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STAT2013/STAT6013-stat代写

时间：2023-05-19

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)

Your solutions to the assignment should be submitted via wattle by 11:59 pm, Friday

19 May (Week 11).

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.

The assignment is to be done alone. Marks may be deducted for any copying.

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.

STUDIES AND STATISTICS

STAT2001/STAT2013/STAT6013 - Introductory

Mathematical Statistics (for Actuarial

Studies)/Principles of Mathematical Statistics (for

Actuarial Studies)

Assignment 2 Semester 1 (2023)

Your solutions to the assignment should be submitted via wattle by 11:59 pm, Friday

19 May (Week 11).

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.

The assignment is to be done alone. Marks may be deducted for any copying.

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.