What are the conditional posterior distributions write a

Question

Question 1 -

Consider a set of data relating two score tests, LSAT and GPA, at a sample of 15 American law schools. Of interest is the correlation θ = cor(lsat, gpa) between these measurements and the variance ratio ψ = var(lsat)/var(gpa).

list(lsat = c(576, 635, 558, 578, 666, 580, 555, 661, 651, 605, 653, 575, 545, 572, 594),

gpa = c(3.39, 3.30, 2.81, 3.03, 3.55, 3.07, 3.00, 3.43, 3.36, 3.13, 3.12, 2.74, 2.76, 2.88, 2.96),

n = 15)

1. Write a function in R called CI.cor that returns 95% bootstrap confidence intervals for the correlation parameter θ using the basic bootstrap interval method and the percentile interval method. Your answer should contain the R function CI.var.cor.

2. Write a function in R called CI.var.ratio. that returns 95% bootstrap confidence intervals for the variance ratio ψ using the basic bootstrap interval method and the percentile interval method.

Your answer should contain the R function CI.var.ratio.

Question 3 -

Consider the following dataset

list(t = c(94.3, 15.7, 62.9, 126, 5.24, 31.4, 1.05, 1.05, 2.1, 10.5),

x = c(5, 1, 5, 14, 3, 19, 1, 1, 4, 22),

n = 10)

on observed failures x_i of n = 10 power plant pumps. Here ti denotes the length of operation time of the pump (in 1000s of hours). The number of failures X_i is assumed to follow a Poisson distribution X_i| θ_i ∼ Poisson(θ_it_i). Consider the hierarchical model

θ_i| α, β ∼ Gamma(α, β)

where

α ∼ Exp(λ), β ∼ Gamma(γ, δ), λ = 1 and γ = δ = 0.01.

1. What are the conditional posterior distributions of

- θ_i| θ_-i, α, β, x;

- β | θ, α, x;

- α | θ, β, x?

Write your answer in a section called Question 3 (a) of the file.

2. Write a function in R called mcmc.pumps that implements a general McMC algorithm with components θ, β, α where θ and β are updated from the posterior conditional distribution and log α is updated via a random walk Metropolis step with normal increments. The function should return samples from the posterior distribution of θ, β and α as well as an estimate of the probability of accepting α.

Your answer must contain the code of the function mcmc.pumps.

3. Write a function in R called predictive.pumps that returns estimates of the predictive distribution of failures for the i-th pump for a given length of operation time t^∗_i, using Monte Carlo integration.

Your answer must contain the code of the function predictive.pumps.

Question 4 -

Let x₁, . . . , x_n be a sample of independent and identically distributed observations assumed to have been generated from a t-distribution with ν degrees of freedom located at µ, i.e.,

f(x | µ) ∝ {1 + 1/ν(x - µ)²}^-(ν+1)/2                                                 x ∈ R.                     (1)

Assume ν = 10 and suppose µ ∼ N(0, 1).

1. Write down, up to proportionality constant, the posterior µ | x. Is this a recognisable density? Write a function in R called mcmc.t that implements an McMC algorithm for evaluating the posterior distribution of µ where the updating is done using random walk Metropolis with a normal candidate generator.

Your answer should contain the posterior density of µ | x in a section called Question 4 (1) of the =matriculationA2math= file and the function mcmc.t separately in and scriptA2.R

2. Using the fact that the t-distribution is a scale mixture of normals, the sampling model in (1) can alternatively be represented as

x_i| µ, z_i ∼ N(µ, 1/z_i)

where z_i are a priori independent of µ and

f(z_i) ∝ z_i^(ν/2)-1 exp{-(ν/2) z_i}.

Using this representation together with the prior distribution µ ∼ N(0, 1), write down, up to proportionality constant, the conditional posterior densities of µ and of each z_i , i = 1, . . . , n and identify the distributions corresponding to each of these densities. Write a function in R called mcmc.gibbs that implements a Gibbs sampling algorithm for computing the posterior distribution of µ.

Your answer should contain the conditional posterior densities of µ and each of the z_is in the file in a section called Question 4 (2) of the =matriculationA2math= file and the function mcmc.gibbs separately in and scriptA2.R

3. Let x^∗ denote a future observable. Write a function in R called predictive.t that returns estimates of the predictive distribution x^∗| x using Monte Carlo integration. Your answer must contain the code of the function predictive.t.

4. Write a function in R called predictive.quantile that estimates the pth quantile of the predictive distribution

x_p : Pr (x^∗ ≤ x_p | x) = p

and use it to construct a 95% predictive interval.

Your answer must contain the code of the function predictive.quantile as well the code used to obtain the 95% predictive interval.

I Need help with questions 1, 3, 4.

Attachment:- Assignment File.rar