STAT-S 420 Final Exam Practice
Chapters 7-9 & 15 (and Chapters 1-6 as review)
1 Let Xbe a binomial RV with n= 4 and parameter p.
We wish to perform a likelihood ratio test of the following hypotheses:
H0:p= 0.6 vs Ha:p= 0.7.
We observe one realization of X.
1. Suppose we observe X= 3. What is the P-value?
2. Suppose a test rejects for X= 3 and X= 4 only. What is the power of this test?
2 Let X1, . . . , Xn
iid
∼Exponential(λ)with the following pdf
f(x;λ) = λe−λx, λ > 0, x > 0.
1. Find E(X) and E(X2) and two corresponding method of moments estimators for λ.
2. Find the maximum likelihood estimator for λ. Is the MLE of λan unbiased estimator?
3. Find the Fisher information for λ,I(λ).
4. What is the asymptotic distribution of the MLE of λ?
5. Find an approximate 90% symmetric confidence interval for λ.
6. If one goal is to estimate τ=P(X1>1).
(a) What is the MLE of τ, ˆτ1?
(b) Consider another estimator, ˆτ2, of τ, where ˆτ2=1
nPn
i=1 I(Xi>1).Use the Central
Limit Theorem to find the asympotic distribution of ˆτ2.
(c) By Delta Method, one can obtain the asymptotic distribution of ˆτ1is Nτ , e−2λλ2
n.
Given this information, how would you compare ˆτ1and ˆτ2? Which of these two estimators
would you choose to use?
7. One type of objective prior is Jeffreys’ prior, which has density proportional to pI(λ), the
square root of the Fisher information for λ.
(a) Use this information to write the Jeffreys’ prior as below:
fJ(λ)∝
(b) Assign Jeffreys’ prior to λand derive the posterior distribution of λ.
(c) Find the Bayesian estimator and a 90% credible interval.
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