Stats 426 Homework 4
统计课业代做代写 Instructions: Solve the problems in the spaces provided and save as a single PDF. Then upload the PDF to Canvas Assignments by the due date.
Instructions: Solve the problems in the spaces provided and save as a single PDF. Then upload the PDF to Canvas Assignments by the due date. The recommended procedure is to download and print the homework. Fill in your solutions. Then scan the document and upload to Canvas Assignments. If this is not feasible, you may solve the problems on your paper, scan your solutions, then upload to Canvas. Neatness and presentation are important.
Late homework not accepted. Show all work. Total points: 30
1) 统计课业代做代写
Consider a random sample of pairs (Xi , Yi), i = 1, . . . , n from a bivariate population with joint pdf
f(x, y|θ) = exp{−θx − y/θ} x > 0 , y > 0 , θ > 0
Find the MLE of θ. (3 points)
2)
Let Y1, . . . , Ynbe iid N(µ, σ2 ). Let µ ∼ N(δ, τ2 ) and treat σ2 , δ, and τ2 as fixed and known. Find the posterior distribution p(µ|y). This distribution will depend on σ2 , δ, and τ2 . Calculations are tedious here. Use the hints given in lecture video recorded 6/8 and follow through. (4 points)
3) 统计课业代做代写
Let Y1, . . . , Yn be iid N(µ, σ2 ). Let σ2 ∼ Inv-Gamma(α/2, ν/2). The notation Inv-Gamma refers to the inverse gamma distribution. Treat µ, α, and ν as fixed and known. Find the
posterior distribution p(σ2 |y) which will depend on µ, α, and ν. (3 points)
b) Find a 95% confidence interval for σ2. (2 points)
5) 统计课业代做代写
Suppose Y ∼ N(µ, 1) is a single observation from the normal density and consider the test
H0 : µ = 0
H1 : µ = 1
a) Find the most powerfult test of size α = 0. (3 points)
b) For the test in part a) find β, the probability of a Type II error. (2 points)
c) Find the test that minimizes the sum of the two errorprobabilities (α + β). (3 points)
7)
Let X1, . . . , Xn be iid Poisson(θ). To conduct the Generalized Likelihood Ratio Test (GLRT) for testing
H0 : θ = θ0
H1 : θ ≠ θ0
calculate the likelihood ratio statistic −2 log λ and explain how you would conduct this test at α = 0.05. Your test statistic will depend on n, θ0, and some function of the data X1, . . . , Xn. (3 points)
8) 统计课业代做代写
A simulation exercise in R: Demonstration of the meaning of a confidence interval. In this simulation experiment, 1,000 data sets are generated from N(5, 1) with n = 16. Each data set is a column in a 16×1000 matrix. We then calculate a 95% confidence interval for µ using the standard formula for each column in the matrix. Since we know that µ = 5, approximately 950, or 95% of the confidence intervals should trap the true value of µ = 5 in the population. Use the following code in R and report the proportion of confidence intervals that trap the true value. Is it close to 0.95? (1 point)
set.seed(12345)
y <- rnorm(16000,mean=5,sd=1) #generate 16,000 N(5,1)
mat <- matrix(y,nrow=16,ncol=1000) #put in 16x1000 matrix
cifun <- function(x) { #create function to
t.test(x)$conf.int } #calculate CIs
conf <- apply(mat,2,cifun) #apply function to columns of matrix
sum(ifelse(conf[1,]<5 & conf[2,]>5,1,0))/1000 #proportion of CIs
#that trap mu=5
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