A Bayesian Approch To Fuzzy Hypothesis Testing For Weibull Distribution

With the deepening of scientific research, the relationship needed to be studiedis more and more complex. In order to describe realistic phenomenon precisely,many kinds of new branch of mathematics generated and developed, so far, themathematical model which is used to deal with the reality has been divided into threecategories: one is certainty mathematical model, another is random mathematicalmodel, and the third one is fuzzy mathematical model. The meaning of the describedthings is determinate, which is the common characteristics of two foregoing typesof model. The cornerstone of their existence–Set Theory, satisfies complementarylaw which is the abstract of either-or clear concept, however much situation weencountered in the real world is ambiguous, couldn't be defined accurately, it's thethird kind mathematical model. Fuzzy set re?ected those ambiguity properly,andit doesn't satisfy complementary law.Hypothesis testing is an important element in statics, in recent years a lot ofscholars introduce fuzziness to hypothesis testing from di?erent aspects. Somebodyhas considered the problem of testing the ordinary(crisp) statistical hypotheses whenthe observations provide not exact but rather fuzzy information, somebody hasconsidered the problem of hypotheses testing when data are ordinary(crisp) and thehypothesis are fuzzy. In this paper, we selected the latter method and discuss theBayesian approach to fuzzy hypotheses testing of Weibull distribution.We give the following lemma:Lemma 1 In the casion of step-by-step fixed failure number test , invertedgamma distribution is the conjugate prior distribution ofθ. Let IGa(a,b) be theconjugate prior distribution ofθ, then the posterior distribution ofθis IGa(a ,b ),and a = a + m,b = b + mLemma 2 In the casion of fixed failure number test , inverted gamma dis-tribution is the conjugate prior distribution ofθ. Let IGa(a,b) be the conju-gate prior distribution ofθ, then the posterior distribution ofθis IGa(a ,b ), andLemma 3 In the casion of fixed failure number test, X1≤X2···≤Xr arethe ordered statistic of Weibull distribution, and (X1 ?ν)βare in dependence with Lemma 4 In the casion of fixed failure number test, the Je?reys prior ofθπj(θ)∝1θ,θ> 0.Regarding multiple fuzzy hypothesis testing (H0,H1,···,Hk), we use the fol-lowing loss functiongi(θ) are arbitrary non-negative function,i = 0,1,···,k. Specifically speaking, theselection of gi(θ)depends on the sensitivity degree of decision-makers to accept Hiwrongly.According toTheorem Regarding multiple fuzzy hypothesis testing (H0,H1,···,Hk), ifthe loss function,x is the sample of observation, the Bayesian rule for testing is that we accept Hi,if and only ifSpecially,if gi≡C(constant), i = 0,1,···,k,then (1) is equivalent toWe get the following deductions:Deduction 1 In the casion of step-by-step fixed failure number test, if theconjugate prior distributionπ(θ) = IGa(a,b), then(1),(2)are equivalent to Deduction 2 In the casion of fixed failure number test, if conjugate priordistributionπ(θ) = IGa(a,b), then(1),(2)are equivalent toDeduction 3 In the casion of fixed failure number test ,if Je?reys prior...