Bayesian Estimation For Parameters Of Several Distribution

In recent years,Bayesian theory,practical applications in many fields have a wide range of applications,especially in the social sciences and economic fields are mostly, Bayesian statistics are statistics of today's scientific research hot spots.First,Bayesian methods of estimation and selection of prior distribution.Bayesian theory is applied to the results of parameter estimation of Bayesian estimation methods will be formed, Bayesian estimation method is to use decision theory 1 research parameter problem, the main idea is:1.When the overall distribution of the unknown parametersθare known, The joint distribution density of the sample p(χ1,χ2,…-,χn|θ)or abbreviated as p(χ|θ).2.Based on previous knowledge of the parameters toθdetermine the prior distributionπ(θ).3. The use of conditional distribution density p(χ|θ) and prior distributionπ(θ), Bayesian formula can be obtained using the posterior distribution of the densityζ(θ|χ1,χ2,…,χn) recorded asζ(θ|χ).4. Use the a posteriori distribution densityζ(θ|χ) of the parametersθto make the inference.Now on the determination of the prior distribution have been some successful methods,this paper is the more commonly used conjugate prior distribution for the distribution method selected.Second,empirical Bayesian estimation methods.Empirical Bayesian method is regarded as EB methods.The basic idea of this method are as follows:Need to estimate the parameters X corresponding to the current sampleθ, X1,X2,…,Xn,it is known the history of the sample, but Xi (i=1,2,…,n) corresponds to the distribution of the parametersθi(i=1,2,…,n)are unknown, but all follow the same prior distributionπ(θ). Empirical Bayes estimates of the role is to find a function to depend on X1,X2,…,Xn,X and estimated parametersθ, and makeπ(θ) known when close to Bayesian estimation.Third, LINEX loss function:L(θ,θ)=L(Δ)=beαΔ-cΔ-b,Δ=θ-θmeanθestimation error. a,c≠0,b＞0, we can infer L(0)=0,loss function in accordance with the meaning if L(Δ) is min whenΔ=0, parameter a,b,c should be ab=c.This type of argument is the loss function a,b,respectively, the scale parameter and shape parameter.Fourth, double exponential distribution scale Bayesian Estimation.Double-exponential distribution is a two-exponential distribution,θ,σrespectively the location parameters and scale parameters,this loss function for scale parameter applied to empirical Bayes estimation.Fifth, Weibull distributed parameter estimation of loss function and risk function Bayes estimates.In statistical decision-making, d=δ(χ) would be estimated as unknown parametersθ, assuming r is estimated that w(θ,d) losses caused by the estimated loss functionL(θ;δ,r) estimation error and r accuracy is estimated to be combined. For the loss function L(θ;δ,r), when r(x) is Eθ[r(x)]≥R(θ,δ)= Eθ[w(θ,δ)],r(x) can be called a conservative estimate.As the decision function of the estimated d=δ(χ) average loss Eθ[r(x)]≥R(θ,δ)=Eθ[w(θ,δ)]is a function about the unknown parametersβα,therefore,it must be estimated.As R(βα,δ)is the average of w(βα,δ),so R(βα,δ)and w(βα,δ)are same.In the square loss,Sixth.Truncated Exponential Distribution under Bayes estimated. Theorem 4.1 Set the overall obey exponential distribution,B(a,b,λ)isλprior distribution,under M1-(n,无,r)(1)If the loss function,taking the square loss,then the parametersλestimated(2)If the loss functi.n is L(λ1,λ)=λ1-2λ1(λ1-λ)2(-∞＜l＜+∞)and Theorem 4.2 Set the overall obey exponential distribution,B(a,b,λ) isλprior distribution,under M2-(n,有,r)(1)If the loss function,taking the square loss,then the parametersλestimated(2) If the loss function is and r+l>0 ,λis...