这个作业是完成统计推断相关的数学问题

AMA529 STATISTICAL INFERENCE
Assignment 2

1. Let X1, . . . , Xn be a random sample from the N(µ, σ2
) distribution, where µ is known, and
σ
2 > 0 is unknown.
(a) Calculate E((X1 − µ)
2
).
(b) Show that n
−1 Pn
i=1(Xi−µ)
2
is the best unbiased estimator of σ
2
. You may use without
proof the fact that Pn
i=1(Xi − µ)
2
is a complete statistic.
2. Let X1, . . . , Xn be a random sample from the Gamma(α, θ) distribution with density
f(x; θ) = θ
α
(α − 1)!x
α−1
e
−θx for x > 0,
where θ > 0 and α is a known positive constant.
(a) Show that Pn
i=1 Xi
is a sufficient statistic for θ.
(b) Find the MLE of 1/θ.
(c) It is known that Pn
i=1 Xi
is a complete statistic. Find the best unbiased estimator of
1/θ.
(d) Find the best unbiased estimator of θ. (Hint: Use the fact that Pn
i=1 Xi ∼ Gamma(nα, θ)
and see the example on p.26 of Chapter 2.)
3. Let X1 and X2 be two independent and identically distributed variables with probability
mass function
f(x; θ) = 1
2
x−θ+1 for x = θ, θ + 1, . . .
where θ ∈ {0, 1, 2, . . .}.
(a) Show that min(X1, X2) is a complete sufficient statistic for θ.
(b) Find P(min(X1, X2) = k) for k = θ, θ + 1, . . . and E(min(X1, X2)).
(c) Find the best unbiased estimator of θ.
4. Let X1, . . . , Xn be a random sample with density
f(x; θ) = θxθ−1
for 0 < x < 1.
Consider testing H0 : θ = 1 versus H1 : θ = 2. Show that the rejection region of the most
powerful test takes the form of R =

x :
Qn
i=1 xi ≥ c

for some constant c.
5. Let X1 and X2 be independent and identically distributed random variables whose probability mass function under H0 and H1 is given by
x 0 1 2
f(x; H0) 0.5 0.3 0.2
f(x; H1) 0.4 0.1 0.5
Use the Neyman–Pearson lemma to find the most powerful test for H0 versus H1 with size
α = 0.24. Compute the type II error probability for this test.


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