MATH5905 代写、R 语言编程代写

日期：2023-03-05 11:09

MATH5905 Term One 2023 Assignment One Statistical Inference

University of New South Wales

School of Mathematics and Statistics

MATH5905 Statistical Inference

Term One 2023

Assignment One

Given: Friday 24 February 2023 Due date: Sunday 12 March 2023

Instructions: This assignment is to be completed collaboratively by a group of at most 3

students. (If you are unable to join a group you can still present it as an individual assignment.)

The same mark will be awarded to each student within the group, unless I have good reasons

to believe that a group member did not contribute appropriately. This assignment must be

submitted no later than 11:59 pm on Sunday, 12 March 2023. The first page of the submitted

PDF should be this page. Only one of the group members should submit the PDF file on

Moodle, with the names, student numbers and signatures of the other students in the group

clearly indicated on this page.

I/We declare that this assessment item is my/our own work, except where acknowledged, and

has not been submitted for academic credit elsewhere. I/We acknowledge that the assessor of

this item may, for the purpose of assessing this item reproduce this assessment item and provide

a copy to another member of the University; and/or communicate a copy of this assessment

item to a plagiarism checking service (which may then retain a copy of the assessment item on

its database for the purpose of future plagiarism checking). I/We certify that I/We have read

and understood the University Rules in respect of Student Academic Misconduct.

Name Student No. Signature Date

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MATH5905 Term One 2023 Assignment One Statistical Inference

Problem One

i) It is known that for independent Poisson distributed random variables X1 ‵ Poisson(竹1)

and X2 ‵ Poisson(竹2) it holds that

X1 +X2 ‵ Poisson(竹1 + 竹2).

Show that ifXi ‵ Poisson(竹i), i = 1, 2, . . . , k are independent then the conditional distribution

of X1 given X1 + X2 + . . . Xk is Binomial and determine the parameters of this Binomial

distribution.

ii) Suppose that the X and Y are components of continuous random vector with a density

fX,Y (x, y) = cxy

2, 0 < x < y, 0 < y < 2 (and zero else). Here c is a normalizing constant.

a) Show that c = 516 .

b) Find the marginal density fX(x) and FX(x).

c) Find the marginal density fY (y) and FY (y).

d) Find the conditional density fY |X(y|x).

e) Find the conditional expected value a(x) = E(Y |X = x).

Make sure that you show your working and do not forget to always specify the support of

the respective distribution.

Problem Two

Let X and Y be independent uniformly distributed in (0, 1) random variables. Further, let

U(X,Y ) = X + Y, V (X,Y ) = Y ?X be a transformation.

a) Sketch the support S(X,Y ) of the random vector X,Y in R

2.

b) Sketch the support S(U,V ) of the random vector (U, V ) in R

2.

c) Determine the Jacobian of the transformation

d) Determine the density of the random vector (U, V )

Justify completely each step.

Problem Three

You are going to the races and want to decide whether or not to bet on the horse Thunderbolt.

You want to apply decision theory to make a decision. You use the information from two

independent horse-racing experts. Data X represents the number of experts recommending

you to bet on Thunderbolt (due, of course, to their belief that this horse will win the race).

If you decide not to bet and Thunderbolt does not win, or when you bet and Thunderbolt

wins the race, nothing is lost. If Thunderbolt does not win and you have decided to bet on

him, your subjective judgment is that your loss would be four times higher than the cost of not

betting but the Thunderbolt does win (as you will have missed other opportunities to invest

your money).

You have investigated the history of correct winning bets for the two horse-racing experts and

it is as follows. When Thunderbolt had been a winner, both experts have correctly predicted

his win with probability 5/6 (and a loss with a probability 1/6). When Thunderbolt has not

won a race, both experts had a prediction of 3/5 for him to win. You listen to both experts

and make your decision based on the data X.

a) There are two possible actions in the action space A = {a0, a1} where action a0 is to bet

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MATH5905 Term One 2023 Assignment One Statistical Inference

and action a1 is not to bet. There are two states of nature 成 = {牟0, 牟1} where 牟0 = 0

represents ※Thunderbolt winning§ and 牟1 = 1 represents ※Thunderbolt not winning§.

Define the appropriate loss function L(牟, a) for this problem.

b) Compute the probability mass function (pmf) for X under both states of nature.

c) The complete list of all the non-randomized decisions rules D based on x is given by:

d1 d2 d3 d4 d5 d6 d7 d8

x = 0 a0 a1 a0 a1 a0 a1 a0 a1

x = 1 a0 a0 a1 a1 a0 a0 a1 a1

x = 2 a0 a0 a0 a0 a1 a1 a1 a1

For the set of non-randomized decision rules D compute the corresponding risk points.

d) Find the minimax rule(s) among the non-randomized rules in D.

e) Sketch the risk set of all randomized rules D generated by the set of rules in D. You

might want to use R (or your favorite programming language) to make this sketch more

precise.

f) Suppose there are two decisions rules d and d∩. The decision d strictly dominates d∩ if

R(牟, d) ≒ R(牟, d∩) for all values of 牟 and R(牟, d) < (牟, d∩) for at least one value 牟. Hence,

given a choice between d and d∩ we would always prefer to use d. Any decision rules

which is strictly dominated by another decisions rule (as d∩ is in the above) is said to be

inadmissible. Correspondingly, if a decision rule d is not strictly dominated by any other

decision rule then it is admissible. Show on the risk plot the set of randomized decisions

rules that correspond to the admissible decision rules.

g) Find the risk point of the minimax rule in the set of randomized decision rules D and

determine its minimax risk. Compare the two minimax risks of the minimax decision

rule in D and in D. Comment.

h) Define the minimax rule in the set D in terms of rules in D.

i) For which prior on {牟1, 牟2} is the minimax rule in the set D also a Bayes rule?

j) Prior to listening to the two experts, you believe that Thunderbolt will win the race with

probability 1/2. Find the Bayes rule and the Bayes risk with respect to your prior.

k) For a small positive ? = 0.1, illustrate on the risk set the risk points of all rules which

are ?-minimax.

Problem Four

In a Bayesian estimation problem, we sample n i.i.d. observations X = (X1, X2, . . . , Xn)

from a population with conditional distribution of each single observation being the geometric

distribution

fX1|成(x|牟) = 牟x(1? 牟), x = 0, 1, 2, . . . ; 0 < 牟 < 1.

The parameter 牟 is considered as random in the interval 成 = (0, 1).

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MATH5905 Term One 2023 Assignment One Statistical Inference

i) If 牟 is interpreted as a probability of success in a single trial in a success-failure scheme, give

an interpretation of the conditional distribution of the random variable X1(You may

use the integrate function in R or another numerical integration routine from your favourite

programming package to answer the question.)