代写MAST 90082、代写c++，Java编程

日期：2023-04-06 08:53

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.