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MAST 90082 Mathematical Statistics
Assignment
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
θˆ =
2
∑n
i=1 I(Xi = −1) +
∑n
i=1Xi
2n+
∑n
i=1Xi
.
(c) Show that θˆ is a consistent estimator of θ, that is, θˆ
p−→ θ as n→∞.
(d) (Optional) Find the asymptotic distribution of θˆ.
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 = (X¯n)
2 and
T2 = {n(n − 1)}−1
∑
1≤i 6=j≤nXiXj as two estimators of µ
2, where X¯n = 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 X¯n = 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 θˆ of θ.
(b) Find a sufficient statistic for θ and prove its sufficiency.
(c) Find the asymptotic distribution of θˆ 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 X¯n = n
−1∑n
i=1Xi.
(a) Show that the conditional distribution of X1 given X¯n is N(X¯n, (1− n−1)σ2).
(b) Find the UMVUE of Pµ(X1 ≤ 1) = Φ
(
1−µ
σ
)
. Hint: you may use the fact that X¯n
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 νˆn be the MLE of ν = p(1− p).
(a) Show that νˆn is asymptotically normal when p 6= 12 .
(b) When p = 1
2
, derive a non-degenerate asymptotic distribution of νˆn.