Homework 04

Instructions

For this homework, you may choose to work with another classmate enrolled in the course. Choose only one member of the pair to submit at SPOC. Otherwise, submit your solutions in the usual way.
Submit your solutions in PDF format to our SPOC website. The file name should be of the form IDNumber1_IDNumber2_HW04.pdf, where IDNumber1 and IDNumber2 are the corresponding university ID numbers of the pair who worked on this homework. If you worked alone, then use the file name IDNumber_HW04.pdf. The file size should be less than 20 MB.
The deadline is on May 21, 2023 at 1700 Beijing time. This deadline is also indicated in the submission form at SPOC. Late submissions will not be accepted, regardless of the validity of the reason. You may still choose to have your solutions checked, but it gets zero credit if submitted late.
Make sure to click on the Submit button after you have uploaded your assignment, so that it does not stay in Draft mode.
You are allowed to discuss the exercises with other classmates, but you have to write up your own solutions. This means you are not allowed to directly copy or even look at solutions from any other source.
Some of the solutions may be chosen for presentation in class. If you or your partner are unable to explain your answers in class, then both of you will get zero credit for this homework.
If you do not provide the table in Exercise A, you automatically will get zero credit for this homework. If you do provide the table in Exercise A, at least you get a 1 point bonus right away.

Exercise A: ID numbers and contributions

Create a table which lists every classmate involved in the development of your solutions. Only put down ID numbers. You will also indicate the contributions made by each member of the pair. You will also indicate what you have discussed with other classmates, along with their ID numbers. If you work alone, then there is no need to be explicit about the contributions. But if you discussed with another classmates, make sure to indicate what you have discussed with them.

The table has the following form:

ID number	Write YES if ID number belongs to submitter.	Contribution

Here is an example, which is not very complete, but conveys the general idea.

ID number	Write YES if ID number belongs to submitter.	Contribution
ID number 1	YES	Worked on R code.
ID number 2	YES	Wrote up the solutions.
ID number 3		Discussed Exercise B Item 3.

If you are working alone, your table might look like:

ID number	Write YES if ID number belongs to submitter.	Contribution
ID number 1	YES
ID number 2		Discussed Exercise B Item 3.
ID number 3		Discussed Exercise A Item 4.

Exercise B: Practice with moment generating functions (mgf)

In this exercise, you will be looking into a different argument to show that \[W_1=\frac{\left(n-1\right)S^2}{\sigma^2}\sim \chi^2_{n-1}.\] Assume that \(Y_1,\ldots,Y_n\) are IID \(N\left(\mu,\sigma^2\right)\), where both \(\mu\) and \(\sigma^2\) are unknown.

(1 point) Show first that \[\sum_{i=1}^n \left(\frac{Y_i-\mu}{\sigma}\right)^2= \frac{\left(n-1\right)S^2}{\sigma^2} + \left(\frac{\overline{Y}-\mu}{\sigma/\sqrt{n}}\right)^2 \]
(2 points) Let \(W=\displaystyle\sum_{i=1}^n \left(\frac{Y_i-\mu}{\sigma}\right)^2\). What is the distribution of \(W_1\)? Write down your argument to support your answer.
(2 points) Let \(W_2=\displaystyle\left(\frac{\overline{Y}-\mu}{\sigma/\sqrt{n}}\right)^2\). What is the distribution of \(W_2\)? Write down your argument to support your answer.
(3 points) Use LM Theorem 4.6.5 to compute the mgfs for \(W\) and \(W_2\). Are there restrictions on the domain of these functions?
(3 points) Can you apply LM Theorem 3.12.3 directly to the mgf for \(W\) and use it to indirectly recover the mgf for \(W_1\)? If it is possible to apply directly, write down your argument to obtain the mgf for \(W_1\). If it is not possible to apply directly, write down what conditions have to be satisfied first in order to use LM Theorem 3.12.3. Do you think these conditions are satisfied? After that, write down your argument to obtain the mgf for \(W_1\).
(2 points) Now that you have obtained the mgf for \(W_1\) in Item 5, can you identify the distribution of \(W_1\)? Cite the theorem in LM that you have to apply to obtain your finding.
(2 points) Use the mgf in Item 5 to obtain the mean and variance of \(W_1\).
(2 points) Use Item 7 to find the mean and variance of \(S^2\).
(4 points) I already showed in class how to work out the details of maximum likelihood estimation of \(\mu\) and \(\sigma^2\) in the IID \(N\left(\mu,\sigma^2\right)\). Revisit those details and determine whether \(S^2\) is an efficient estimator of \(\sigma^2\).

Exercise C: Confidence intervals for \(\sigma^2\)

You will be using a Monte Carlo simulation to evaluate the performance of two different \(100\left(1-\alpha\right)\%\) confidence intervals for \(\sigma^2\).

(3 points) Use the finding in Item 6 in Exercise B to construct a 90% conservative confidence interval for \(\sigma^2\) based on Chebyshev’s inequality. Make sure to provide expressions for \(L\) and \(U\) and show that \(\mathbb{P}\left(L\leq \sigma^2 \leq U\right) \geq 0.9\) is satisfied.
(3 points) Exercise B made you go through the argument to show that \(\displaystyle\frac{\left(n-1\right)S^2}{\sigma^2}\sim \chi^2_{n-1}\). Use this to construct an exact 90% confidence interval for \(\sigma^2\). Make sure to provide expressions for \(L\) and \(U\) and show that \(\mathbb{P}\left(L\leq \sigma^2 \leq U\right) = 0.9\) is satisfied.
(3 points) Use the intervals constructed in Items 1 and 2 to write R code to conduct a Monte Carlo simulation where you draw \(n=10\) random samples from \(N\left(1,4\right)\) and then you calculate the two confidence intervals obtained in Items 1 and 2. Compare the coverage properties and the average lengths of the two intervals. Show your code and discuss your findings. You may find the command qchisq() useful.
(3 points) Modify the R code in Item 3 using the same condition that the population mean is 1 and the population variance is 4 but the shape of the distribution is not normal. You are free to choose a distribution that satisfies the conditions mentioned, provided that you can draw random samples from that distribution in R. Compare the coverage properties and the average lengths of the two intervals. Show your code and discuss your findings. You may find the command qchisq() useful.
(1 point) Will increasing sample size from \(n=10\) to \(n=1000\) help in Item 4? Discuss your findings.

Exercise D: The connection between \(\chi^2\) and \(F\) distributions in large samples

Let \(V \sim \chi^2_m\), \(U\sim\chi^2_n\), and \(V\) and \(U\) are independent. From LM Theorem 7.3.3, you already know that \(F=\left(V/m\right)/\left(U/n\right)\) has an \(F\) distribution with \(m\) numerator degrees of freedom and \(n\) denominator degrees of freedom. Your task is to apply the asymptotic tools to show that \[m\times F =m\times \dfrac{V/m}{U/n}= V \times \frac{1}{(U/n)} \overset{d}{\to} \chi^2_m.\]

(2 points) It is possible to express \(U/n\) as a sample average of IID random variables. Show how to do this. What is the common distribution of these random variables?
(2 points) Apply the law of large numbers to \(U/n\) and show that it converges to a known constant. Make sure to determine the value of this constant and how you got this value.
(3 points) Show how to apply Theorem 1 to obtain the desired result. Make sure to write a complete argument and cite which parts of Theorem 1 you have used.

Exercise E: Confidence intervals for \(\sigma^2_X/\sigma^2_Y\)

Suppose you have \(X_1,X_2,\ldots,X_m\) are IID \(N\left(\mu_X,\sigma^2_X\right)\) and \(Y_1,Y_2,\ldots,Y_n\) are IID \(N\left(\mu_Y,\sigma^2_Y\right)\). Assume that these two sets of random samples are independent of each other.

(2 points) What is the distribution of \(\dfrac{\left(m-1\right)S^2_X}{\sigma^2_X}\)? Show your work and cite the results needed to support your argument.
(2 points) What is the distribution of \(\dfrac{\left(n-1\right)S^2_Y}{\sigma^2_Y}\)? Show your work and cite the results needed to support your argument.
(2 points) Why are \(\dfrac{\left(m-1\right)S^2_X}{\sigma^2_X}\) and \(\dfrac{\left(n-1\right)S^2_Y}{\sigma^2_Y}\) independent random variables? Show your work and cite the results needed to support your argument.
(3 points) Using the results in Items 1 to 3, propose a pivotal quantity which can be used to construct a \(100\left(1-\alpha\right)\%\) confidence interval for \(\sigma^2_X/\sigma^2_Y\). After proposing this quantity, construct the a \(100\left(1-\alpha\right)\%\) confidence interval. Provide the appropriate \(L\) and \(U\).