Mathematical Statistics
Fall 2024
Midterm 2
1. [20 pts] Let X1, . . . , Xn be IID, N(0, σ2) random variables. You may find it useful to recall that if X ∼ N(0, σ2), then Var(X2) = 2σ
4
.
(a) [5pts] Show that the maximum likelihood estimator of θ = σ
2
is ˆσ
2 = n
1 ∑n
j=1 Xj
2. (You may assume that any critical point is a global maximum, without checking second deriva-tives).
(b) [5pts] Find the Fisher information, I(σ2).
(c) [5 pts] σˆ
2
is an unbiased estimator of σ2
. What is the smallest possible variance among all possible unbiased estimators of σ2?
(d) [5pts] Assume that n = 10. Find an exact 95% confidence interval for σ2
of the form. σ2
∈ [b,∞) for some b. For partial credit you can find an asymptotic (as n → ∞) 95% confidence interval of the same form. instead (do not find both).
2. [10 pts] Suppose that Y is a uniform. random variable on (0,1) and Z is Bernoulli(1/θ). Y and Z are independent and X = Y Z. We are testing H0 ∶ θ = 2 versus H1 ∶ θ = 1. Consider the test that rejects H0 when X > 1/2.
(a) [5 pts] What is the probability of type 1 error?
(b) [5 pts] What is the probability of a type 2 error?
3. [10 pts] Suppose we ask 10 randomly chosen individuals what their favorite candy bar is among two candy bar options (A and B). The observed counts of reported favorites are given in the table below:
Category A B
Observed Count 4 6
We want to test whether these preferences are uniformly distributed across the categories. The null hypothesis H0 is that an individual prefers any one of the candy bars with equal probability (p0 = 1/2). Their alternative hypothesis is that some candy bars are preferred over others. Recall that the MLE of the true probability p that A is preferred is ˆp = 2/5.
(a) [5 pts] Write the likelihood ratio test statistic Λ for this test and data. Use the numbers in the table, but an answer with fractions and powers is fine.
(b) [5 pts] Determine a value c so that rejection when {Λ < c} results in a test with approximately 0.05 significance level.