Bayes Estimators For Parameters Of The Two-parameters Weibull Model

The Weibull model is one of the most common survival time distribution and usually used in reliability.It is necessary to research it,in this text we mostly consider the two-parameter Weibull model and give the Bayesian estimators of two parameters. It is the density of Weibull distribution below. Difinition:Suppose the random variable X has the density f ( x ), where bothβandηare unknown parameters andβ> 0andη> 0.Supposeθ=ηβin order to simplify the later problem,this does not alter the number of parameters and the Weibull distribution density function.Then the density function becomes f ( x ) = (βx (β?1)/θ)exp( ? xβ/θ), x >0. The main purpose of this paper is the estimation of the two parameters of this model. For Bayesian estimation, the main two questions are the choices of a prior distribution and loss function,when the selection of loss function is the square loss,the theorem as follows establishes.Theorem:In the occasions of the vector parametersθ= (θ1 ,...,θp)',for a given prior distributionπ(θ) and the quadratic loss function L (θ,δ) = (δ?θ) 'Q(δ?θ),where is a positive definite matrix,the Bayessian estimators of Qθis the mean vector of the posterior distributionπ(θ| x),that isSuppose the distribution function of random variable X is the two-parameters Weibull model. X 1 , X 2,..., X n is the i.i.d. samples,the parametersθandβare unknown.Assumimg that the unknown parametersθandβhave the prior distributions IG (α,λ) and g (β),they are independent of each other,then their joint density function is By the above theorem,we can see that when the loss function is the quadratic loss,the Bayesian estimators of the unknown parametersθandβare the the mean vector of the posteriorr distribution,through the integral calculation we getFor smaller n, the above integral using numerical solution can be calculated, if the n value is larger, it will give the estimated value calculated difficult. In order to avoid complex calculation, this paper parameter estimation was improved based on Bayesian methods , we know that when the number of samples is large enough , the empirical distribution function and distribution functions should not differ too much. This difference is not too much can use distance to measure.We use following distance criteria in our analysis.At two-parameter Weibull distribution,Assumpting theβis known, if no prior information is available about the parameterη,its prior density isπ(η) = 1/η, letθ=ηβ,so that in the square loss, the Bayesian Estimation of the parameterθis its posterior mean, that is,We putθinto 2 1, it is about the function ofβ, our aim is to derive the estimated value ofβmaking A2to a minimum. This can estimate two parameters of the Weibull distribution. No matter what the form of prior distribution, we can pass through the above methods to estimate both parameters.