Testing Statistical Hypotheses with Fuzzy Data

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1 Iteratioal Joural of Statistics ad Systems ISS Volume 6, umber 4 (), pp Research Idia Publicatios Testig Statistical Hypotheses with Fuzzy Data E. Baloui Jamkhaeh ad A. adi Ghara Departmet of Statistics, Qaemshahr Brach, Islamic Azad iversity, Qaemshahr, Ira e_baloui8@yahoo.com Mazadara Medical Sciece iversity, Ira statistic.adi@gmail.com Abstract I traditioal hypotheses test, the sample data ad hypotheses are crispess. I this paper we cosider hypotheses test for mea i ormal populatios with fuzzy data whe variace of populatio is ukow. I this fuzzy test, we will make a fuzzy decisio for rejectio or acceptace ull hypothesis. This fuzzy decisio shows a degree of acceptability ad degree of rejectio of the ull hypothesis. Our fuzzy test is well-defied sice if the data are precise, we get a classical statistical test. Fially we give some example. Mathematics Subject Classificatio: 6F3. Keywords: Hypothesis testig, Fuzzy data, Fuzzy test. Itroductio Test hypotheses cocerig some populatio parameters are a importat part of statistical aalysis. I traditioal mea testig, the observatio of sample is geerally assumed to be a crisp value ad to satisfy some relevat assumptios. Sice, the real data are usually vague, difficulties arise whe testig are traditioal methods. I traditioal testig, a statistical test leads to the biary decisio, but decisio i our fuzzy test is a fuzzy set. The fuzzy set theory is a powerful ad kow tool i modelig ad aalysis of imprecise ad subjective situatios where exact aalysis is either difficult or impossible. We are goig to apply fuzzy sets theory to the statistical test by cosiderig fuzzy umber for vague observatio. Some previous studies have preseted approaches to fuzzy testig for a wider rage of applicatios. Casals et al. [3], So et al. [], Grzegorzewski [5], ad Wu, H. C. [5] cosidered statistical hypothesis with vague data. Delgado et al. [4], Saade ad Schwarzlader [9], Saade

2 44 E. Baloui Jamkhaeh ad A. adi Ghara [8], ad Moteegro et al. [7] cosidered fuzzy hypothesis testig with vague data. Kruse ad Meyer [6] discussed the problem of cofidece iterval uder fuzzy radom variables. Wataabe ad Imaizumi [] discussed a fuzzy hypothesis ad proposed a method for testig a fuzzy hypothesis with radom data. Arold [] coducted statistical tests uder fuzzy costraits o the type I ad type II errors. Taheri ad Behboodia (999) formulated the problem of testig fuzzy hypotheses whe the observatios are crisp. Buckley (5) proposed aother approach for testig hypotheses, i which he used a set of cofidece itervals to produce a fuzzy statistical test. Testig statistical hypotheses for compare meas with vague data was studied by Baloui Jamkhaeh ad adi Ghara [3]. This work proposes a fuzzy testig of statistical hypotheses usig fuzzy observatio for mea i ormal populatios whe variace of populatio is ukow. Tests ad cofidece itervals are dual otios. Hece, we have makig fuzzy test based o fuzzy cofidece iterval due to Kruse ad Meyer [7]. This paper is orgaized as follows. We provide some defiitio i the ext sectio. Fuzzy tests are itroduced i sectio 3. I sectio 4 we itroduce P-values. Fially, i sectio 5 we deal some of umerical examples. Prelimiaries Suppose a radom experimet is described as usual by a probability space ( Ω, Α, Ρ) where Ω is a set of all possible outcomes of the experimet, Α is a σ algebra of subset of Ω ad Ρ is a probability measure. Defiitio : [] The fuzzy subset of real lie IR, with the membership fuctio μ : IR [,] is a fuzzy umber if ad oly if (a) is ormal (b) is fuzzy covex (c) μ is upper semi cotiuous (d) supp ( ) is bouded. Defiitio : [] The -cut of a fuzzy umber is a o-fuzzy set defied as [ ] = { x IR; μ ( x) }. Hece we have [ ] = [ ], where = if { x IR; μ ( x) }, { μ } = sup x IR; ( x). The set of all these fuzzy umbers is deoted by F (IR). Defiitio 3: [5] A mappig : Ω F( IR) is called a fuzzy radom variable, if it satisfies the followig properties: (a) { ; [,] } is a set represetatio of (ω) for all ω Ω (b) For each [,] both = = if, = = sup are usual real valued radom variables o ( Ω, Α, Ρ). Thus, a fuzzy radom variable is cosidered as a perceptio of a ukow usual radom variable V : Ω IR called a origial of. et χ deote a set of all possible origials of. We ca cosider a fuzzy set o χ, with a membership fuctio ν : χ [,] give as follows:

3 Testig Statistical Hypotheses with Fuzzy Data 443 { μ ( ( ω } ν V ) = if ω )), ω Ω () ( ( ) V I fact this iterpretig is used whe the origi of radom variable is defied as classic, but the observatio of radom variable is imprecise. Similarly -dimesioal fuzzy radom sample,,..., may be treated as a fuzzy perceptio of the usual radom sample V, V,..., V (wherev, V,..., V are idepedet ad idetically distributed crisp radom variables). A set χ of all possible origials of that radom sample is, i fact, a fuzzy set with membership fuctio: ν V,..., V ) = mi = if μ ω ( V ), ω Ω () i ( i,.., ( ) i { } Fuzzy tests et,,..., deote a fuzzy sample, i.e., fuzzy perceptio of the usual radom samplev, V,..., V, from the populatio with the distributio ormal with parameter ( μ, σ ), where σ is ukow. Defiitio 4: [5] A fuctio : ( F( IR)) F{, } ϕ is called a fuzzy test for the hypothesis H, o the sigificace level α (, ), if sup [,] P ω Ω, ϕ ( ) { } H (3) { } α where ϕ is the cut set of ϕ ( ). Theorem : et Π defie Π = [ Π, Π ] for all α [, ] as t α t α,, Π = s(,..., ) ad Π = + s (,..., ) ( ) ( ) where S (,..., ) = Max { ( ) + ( ) A i i i A i {,..., }\ A [ ( ) ( ) ] i i A {,..., }} i A i {,... }\ A + (4) The fuctio ϕ : ( F( IR)) F{, } with followig cuts

4 444 E. Baloui Jamkhaeh ad A. adi Ghara i = = (5) {,}, μ Π ( Π ( )) φ (,, i,,..., ) {}, μ Π ( ( Π)) {}, μ (( Π ) Π ), μ ( Π ( Π)) is a fuzzy test for H : μ = μ agaist H : μ μ o the sigificace levelα. Proof: Kruse ad Meyer [7] itroduced the otio fuzzy cofidece iterval for the parameterθ. Their costructio ca be described by the followig lemma. emma : et [ Π, Π ] deote two-sided symmetrical cofidece iterval for μ o the level α. et i, i =,.., deote fuzzy radom sample which are fuzzy perceptio of the crisp radom samplesv i, i =,..,. The a covex ad ormal fuzzy set Π with the membership fuctio μ x ) sup I ( x), [,] (6) { } Π ( = [ Π, Π ] where Π ad Π are defied by (4), is called a fuzzy cofidece iterval for μ o the cofidece level α, i.e. { ω Ω, Λ ( μ) Π } α, [,] P (7) where Π = [ Π, Π ], ad Π = Π ( ( ω),...,, α ), Π = Π ( ( ω),...,, α ) ad Λ (μ ) is a fuzzy perceptio of the parameter do give by (8 ). Proof: For the proof of lemma we refer the reader to [6]. et us take ay [, ]. The we have P{ ω Ω, ϕ ( ) {} H }= P{ ω Ω, ϕ ( ) { Φ,{}} H } { ω Ω, ϕ ( ( )) {{},{,}} } { ω Ω, ϕ ( ( )) {{},{,}} } = P ω H = P ω H { ω Ω, μ Π ( ω } = P ) H (8) If the ull hypothesis H : μ = μ is fulfilled the μ ( Λ μ) for all [, ]. By P ω Ω d Π. Thus, from (8) we get usig from (7) we get { } α sup, { ω Ω, ϕ ( ) { H } α P (9) [,] }

5 Testig Statistical Hypotheses with Fuzzy Data 445 which completes the proof. By Theorem give above we get a membership fuctio of the fuzzy test cosidered above for the hypothesis H : μ = μ agaist H : μ μ, as μ ϕ t) μ ( μ ) I ( t) + μ ( μ ) I ( t) = μ ( μ ) I ( t) + ( μ ( μ )) I ( t), t {, } () ( = Π {} Π {} Π {} Π {} It is easily see that our fuzzy set, lead to the fuzzy decisio. We get the set fuzzy { η ϕ =, η }, where η = μ Π ( μ ) [,], (i) if η = the we should accept H, (ii) ifη = the we should reject H, (iii) otherwise, we should accept (with degree η ) or reject (with degree η ) the hypothesis H. Fig shows this idea. Figure : ( η ) { η ϕ x =, }. Similarly, we ca obtai fuzzy tests for oe-sided hypothesis of H : μ μ agaist H : μ > μ ad H : μ μ agaist H : μ < μ, sufficiet that Π i the Theorem ad emma will substitute with Π ad Π, respectively, that Π = [ Π, ], Π = ad Π = [, Π ], Π = + t, α s ( ) t, α s ( ) (,..., ), () (,..., ), () where Π ad Π are the oe-sided fuzzy cofidece iterval for μ o the cofidece level α.

6 446 E. Baloui Jamkhaeh ad A. adi Ghara Remark : If the data are precise, our fuzzy tests reduce to covetioal tests. P-values I the traditioal approach to hypotheses testig the P-value (critical level) is the smallest (largest) sigificace level at which the ull hypothesis would be rejected (accepted) for the give observatio. O the other had, P-value is the largest sigificace level that we will accept ull hypothesis. That is reject H P value < α, accept H P value α The more P-value idicates the more cofirmatio of ull hypotheses by observatio. However, the smallest sigificace level for reject H with η = ad largest sigificace level for accept H with η = are ot equal i the case of fuzzy test. Therefore, defie these critical levels as follows: Defiitio 5: A real umber P R valueis called a critical level of reject H, if P R value = if{ α (,), η = }, the P R valuedeotes the smallest sigificace level of the fuzzy test for rejectig H with the degree of membership equal to oe. Defiitio 6: A real umber P A valueis called a critical level of acceptig H, if P A value = sup{ α (,), η = }, the P A valuedeotes the largest sigificace level of the fuzzy test for rejectig H with the degree of membership equal to zero. I geerally for everyα RH ; if PR value < α RH with deg ree η decisio = ; if PA value α < PR value AH with deg ree η AH ; if α PA value Remark : The ucertaity degree of data is oe of the factors that iterval betwee critical levels depeds o it. The less ucertaity value results i less iterval, ad if data gets crisp values, critical level of reject ad critical level of acceptig will become equal, which that P-value is i classic state.

7 Testig Statistical Hypotheses with Fuzzy Data 447 umerical example A car maufacturer compay is producig less cosumptio cars ad is claimed that, this ew geeratio of these cars, will cosume equal to the maximum 4.8 liter i per kilometers. But it seems that the customer of these cars is ot accepted this claim. I order to ivestigate this, we chose a forth samples of these car, s owers ad we collected followig data that are as triagular fuzzy umbers. Suppose the radom variable has distributio ormal ad the variace of both populatios are kow ad equal with oe. We ivestigate the claim o cosumptio of the petrol i sigificat level.. The ( 4,5,6), ( 4.5,5,5.5), (5,6,7), (6,6.5,7) H μ 4.8, H : μ 4.8, : > After simple calculatio we obtai a fuzzy sample average with membership fuctio x 4.875, x < μ = x, 5.65 x < By usig (4) ad () we get S = t, Π = [ Π, ], Π α = (3) ( ) where t, α = t3,.9 =.64, = 4, ad substitutig μ ito () ad usig (5) we get ϕ ( x ) = {, }, that is, we should accept (with degree.85) or reject (with degree.49) the hypothesis H, also may be iterpreted as approximately H should be accepted. Also, the critical levels are: P R value =.468, P A value =.59, therefore RH ; if.468 < α RH with deg ree η decisio = ; if.59 α <.468 AH with deg ree η AH ; if α.59

8 448 E. Baloui Jamkhaeh ad A. adi Ghara Figure : ϕ ( x ) = {, } Coclusios I this paper, the fuzzy cofidece iterval attributed to Kruse ad Meyer is used to itroducig a fuzzy test suitable for a situatio havig both probabilistic ucertaity (radomess) ad possibilistic ucertaity (impreciseess). For fixedα, we have make a fuzzy decisio based o fuzzy set ad for everyα, we have also preseted a rule decisio makig by usig critical values rejectio ad acceptig. Our defiitio makes covetioal P-value a special case of critical values of fuzzy tests. Thus our fuzzy theory is a geeralizatio of customary theory. Refereces [] B.F. Arold, Statistical tests optimally meetig certai fuzzy requiremets o the power fuctio ad o the sample size, Fuzzy Set ad System, 75 (995), [] E. Baloui Jamkhaeh, A. ade Ghara, Statistical hypotheses for compare meas with vague data, Iteratioal Mathematical Forum, 5 (), [3] J.J. Bukley, Fuzzy statistics: hypothesis testig, Soft Comput, 9 (5), [4] R. Casals, M.A. Gill, A ote o the operativeess of eyma-pearso tests with fuzzy iformatio, Fuzzy Set ad System, 3 (989), [5] M. Delgado, J.. Verdegay ad M.A. Vila, Testig fuzzy hypotheses: A Bayesia approach, i: M.M. Gupta et al.(eds.), Approximate Reasoig i Expert Systems, Elsevier, Amsterdam, (985), [6] P.Grzegorzewski, Testig statistical hypotheses with vague data, Fuzzy Set ad System, (), 5-5.

9 Testig Statistical Hypotheses with Fuzzy Data 449 [7] R. Kruse, K.D. Meyer, Cofidece itervals for the parameter of a liguistic radom variable, i: J. Kasprzyk, M. Fedriz(Eds.), Combiig Fuzzy Imprecisio with Probabilistic certaity i Decisio Makig, Spriger, Berli, (988), 3-3. [8] M. Moteegro, M.R. Casals, M.A. ubiao ad M.A. Gil, Two-sample hypothesis tests of meas of a fuzzy radom variable, Iformatio Scieces, 33 (), [9] J. Saade, Extesio of fuzzy hypothesis testig with hybrid data, Fuzzy Set ad System, 63 (994), [] J. Saade, H. Schwarzlader, Fuzzy hypothesis testig with hybrid data, Fuzzy Set ad System, 35 (99), [] J. CH. So, I. Sog, H.Y. Kim, A fuzzy decisio problem based o the geeralized eyma-pearso criteria, Fuzzy Set ad System, 47 (99), [] S. M., Taheri, J. Behboodia, Fuzzy hypotheses testig with fuzzy data: A Bayesia Approach,.R. Pal amd M. Sugeo (Eds.): AFSS, Physica- Verlag Heidelberg, (), [3] S.M. Taheri, J. Behboodia, eyma-pearso emma for fuzzy hypotheses testig, Metrika, 49 (999), 3-7. [4]. Wataabe, T. Imaizumi, A fuzzy statistical test of fuzzy hypotheses, Fuzzy Set ad System, 53(993), [5] H.C., Wu, Aalysis of variace for fuzzy data, Iteratioal Joural of System Sciece, 38(3) (7),

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