闪电代写 -代写CS作业_CS代写_Finance代写_Economic代写_Statistics代写_代码代做_IT代写_加急帮助

Hello, dear friend, you can consult us at any time if you have any questions, add WeChat: daixieit

STATS 310/732

2022

Assignment 4

In general, you may use any results given in the course book or lectures to solve an assignment problem (unless the problem is about establishing the result itself).

1. [18 + (4) marks] Let X be a random variable with density

f (x; θ) = 2θx exp(-θx2 ), x > 0, θ > 0.

We wish to use a single value X = x to test the null hypothesis

H0 : θ = 1

against the alternative hypothesis

H1 : θ = 2.

Denote by C = {x : x < aα } the critical region of a test at the signiﬁcance level of α = 0.05.

(a) [2 marks] Show that the cdf of X is given by F (x) = 1 - exp(-θx2 ), x > 0.

(b) [2 marks] What is the sample space 5, the parameter space Θ and the null parameter

space Θ0 of the test?

(d) [2 marks] Compute the power of the test.

(e) [2 marks] Compute the probability of Type II error.

(f) [2 marks] Show that the test is the most powerful at level α .

(g) [2 marks] Show that the test is also the uniformly most powerful (UMP) test when

the alternative hypothesis is replaced with H1 : θ > 1.

(h) [2 marks] Show that there exists no UMP test when the alternative hypothesis is

replaced with H1 : θ 1.

(i) [2 marks] Extend the above result to the more general situation where X1 , . . . , Xn are an iid sample of the distribution. Show that the UMP test for testing H0 : θ = 1 against H1 : θ > 1 exists and has the critical region of the form

Cα = {x : xi(2) < bα }.

i=1

(j) [(4) marks] BONUS: Compute the value of bα , when n = 10 and α = 0.05. (Hint: What is the distribution of Xi(2) and Xi(2) under H0 ?)

2. [12 marks] Let X1 , . . . , X5 have a multinomial distribution with parameters p1 , . . . , p5 and joint probability function

f (x, p) = p1(x)à p2(x)女p3(x)gp4(x)』p5(x)扌 , pi > 0, xi > 0, i = 1, . . . , 5

where x = (x1 , . . . , x5 )T , p = (p1 , . . . , p5 )T , n = xT 1, and pT 1 = 1.

(a) [3 marks] Show that the maximum likelihood estimator of p is given by x/n.

(b) [3 marks] Show that under the assumption p1 = p2 = p3 and p4 = p5 , the maximum

likelihood estimator 0 of p is given by

╱ , , , , 、T .

H0 : p1 = p2 = p3 = p4 = p5

against the alternative hypothesis

H1 : p1 = p2 = p3 p4 = p5 .

Let x = (171, 189, 200, 226, 214)T . Compute the p-value of the test.

(d) [3 marks] Consider using - log(LR) to test the goodness-of-ﬁt of the model p1 = p2 = p3 p4 = p5 . Write down H0 and H1 in a form speciﬁc to this problem and compute the p-value of the test using the same x as in part (c).

3. [20 marks] An experimenter observes independent observations

Y11 , Y12 , . . . , Y1n

Y21 , Y22 , . . . , Y2n

where E(Y1j) = α1 + β1 xj and E(Y2j) = α2 + β2 xj + γzj , xj and zj being the jth values of numerical explanatory variables with sample means 0 and zero empirical correlation, i.e. x = 0, z = 0, xT z = 0. Denote by eij = Yij -E(Yij ) the errors, and assume eij N(0, σ2 ) for all i and j. Note that σ 2 is common to all errors.

Further, let yi = (Yi1, Yi2, . . . , Yin )T and 卡i = (ei1, ei2, . . . , ein )T , for i = 1, 2, x = (x1 , x2 , . . . , xn )T , and z = (z1 , . . . , zn )T . Also, 0n and 1n are vectors of length n with elements of 0, and 1, respectively.

(a) [4 marks] Show that this model can be expressed as

y = ╱y(y)2(1)、 = ╱ 0(1)n(n)

╱α 1 、

z(0n)、 + ╱ 卡(卡)2(1)、 .

．γ ．

(b) [4 marks] Show the least squares estimator of ← = (α1 , β1 , α2 , β2 , γ)T is

← = ╱Y1 , , Y2 , , 、T

where Yi = n1 Yij .

╱．

Cov(←) = σ 2 ．(．) 0

． 0

0 (xT x)≥ 1

(xT x)≥ 1 0

0(0) 、．

0 ．(．) .

0 ．

(d) [4 marks] Verify that the estimate of σ 2 is

j(n)=1 {Y1j - Y1 - 1 (xj - x)}2 + j(n)=1 {Y2j - Y2 - 2 (xj - x) - (zj - z)}2

2n - 5 .

(e) [4 marks] If one would like to ﬁnd the least squares estimate under the assumption

that α 1 = α2 and β1 = β2 , one can rewrite the model using only three parameters, e.g., ← ≥ = (α, β, γ)T , in the form

y = X ≥ ← ≥ + 卡,

where 卡 = (卡1(T) , 卡2(T))T . Write down the new design matrix X ≥ .

** Extra Questions for STATS 732 Only **

4. [20 marks] The table below gives the number of failures xi and the length of operation time ti (in 1000s of hours), i = 1, . . . , 10, of ten power plant pumps. We assume that the number of failures are independent Poisson random variables, i.e.

Xi |θ, ti ～ Poisson(θti ), i = 1, . . . , 10.

with probability density function

f (xi |θ, ti ) = 会 e ≥θt会 for xi = 0, 1, 2, . . . .

where θ is the overall failure rate. As prior distribution for θ, assume the Gamma(α, β) distribution which is the conjugate distribution, with probability density function

f (θ) = θα ≥ 1 e ≥βθ

for θ > 0, α > 0, β > 0.

The Gamma(α, β) distribution has mean and variance β女(α) .

Pump ti xi

94.50 15.70 62.90 126.00 5.24 31.40 1.05 1.05 2.10 10.50

Answer questions a)- f) for general α , β and x1 , . . . , xn , t1 , . . . , tn .

(a) [2 marks] Derive the posterior distribution of θ|x1 , . . . , xn .

(b) [1 marks] What is the Bayes estimator, dB (x1 , . . . , xn ), under the squared error loss

function?

(d) [2 marks] Show that the Bayes estimator in part b) is a weighted average of prior mean and MLE.

(e) [4 marks] Calculate the risks of dB (X1 , . . . , Xn ) and dMLE (X1 , . . . , Xn ) under the

squared error loss function.

(f) [3 marks] Calculate the Bayes risk of dB and dMLE .

(g) [2 marks] If we have prior information that θ is expected to be about 0.8 with a

standard deviation of 0.4, what is the corresponding Gamma(α, β) prior distribution?

(h) [2 marks] With the values for α and β in part g), and the observations given in the

table above, ﬁnd the posterior probability that θ s 0.15.

(i) [2 marks] With the values for α and β in part g), and the observations given in the table above, give a 95% central posterior credible interval for θ .