b if b x c c x c b if c x c The following theorem gives the credibility rejection region obtined on using membership rtio criterion nd proves tht the resulting rejection is the best credibility rejection region with significnce level when the lterntive credibility hypothesis specifies tringulr credibility distribution which is right shift of the tringulr credibility distribution specified under null credibility hypothesis for. Theorem: 4. If Tri(, b, c) is right shift of Tri(, b, c), then for testing the null credibility hypothesis H : ~ Tri(, b, c) ginst the lterntive credibility hypothesis K : ~ Tri(, b, c), () membership rtio criterion credibility rejection region of level is given by C x x c c b (b) the credibility rejection region C x x c c b nd is the best credibility rejection region of level.
Journl of Uncertin Systems, Vol.9, No., pp.6-74, 5 69 Proof: () From Tble, it is observed tht whenever the credibility distribution Tri(, b, c ) is right shift of the distribution Tri(, b, c ), the rtio ( x) ( x) is non decresing in x. Hence the set of vlues obtined using the membership rtio criterion given by ( x) x k ( x) reduces to C x x c where c is selected such tht This implies Hence Solving for c, we get Thus we hve proved (). (b) In order to prove tht C x x c level, we must show tht where C stisfies Cr C H. Note tht the vlue of Cr C K intervl, b or in Cse : c, b In this cse, Therefore, Cr C H. ( ) ( ). Sup x Sup x xc xc with c c. c b c c ( c b ). (4.) c s defined in (4.) is the best credibility rejection region of Cr C K Cr C K (4.) depends on the position of the cutoff point c. It cn lie either in the b, c. The two cses re considered below. c Cr C K Cr c K. b c Cr C K. (4.4) b Now we shll prove (4.) by considering C of different forms s identified in F, F nd the unions of members of F nd/or F where F x x( c, x ), x c (4.5) nd with c c ( c b ) nd ( b ). C F x x( c, x ), x c. Consider the cse where (4.3) F x x( x, ), x. (4.6) It my be noted tht two cses rise regrding the position of x when the credibility vlue is computed under K, nmely, x (, b ) nd x ( b, c ). Consider the cse, where x (, b ) s shown in the Figure 4.. Here, Therefore, x Cr C K Sup x Sup x ( ) ( ). x( c, x ) x( c, x ) b
7 S. Smpth nd B. Rmy: Credibility Hypothesis Testing of Fuzzy Tringulr Distributions x c Cr C K Cr C K. ( b ) Figure 4.: x, c, b On the other hnd, if x ( b, c ) s shown in Figure 4., we hve Cr C K ( ) ( ) Sup x Sup x x( c, x ) x( c, x ) c c x mx, b c b c c c x, if b b c b c x c x c, if c b c b b (4.7) Figure 4.: x ( b, c ), c (, b ) Using (4.4) nd (4.7) we get c c x if b c b Cr C K Cr C K c x c c x c if. c b b c b b Thus we hve proved irrespective of the position of x, for every C F, Cr C K Cr C K.
Journl of Uncertin Systems, Vol.9, No., pp.6-74, 5 7 Now we shll prove the inequlity (4.) by considering credibility rejection regions CF F x x( x, ), x. where Figure 4.3: x,, c (, b ) x (, b ) s shown in the Figure 4.3 then we hve If Hence Therefore, Cr C K Sup x Sup x ( ) ( ). x( x, ) x( x, ) b c Cr C K Cr C K. b b Cr C K Cr C K. The cse x ( b, c ) will not rise becuse x, b. Thus we hve proved (4.) for ll C F. By the mximum property of credibility mesure, we observe tht when the rejection region C of size is constructed by tking the union of two or more regions belonging to F nd /or F, the inequlity Cr C K Cr C K continues to hold good. Thus we hve proved under ll possible scenrios existing under cse, Cr C K Cr C K. Cse : c b, c nd In this cse, c c Cr C K Cr c K Sup ( x) Sup ( x). (4.8) xc xc c b Therefore, c c Cr C K. c b Now we shll prove the inequlity (4.) by considering different possibilities s in the previous cse. Let C F where F x x( c, x ), x c. If x ( b, c ) s shown in Figure 4.4 then we hve Hence Cr C K Cr C K c c Cr C K Sup x Sup x ( ) ( ). x( c, x ) x( c, x ) c b. Thus we hve proved (4.) is stisfied with equlity sign. The cse x (, b ) does not rise in this sitution.
7 S. Smpth nd B. Rmy: Credibility Hypothesis Testing of Fuzzy Tringulr Distributions Let C F where F x x x x Figure 4.4: x, c ( b, c ) (, ),. If x (, b ) s shown in the Figure 4.5, then Cr C K Sup x Sup x By Propositions nd 3, we hve c c. b c b ( ) ( ). x( x, ) x( x, ) b Figure 4.5: x (, b ), c ( b, c ) Under the condition, b b nd c c, ( ) (refer Figure 4.5) b b which gives. b Therefore, Cr( C K). b Using similr rgument, we get c c Cr( C K). c b Hence Cr C K Cr A K.
Journl of Uncertin Systems, Vol.9, No., pp.6-74, 5 73 As pointed out in the previous cse, by the mximlity property of Credibility mesure when the rejection region C of size is constructed by tking the union of two or more regions belonging to F nd /or F, the inequlity Cr C K Cr C K continues to hold good. Thus we hve proved (b) of the theorem. Proceeding s in the cse of Theorem 4., for the cse of left shift tringulr credibility distributions, the best credibility rejection region cn be developed esily. The region identified by the membership rtio criterion nd its optiml property re stted in the theorem 4. without proof. Theorem: 4. If Tri(, b, c) is left shift of Tri(, b, c), then for testing the null credibility hypothesis H : ~ Tri(, b, c) ginst the lterntive credibility hypothesis K : ~ Tri(, b, c), () membership rtio criterion credibility rejection region of level (b) the credibility rejection region C x x b is given by C x x b nd is the best credibility rejection region of level. Illustrtion It is believed tht the menstrul cycle length of young women is lunr cycle length. However, it is suspected tht the chnge in life style nd living conditions hve ffected the menstrul cycle length in group of women. A gynecologist believes tht the cycle length of women in the group under study hs tringulr credibility distributions 9,3,33 Tri 6, 8,3 which supports the lunr cycle length Tri insted of tringulr credibility distribution belief. In order to choose proper set of vlues it is decided to tret this s credibility hypotheses testing problem H : ~ Tri 6, 8,3 K : ~ Tri 9,3,33 with significnce level where the null nd lterntive hypotheses re nd.5. Here we hve 6, b 8, c 3, 9, b 3 nd c 33. For the given dt we note tht the distribution K : ~ Tri 9,3,33 H : ~ Tri 6,8,3. c c b 9.8. mentioned under is right shift of Further, Hence by Theorem 4., we get the membership rtio criterion test s Reject the null credibility hypothesis if the observed vlue of the fuzzy vrible is n element of the set x x 9.8. Hence, the best credibility rejection region for the given testing problem is C x x 9.8. For exmple, if it is observed tht the menstrul cycle length of young womn is 3 then we reject the null credibility hypothesis nd ccept the lterntive credibility hypothesis with level.5. Tht is, we conclude tht the belief of the gynecologist cn be ccepted with level.5. Comprison of Best Credibility Region with Other Region with Sme Levels In the bove exmple, the power of the test defined by the best credibility criticl region under the lterntive credibility hypothesis is 9.8 9.8 Cr C K.8. 3 9 4 A x x 6. x x 3. Evidently the set A stisfies the condition Now consider the set A H.5 6. 3 Cr nd hence it is member of the regions of level.5. The power under lterntive of the set A x x x x is 3 9 Cr A K.75 3 9 4 Cr( x x 9.8 K).8. This clerly supports the use of the which is obviously smller when compred to test defined by the best credibility test. 5 Conclusion In the previous section of this pper, best credibility rejection regions hve been identified for testing credibility hypotheses relted to fuzzy tringulr distributions where the distribution under the lterntive credibility hypothesis is right (left) shift of the tringulr credibility distribution under null credibility hypothesis. Further, it is importnt to note tht the membership rtios corresponding to the cses 4, 7, 9 nd listed in Tble do not possess the monotonicity. The rtios re concve downwrds nd hence the credibility rejection regions identified by the
74 S. Smpth nd B. Rmy: Credibility Hypothesis Testing of Fuzzy Tringulr Distributions membership rtio criterion will be of the form x x, where nd re constnts selected such tht the credibility rejection region hs size. In such cses, the credibility rejection region of the given size cn be determined in more thn one wy. The following re vrious choices for nd yielding credibility rejection regions of size., (i) c, c (ii) x,, x (iii) c, x, x c (iv) where ( b ) nd c c ( c b ) re s identified in Theorem 4.. Existence of more thn one set of vlues for nd mkes the process of identifying the best credibility rejection region difficult one. Further, in cse 8 of Tble, the rtio hs more thn one turning point which complictes the process of identifying the rejection region. Hence further studies re needed for developing optiml tests in these cses. The tests developed in this pper re ment only for testing simple null credibility hypothesis ginst simple lterntive credibility hypothesis. Tht is, the underlying hypothesis is bout only one distribution. Hence, there is good scope for future work in developing test procedures when more thn one distribution is involved in the credibility hypothesis being considered. The entire pper is focused only on tringulr credibility distribution. The uthors re working on other types of credibility distributions like, exponentil, norml nd trpezoidl s well. References