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STA130B Problem 1 (2)
发布时间:2022-02-23
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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.