ST 522-002: Weekly Review #10

Prepared by Chen-Yen Lin

Mar. 23, 2011

1. Concept Review:

• Bayes Estimator

• Asymptotic Evacuation

• Likelihood Ratio Test

2. Exercises

(a) Let X1, . . . , Xn be iid random variable from geometric distribution with successprobability p. Suppose the prior for p has a Beta(α, β), then

i. Derive the posterior density function of p given X1, . . . , Xn.

ii. Obtain the Bayes estimator with respect to the squared error loss function.

iii. Show that as n → ∞, the Bayes estimator above approaches to the MLE ofp.

(b) Let X1, . . . , Xn be iid random variables from U(0, θ).

i. Derive the MLE θ̂ and UMVUE θ̃ of θ.

ii. Show that n(θ − θ̂) d→ exp(θ)

iii. Compute the asymptotic relative efficiency of the MLE with respect to theUMVUE.

(c) Let X1, . . . , Xn be iid random variable from Beta(θ, 1). What is the likelihoodratio test of H0 : θ = θ0 versus H1 : θ 6= θ0.

3. Open for questions.

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