RESEARCH SCHOOL OF FINANCE, ACTUARIAL
STUDIES AND STATISTICS
STAT2001/STAT2013/STAT6013 - Introductory
Mathematical Statistics (for Actuarial
Studies)/Principles of Mathematical Statistics (for
Actuarial Studies)
Assignment 2 Semester 1 (2023)
ˆ Your solutions to the assignment should be submitted via wattle by 11:59 pm, Friday
19 May (Week 11).
ˆ The assignment is out of 20 and is worth 15% of your overall course mark. Problem
1 is worth 6 marks in total and Problems 2 and 3 are each worth 7 marks. Questions
within a problem may not be given equal weight in marking.
ˆ The assignment is to be done alone. Marks may be deducted for any copying.
ˆ Always show your full reasoning when writing your solution to each assignment ques-
tion. Unjustified correct answers may not receive full marks.
1
Problem 1
Let Y1 ∽ N(µ1, 1), Y2 ∽ N(µ2, 1), and Y3 ∽ N(µ3, 1) and also assume that these three
random variables are mutually independent. The observed sample values are y1 = 2, y2 = 0
and y3 = −1, respectively. We are interested in doing inference on the parameter
θ = 0.1µ1 + 0.2µ2 + 0.7µ3
(a weighted average of the population means).
(a) Derive an exact lower range 90% confidence interval for θ. Calculate this interval for
the given sample.
(b) Now suppose that µ2 = 0 and µ3 = 0 are known. Derive an exact upper range 90%
confidence interval for θ taking this additional information into account. Calculate this
interval for the given sample.
Problem 2
Suppose that Y1, Y2, Y3 denote a random (independent) sample of size n = 3 from a distri-
bution with parameter λ defined by the following probability density function:
f(y) =
{
λyλ−1, 0 < y < 1, λ > 0
0, elsewhere.
The observed values of the sample are: y1 = 0.1, y2 = 0.2 and y3 = 0.3.
(a) Find the method of moments estimate of λ.
(b) Find the maximum likelihood estimate of λ.
(c) LetWi = − log (Yi), i = 1, 2, 3, where the function log () denotes the natural logarithm.
Find the distribution of each Wi.
(d) Derive the distribution of U = 2λ(W1 +W2 +W3).
(e) Using (d), or otherwise, find an exact central 95% confidence interval for λ and evaluate
it for the sample.
2
Problem 3
Suppose that Y1, Y2, Y3 denote a random (independent) sample from an exponential distri-
bution with probability density function
f(y) =
{
1
θ
e−y/θ, y > 0
0 otherwise.
Consider the following five estimators of θ:
θˆ1 = Y3,
θˆ2 =
Y1+Y2
2
,
θˆ3 =
Y1+2Y2
3
,
θˆ4 = min(Y1, Y2, Y3),
θˆ5 =
Y1+Y2+Y3
3
.
(a) Calculate the bias of each of the above 5 estimators of θ.
(b) Calculate the variance of each of the above 5 estimators of θ. Among the unbiased
estimators, which has the smallest variance?
(c) Consider the estimator θˆ6 = cθˆ1 + (1 − c)θˆ2, where θˆ1 and θˆ2 are defined previously.
How should the constant c be chosen in order to minimise the variance of θˆ6?
(d) Calculate the efficiency of θˆ6 relative to θˆ1.