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这是一篇是用R语言建立参数模型并进行分析
STAT 440/840 – CM 761 – Assignment 4
问题一-25分
形状参数α> 0和比例参数θ> 0的威布尔分布由下式给出
（
θ，α）=（
α
θ
X
θ
α-1

（x /θ）
α
x≥0，
0 x <0， 使用γ=θ重新参数化模型 α。假设我们获得以下数据； set.seed（440） x = rweibull（24，形状= 1/2，比例= 64） （ γ，α）=（ αxα-1 γ -x α/γx≥0， 0 x <0， 注意：这里我们对γ=θ的推论感兴趣 α以α为条件。 a）[4分]显示γ的共轭先验是具有超参数的反伽马分布 由invΓ（λ，β）表示的（λ，β）。密度由下式给出 （ λ，β）=（β λ Γ（λ） （1 / x） λ+ 1 exp（-β/ x）x≥0， 0 x <0， b）[6分]使用以下每个先验绘制先验和后验，然后计算可信度 γ的间隔 i）λ= 1和β= 10之前的反伽马， ii）λ= 10且β= 1的先验反伽马，以及 iii）p（γ）= 1的不当先验 请注意，R包invgamma可能会有所帮助。 c) [4 Marks] Calculate a confidence interval using the log-likelihood ratio. d) [2 Marks] Compare and discuss the intervals generated in b) and d). e) In b) we can calculate the credible intervals exactly. Instead use MCMC to estimate the credible interval from b iii). i) [1 Mark] Construct a R function that generates a random walk MCMC algorithm to sample from the posterior. The input is the random walk length, the standard deviation for the random walk density σ, and the initial position. ii) [8 Marks] Run a random walk MCMC using σ = 0.1, 1, 5, 20, T = 104 iterations and starting state γ (0) = 8.9. Then for each MCMC provide traceplot, autocorrelation, summary table with the acceptance rate, estimate of the posterior mean, naive estimate of the credible interval, and a measure of mixing using 1 T X T t=1 (xt xt1) 2 and then comment on the results. Question Two – 16 Marks Here we are interested in Bayesian inference for α conditional on the scale scale parameter being known and equal to θ = 64. Use an improper of p(α) = 1 for α ≥ 0. set.seed(440) x = rweibull(n=24, shape=1/2, scale=64) a) [2 Mark] Write a R function that is proportional to the posterior on the log-scale. b) Write R functions that generates a MCMC sample from the posterior using the following the proposals or candidate densities; i) [2 Marks] follows a gamma with shape equal to 1 and rate equal to 2 ii) [2 Marks] follows a gamma with shape equal to 1 and rate equal to 1/Xt iii) [2 Marks] follows a N(Xt, 0.1), and iv) [2 Marks] follows a N(α, b O(αb) 1 ). i.e. an Gaussian independence sampler with mean equal to the MLE and variance equal to the inverse observed information. v) [6 Marks] Generate a sample path of length 10, 000 using each of the above MCMCs and the initial value equal to 1/2. Summarize and comment on the traceplots, autocorrelations, histogram of the posterior from each MCMC. Use a table to summarize the acceptance ratio, the estimate of the posterior average & variance and the mixing criteria, 1 T PT t=1 (xt xt1) 2 .