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MATH5905, T1 2020 Assignment Two Statistical Inference
Department of Statistics
Term 1, 2020
MATH5905 – Statistical Inference
Assignment 2
Instructions: This assignment must be submitted no later than 6 pm on Thursday, 23rd April
2020 (week 10). Please, declare on the first page that the assignment is your own work, except where
acknowledged. State also that you have read and understood the University Rules in respect to Student
Academic Misconduct. You need to submit the PDF file on Moodle using the assignment submission
link by 6 pm on Thursday 23rd April 2020 (week 10).
Problem 1
Let X = (X1, X2, . . . , Xn) be sample of n i.i.d. random variables, each with a density
f(x, θ) = 1
x

2πθ
exp 

1

log2
(x)

when x > 0 otherwise zero and where θ > 0 is a parameter.
a) Find the distribution of Yi = log Xi and hence or otherwise compute E(log2 Xi).
b) Find the Fisher information about θ in one observation and in the sample of n observations.
c) Find the Maximum Likelihood Estimator (MLE) of θ and show that it is unbiased.
d) Does the variance of the MLE attain the Cramer Rao bound?
Note: The χ
2
k
distribution has mean k and variance 2k.
e) Determine the asymptotic distribution of the MLE of θ and also the asymptotic distribution of
h(θ) = e
θ
.
Problem 2
Suppose X = X1, X2, . . . , Xn is a sample of n i.i.d. random variables from a population with a density
f(x; θ) = (
τθτx
−(τ+1) if x > θ
0 if otherwise
.
where τ > 0 is a known constant and θ > 0 is an unknown parameter.
a) Show that the family {L(X, θ}, θ > 0 has a monotone likelihood ratio in the statistic T = X(1).
b) Show that the density of T = X(1) is
fX(1) (x) = (
τnθτnx
−(τn+1) if x > θ
0 if otherwise
.
Hint: It might provide useful to consider the following
P(X(1) ≤ x) = 1 − P(X1 ≥ x, X2 ≥ x, . . . , Xn ≥ x).
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MATH5905, T1 2020 Assignment Two Statistical Inference
c) Find the uniformly most powerful α-size test ϕ
∗ of
H0 : θ ≥ τ versus H1 : θ < τ.
d) Calculate the power function of ϕ

.
e) Compute the value of the power function at zero, τ and at the threshold constant of the test.
Then, sketch a graph of the power function as precisely as possible.
Problem 3
Suppose that X is uniform on [0, θ] with density f(x, θ) = 1/θ, 0 < x < θ, zero elsewhere.
a) Let X = (X1, . . . , Xn) be a sample of n i.i.d. observations from this distribution. Similar to the
first question it can be shown that the family {L(X, θ)} has a monotone likelihood ratio in the
statistic T = X(n)
.
i) Compute the density and distribution function for T = X(n)
.
ii) Determine the uniformly most powerful α-size test of
H0 : θ ≥
1
2
versus H1 : θ < 1
2
.
iii) Show that the random variable
Wn = n

1 −
X(n)
θ

converges in distribution to the exponential distribution with mean one as n → ∞.
iv) Hence or otherwise justify that X(n)
is a consistent estimator of θ.
b) Now suppose that X(1), X(2), X(3), X(4), X(5) are the order statistics of a random sample of size
five from this distribution. Let the observed value of X(5) be x(5). The test rejects H0 : θ = 2 and
accepts H1 : θ 6= 2 when either x(5) ≤ 1 or x(5) > 2.
i) Find the power function γ(θ) for 0 < θ of this particular test.
ii) Plot the power function γ(θ) for all values θ > 0.
Problem 4
Suppose X1, . . . , Xn are an i.i.d. sample with density
f(x, , θ) = θ(1 − x)
θ−1
0 < x < 1,
zero elsewhere and θ > 0.
a) Find the form (you do not have to find the threshold constant) of the uniformly most powerful
test of
H0 : θ = 1 against H1 : θ > 1.
b) Find the likelihood ratio for testing
H0 : θ = 1 against H1 : θ > 1.
2
MATH5905, T1 2020 Assignment Two Statistical Inference
Problem 5
Suppose X(1) < X(2) < X(3) < X(4) < X(5) are the order statistics based on a random sample of size
five from the standard exponential density f(x) = e
−x
, x > 0.
a) Find E(X(4)).
Hint: Use a computer package to calculate the integral.
b) Find the density of the midrange A =
1
2
(X(1) + X(5)).
Hint: Use the following expansion
(e
−v − e
−2u+v
)
3 = −3e
−2u−v + 3e
v−4u − e
3v−6u + e
−3v
.
c) Using this result (or otherwise), find P(A > 2).
Hint: Use a computer package to calculate the integral.
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