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MATH3811/3911 Statistical Inference
创建日期：2024-05-25 14:22:21
所属分类：数学mathematics
标签： MATH3811
Statistical Inference

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MATH3811/3911 Statistical Inference

Time Allowed: 50 minutes
(*) subquestions are only for MATH3911 students
Every subquestion is evaluated separately. Statements of previous subquestions can be
used even if these subquestions were not successfully proven.
Problems
1. Let X = (X1, X2, . . . , Xn) be a random sample of size n each with density
f(x; θ) =
√
2√
piθx
exp
(
− log
2(x)
2θ2
)
, 0 < x < 1,
where θ > 0.
(a) [2 marks] Use any argument to claim that T =
∑n
i=1 log
2(Xi) is a complete
and sufficient statistic for θ.
(b) [2 marks] Show that Y = θ−2 log2(X) ∼ χ21 with density
fY (y) =
1√
2pi
y−1/2e−y/2, y > 0.
Recall the density transformation formula
fY (y) = fX(x(y))
∣∣∣dx
dy
∣∣∣.
(c) [3 marks] Show that the expected Fisher Information about θ contained in
the statistic T is
IT (θ) =
2n
θ2
.
Hint: E(χ21) = 1.
(d) [2 marks] Find the umvue of g(θ) = θ2.
(e) [2 marks] Does the variance of the umvue of g(θ) = θ2 attain the Cramer-Rao
lower bound?
(f) [2 marks] Let gˆ denote the MLE of g(θ). Find the asymptotic distribution
of
√
n(gˆ − g). You do not need to calculate the MLE only its asymptotic
distribution.
MATH3811/3911: Midterm Test
(g) [3 marks] The statistic T =
∑n
i=1 log
2(Xi) also has the MLR property. Find
the UMP-α sized test for testing
H0 : θ ≤ 1 and H1 : θ > 1.
2. Suppose that X = (X1, X2, . . . , Xn) are independent random variables each with
density
f(x; θ) =
1
θ
e−x/θ, x > 0,
where θ > 0. Note that E(Xi) = θ and Var(Xi) = θ2. Consider testing
H0 : θ = θ0 vs H1 : θ 6= θ0.
(a) [2 marks] Show that the likelihood ratio test statistic can be given by
λ(X) =
(
x¯
θ0
)n
e
−n( x¯
θ0
−1)
.
(b) [2 marks] Below is a plot of λ(X) as a function of X¯. Argue that the rejection
region can be simplified to rejecting H0 when
X¯ < k1 or X¯ > k2.
You do not need to find k1 or k2.
0 10 20 30 40
0e
+0
0
2e
−1
0
4e
−1
0
6e
−1
0
X
λ(X
)
(c) [2 marks] Show that to (approximately) exhaust the level the constants k1
and k2 must satisfy
Φ
(√
n(k1 − θ0)
θ0
)
+ 1− Φ
(√
n(k2 − θ0)
θ0
)
= α,
where Φ(·) is the CDF of a standard normal.
(d) (*) [4 marks] Suppose we observe another random sampleY = (Y1, Y2, . . . .Ym)
of size m with parameter µ. Find the likelihood ratio test statistic for testing
H0 : θ = µ vs H1 : θ 6= µ.
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