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TAKE HOME EXAM
发布时间：2024-05-27
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TAKE HOME EXAM
PART A (30 MARKS)
Are the following statements TRUE or FALSE. Justfy your answer. (3 marks per statement. 1 mark for true/false, 2 marks for justification).
1. The events T1 , T2 , T3 are independent if and only if all of the following hold
P(T1 ∩ T2 ) = P(T1 )P(T2 ), P(T1 ∩ T3 ) = P(T1 )P(T3 ), P(T2 ∩ T3 ) = P(T2 )P(T3 ).
2. Given a random sample X1 , X2 ,..., Xn from a distribution with finite mean µ and strictly positive but finite variance σ 2 ,
3. Given X1 ∼ Normal(µ,σ2 ) and X2 ∼ Normal(µ,σ2 ) we have
4. When doing a hypothesis test we use the normal distribution instead of the t distribution because the normal distribution often gives better control of the type I error probability in small samples.
5. The Wald test can be applied to more situations than the likelihood ratio test.
6. The OLS estimator is an example of a method of moments estimator.
7. Given a random variable X with moment generating function mX (t),
8. If Y |X = x ∼ Normal(2x,x) and X ∼ Uniform(0, then V[Y] = 6/5.
9. If ∂E[θ(ˆ)]/∂θ = 1 then θ(ˆ) is unbiased.
10. The 1 − α confidence interval is unique.
PART B (70 MARKS)
1. Consider
where δ > 0, 0 < γ < 1 and 0 < δγ < 1.
(a) Verify that this is a proper density function. (3 marks).
(b) Find the distribution function FY (y). Explain carefully how it depends on γ .
Evaluate it at y = δ = γ = 0.5. (5 marks).
(c) Find the moment generating function mY (t). Evaluate it at t = δ = γ = 0.5. (4 marks).
(d) Find the expectation, median and mode of Y when δ = γ = 0.5. (5 marks).
2. Consider
(a) Show that we need c = 2(1 − β) and 0 ≤ β ≤ 2 for this to be a proper density function. Use these conditions for the rest of this question. (4 marks).
(b) Given a random sample X1 , X2 ,..., Xn , derive the method of moments estimator of β . (3 marks).
(c) Is the method of moments estimator of β consistent? Justify your answer. (3 marks).
(d) Given n = 50 and data verifying x = 7/12, perform a z-test of the null hypothesis that the random sample is drawn from a standard uniform distribution. The alternative should be the complement of the null. You should use α = 0.05. (10 marks).
(e) For n = 2, and given data x1 = 0.1, x2 = 0.4, find the maximum likelihood estimate of β . (5 marks).
3. Consider a random sample from Y ∼ Binomial(2, p) and the hypothesis test
H0 : p = 1/4, H1 : p ≠ 1/4
with α = 0.0625. You must use this value of α in all parts below.
(a) Calculate the probability function pY (y|H0 ). (3 marks).
(b) Derive the method of moments estimator of p. (3 marks).
(c) Derive the maximum likelihood estimator of p. (5 marks).
(d) For a random sample of size n = 100, find the test statistic and rejection region for the z-test. Would you reject H0 if the data satisfy y = 1? (7 marks).
(e) For a random sample of size n = 1, find a test statistic and a rejection region for the likelihood ratio test. Would you reject H0 if the data is y = 1? (10 marks).
