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The questions for the first part of an m.phil. Exam in statistical theory. The exam covers topics such as symmetric matrices, maximum likelihood estimation, and linear regression models. Students are required to attempt no more than three questions out of four, which carry equal weight. The document also includes instructions for stationery requirements and special instructions for the invigilator.
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Monday, 1 June, 2009 1:30 pm to 3:30 pm
Attempt no more than THREE questions.
There are FOUR questions in total.
The questions carry equal weight.
Cover sheet None
Treasury Tag
Script paper
You may not start to read the questions
printed on the subsequent pages until
instructed to do so by the Invigilator.
2
1 (a) Let A be a symmetric n × n matrix of rank n − p, and let B be a p × n matrix of
rank p. Suppose that BA = 0. You are given that we can write A = LL
T , where L is an
n × (n − p) matrix of rank n − p. Show that L
T L is positive definite, and by considering
T L(L
T L)
− 1 , show that BL = 0.
Let Y ∼ Nn(μ, σ
2 I). Find the distribution of the random vector Z =
T Y
, and
deduce that BY and Y
T AY are independent.
(b) Consider the linear model Y = Xβ + ǫ, where X is an n × p design matrix of full
rank p (< n), β ∈ R
p is an unknown vector of regression coefficients and ǫ ∼ N n
(0, σ
2 I).
Write down expressions for the maximum likelihood estimators
β and ˆσ
2 , and also write
down their marginal distributions.
Using the result of part (a), or otherwise, show carefully that
β and ˆσ
2 are indepen-
dent.
2 For n = 1, 2 ,.. ., let Y = (Y 1
n
T have independent and identically distributed
components with density f (·; θ) for some θ ∈ Θ ⊆ R
d on a sample space Y. Let
θ 0
denote the true value of θ. Assume Θ is closed and bounded and that for each
y ∈ Y, the likelihood L(θ; y) is a continuous function of θ. Suppose that, for each n, the
maximum likelihood estimator
θn based on Y 1 ,... , Yn is unique, the model is identifiable
and E θ 0
sup θ∈Θ
| log f (Y 1
; θ)|
Prove that
θ n
is consistent; i.e.
θ n
p
→ θ 0
as n → ∞.
[You may use the fact that Eθ 0
{log f (Y 1 ; θ)} is a continuous function of θ and
sup
θ∈Θ
n
n ∑
i=
log f (Yi; θ) − Eθ 0
{log f (Y 1 ; θ)}
p
as n → ∞].
Under regularity conditions that you need not specify, state a result about the
asymptotic normality of
θ n
Now let Y 1 ,... , Yn be independent U [0, θ] random variables. Find
θn and prove
from first principles that
θ n
is consistent. By considering the distribution function of
n(θ −
θn)/θ, show that
θn = θ + op(n
− 1 / 2 ) as n → ∞. Give one regularity condition for
your asymptotic normality result that is violated in this case.
Statistical Theory
