STA130B Problem 1 （2）
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Problem 1 (15 points)
Let Xl , ..., Xn 
Rayleigh (β) with pdf
f (xlβ) = βxe− _ 
、 (x > 0 ; β > 0)
NOTE: For this problem, denote the αth-quantile of the standard normal distribution as za , i.e. P (Z < za ) = α where Z ~ N (0, 1).
1. (7 pt) Compute the Fisher information I(β)
2. (3 pt) Find the asymptotic distribution of the MLE of β .
3. (5 pt) Give an approximate 90% confidence interval for β based on the asymptotic distri- bution of the MLE.
Problem 2 (20 points)
Let Xl , ..., Xn 
Rayleigh (β) with pdf
f (xlβ) = βxe− _ 
、 (x > 0 ; β > 0)
NOTE: The pdf of the Gamma (α, λ) distribution is
xa − l e −Az x > 0; α > 0; λ > 0
1. (6 pt) Find a sufficient statistic for β .
2. (10 pt) Let the prior distribution of β be Gamma(α, λ) where α and λ are known. Find the posterior distribution of β given an iid sample Xl , ..., Xn .
3. (4 pt) Compute the posterior mean.
Problem 3 (15 points)
Let Xl , ..., Xn 
Gumbel (µ) with pdf
f (xlµ) = exp(_x + µ _ e −z＋u) = e(−z＋u −e −z＋u) (x e R ; µ e R)
For µl > µ0 , suppose we are testing the simple hypotheses
H0 : µ = µ0 vs. Hl : µ = µl
1. (3 pt) State the Neyman-Pearson Lemma. Define all quantities carefully and state the result precisely.
2. (2 pt) For a fixed significance level α, what type of test will have the most power?
3. (6 pt) Show that a likelihood ratio test which rejects for Λ < C is equivalent to a test n
which rejects for （ e −Xi < K .
i=l
4. (4 pt) What are the two types of errors that can arise from the above hypothesis testing. Describe what these errors would be in the context of testing H0 vs. Hl above.
2022-02-23