Math Stats Midterm 2 Review Problems

1. Let X1, . . . , Xn be i.i.d. samples from a Rayleigh distribution with parameter θ > 0

(a) Find the method of moments estimate of θ.

(b) Find the mle of θ.

(c) Find the asymptotic variance of the mle.

(d) Construct an approximate 95% confidence interval for the mle.

2. Let Xi ∼ bin (ni , pi), for i = 1, . . . , m, be independent.

(a) Set up hypotheses that all the pi are equal.

(b) Derive a likelihood ratio test for the hypotheses.

(c) What is the large sample distribution of the test statistic?

3. Let X1, . . . , Xn be a random sample from an exponential distribution with the density function f(x | θ) = θ exp[−θx].

(a) Derive a likelihood ratio test of H0 : θ = θ0 versus HA : θ ≠ θ0.

(b) Show that the rejection region is of the form. {¯X exp [−θ0¯X] ≤ c}.

4. Let P have a uniform. distribution on [0, 1], and conditional on P = p, let X have a Bernoulli distribution with parameter p. Find the conditional distribution of P given X = x.

5. Show that the gamma distribution is a conjugate prior for the exponential distribution.