STATS 310/732辅导、讲解R编程设计、辅导R语言、讲解data 辅导Database|讲解留学生Processing

日期：2020-06-14 10:01

STATS 310/732, 2020 Assignment 4 Due: 3pm Wed 10 June
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). Some questions require
finding probability values or quantiles of a distribution. You may use R or any other statistical
software to obtain them.
1. [18 marks] Suppose X ～ Gamma(2, θ). 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.
(Note: Our scale parameter θ is the rate argument in the R functions for the Gamma
distribution.)
(a) [2 marks] What is the sample space S, the parameter space Θ and the null parameter
space Θ0 of the test?
(b) [2 marks] Compute aα.
(c) [2 marks] Compute the power of the test.
(d) [2 marks] Compute the probability of Type II error.
(e) [2 marks] Show that the test is the most powerful at level α.
(f) [2 marks] Show that the test is also the uniformly most powerful (UMP) test when
the alternative hypothesis is replaced with H1: θ > 1.
(g) [2 marks] Show that there exists no UMP test when the alternative hypothesis is
replaced with H1: θ 6= 1.
(h) [2 marks] Extend the above result to the more general situation where X1, . . . , Xn
iid～
Gamma(2, θ). 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 : x < bα}, where x = n
?1 Pn
i=1 xi
.
(i) [2 marks] Compute the value of bα, when n = 10 and α = 0.05.
(Hint: What is the distribution of Pn
i=1 Xi under H0?)
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2. [12 marks] Let x ～ Multinomial(n1, p) and y ～ Multinomial(n2, q) independently,
where p = (p1, p2, p3)T
and q = (q1, q2, q3)T. Denote θ = (pT, qT)T
and its MLE by θb.
(a) [3 marks] Show that θb = (xT /n1, yT /n2)T.
(b) [3 marks] Find the MLE θb under the restriction p1 = q1.
(c) [6 marks] For both n1 and n2 large, what are the approximate null distributions
of ?2 log(LR) for the following tests? You do not need to derive expressions for
?2 log(LR).
(i) H0 : p1 = q1 against H1 : p1 6= q1;
(ii) H0 : p = q against H1 : p 6= q.
(iii) H0 : p1 = p2 = p3 = q1 = q2 = q3 against H1 : At least one is different.
3. [20 marks] An experimenter obtains observations yij of independent random variables
Yij , for i = 1, 2 and j = 1, . . . , n, where
E(Yij ) = αi + β(xj ? x),
xj being the jth value of a numerical explanatory variable with sample mean x. Denote
by ?ij = Yij ? E(Yij ) the errors, and assume ?ij
iid～ N(0, σ2
) for all i and j. Note that σ
2
is common to all errors.
Further, denote yi = (yi1, . . . , yin)
T and ?i = (?i1, . . . , ?in)
T
, for i = 1, 2, and z = (x1 ?
x, . . . , xn ? x)
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
(b) [4 marks] Show the least squares estimator of β = (α1, α2, β).
(c) [4 marks] Show that the covariance matrix of βb is.
(e) [4 marks] If one would like to find the least squares estimate under the assumption
α1 = α2,
** Extra Questions for STATS 732 Only **
4. [6 marks] Show that the Bayes estimator of θ under loss
l(θ, θ b ) = |θb? θ|
is the median (any median, if more than one) of the posterior density π(θ|x).
5. [14 marks] Assume that X1, . . . , Xn
iid～ N(0, θ?1
) and the prior distribution of θ is
Gamma(k, λ).
(a) [4 marks] Show that the posterior distribution of θ is Gamma
For the following parts, let k = 5, λ = 3, n = 10 and y = 1.
(c) [2 marks] Compute the Bayes estimate of θ under the squared error loss.
(d) [2 marks] Compute the central 95% credible interval for θ.
(e) [2 marks] Compute the narrowest 95% credible interval for θ.
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