MAST 90082 Mathematical Statistics
Assignment 1
Due: 5:00 pm, Wednesday, 19th April, 2023
Please present your solution in details, the mark is distributed to essential steps.
1. Let X1, . . . , Xn
i.i.d.～ Gamma(α, λ), α > 0 and λ > 0. Find the method of moments esti-
mator (MME) for (α, λ) and 2/
√
α (the skewness of X1). Hint : the pdf of Gamma(α, λ)
is
f(x|α, λ) =
{ 1
Γ(α)λα
xα?1e?x/λ, x > 0,
0, x ≤ 0,
where Γ(r) =
∫∞
0
xr?1e?xdx is the gamma function.
2. Let X1, . . . , Xn be a random sample from the uniform distribution on the interval [0, θ],
θ ∈ Θ = [1,∞) is unknown. Find the maximum likelihood estimator (MLE) of θ. Hint:
the parameter space is [1,∞) and does not include all value on the positive real line.
3. Let X1, . . . , Xn be a random sample from a discrete distribution with pmf
f(x|θ) =
{
θ, x = ?1;
(1? θ)2θx, x = 0, 1, 2, . . . ,
where 0 < θ < 1.
(a) Show that E(X1) = 0 and find Var(X1). Hint: finding the variance is optional.
(b) Show that the maximum likelihood estimator (MLE) of θ is
θ? =
2
∑n
i=1 I(Xi = ?1) +
∑n
i=1Xi
2n+
∑n
i=1Xi
.
(c) Show that θ? is a consistent estimator of θ, that is, θ?
p?→ θ as n→∞.
(d) (Optional) Find the asymptotic distribution of θ?.
1
4. Let Xi,1, . . . , Xi,ni be independently distributed as N(μi, σ
2) for i = 1, . . . ,m. Find the
MLE of θ = (μ1, . . . , μm, σ
2)T . (You need to check the corresponding Hessian matrix.)
5. (Optional) Let X1, . . . , Xn be a random sample from N(μ, 1). Define T1 = (Xˉn)
2 and
T2 = {n(n ? 1)}?1
∑
1≤i 6=j≤nXiXj as two estimators of μ
2, where Xˉn = n
?1∑n
i=1 Xi.
Compare T1 and T2 in terms of their biases, variances, and mean squared errors.
6. Let X1, . . . , Xn be a random sample from a population with pdf
f(x | μ, σ) =
{
σ?1e?(x?μ)/σ, x ≥ μ,
0, otherwise,
where μ ∈ R and σ > 0. Find the MLE of (μ, σ). Hint: consider fixing σ first.
7. LetX1, . . . , Xn
i.i.d.～ Exponential(θ), θ > 0. Show that the variance of Xˉn = n?1
∑n
i=1Xi
attains the Cramer-Rao Lower Bound for estimating θ. Hint : the pdf of Exponential(θ)
is
f(x|θ) =
{
θ?1e?x/θ, x > 0,
0, x ≤ 0.
8. Let X1, . . . , Xn be a random sample from a population with pdf
f(x|θ) =
{
2θ2x?3, x ≥ θ,
0, otherwise,
where θ > 0.
(a) Find the MLE θ? of θ.
(b) Find a sufficient statistic for θ and prove its sufficiency.
(c) Find the asymptotic distribution of θ? derived in part (a).
9. (Optional) Let X1, . . . , Xn
i.i.d.～ N(μ, μ2), μ ∈ R. Show that T = (∑ni=1 Xi,∑ni=1X2i )
is not complete for {N(μ, μ2) : μ ∈ R}.
2
10. Let X1, . . . , Xn be a random sample from the Pareto distribution with pdf
f(x|θ) =
{
3θθx?(θ+1), x ≥ 3,
0, otherwise.
(a) Show that T =
∑n
i=1 logXi is complete and sufficient for θ.
(b) Show that Y1 = log(X1/3) follows an exponential distribution with scale param-
eter 1/θ, and find E
{
1∑n
i=1 log(Xi/3)
}
.
(c) Find the UMVUE of θ.
11. (Optional) Let X1, . . . , Xn
i.i.d.～ N(μ, σ2), μ ∈ R is unknown and σ2 > 0 is known.
Denote Xˉn = n
?1∑n
i=1Xi.
(a) Show that the conditional distribution of X1 given Xˉn is N(Xˉn, (1? n?1)σ2).
(b) Find the UMVUE of Pμ(X1 ≤ 1) = Φ
(
1?μ
σ
)
. Hint: you may use the fact that Xˉn
is sufficient and complete for μ without proving it.
12. Let X1, . . . , Xn
i.i.d.～ Uniform(0, θ), θ ∈ Θ = (0,∞). Consider estimators of θ of the
form Tb = bX(n), where X(n) = max{X1, . . . , Xn}.
(1) Use the loss function L(θ, t) = (t? θ)2, compute the risk R(θ, Tb) and determine b
to give the smallest risk for all values of θ.
(2) Use the loss function L(θ, t) = t/θ ? 1 ? log(t/θ), compute the risk R(θ, Tb) and
determine b to give the smallest risk for all values of θ.
13. Let X1, . . . , Xn be a random sample from the following discrete distribution:
P (X1 = 0) =
1? θ
2? θ , P (X1 = 1) =
1? θ
2? θ , P (X1 = 2) =
θ
2? θ ,
where θ ∈ (0, 1) is unknown.
(a) Obtain the method of moment estimator (MME) of θ (denoted as θ?).
(b) Show that θ? → θ in probability.
(c) Find the asymptotic distribution of θ?.
3
14. Let X1, . . . , Xn be i.i.d. from Bernoulli(p) distribution, where p = P(X1 = 1) ∈ (0, 1)
is unknown. Let ν?n be the MLE of ν = p(1? p).
(a) Show that ν?n is asymptotically normal when p 6= 12 .
(b) When p = 1
2
, derive a non-degenerate asymptotic distribution of ν?n.