DEPARTMENT OF APPLIED MATHEMATICS

AMA529 STATISTICAL INFERENCE

Assignment 2

Due at 11:59pm on 7 Mar, 2025

1. Let X1, . . . , Xn be a random sample from the N(µ, σ2) distribution, where µ is known, and σ2 > 0 is unknown.

(a) Calculate E[(X1 − µ) 2 ].

(b) Show that  is the best unbiased estimator of σ2. You may use without proof the fact that  (Xi − µ)2 is a complete statistic.

2. Let X1, . . . , Xn be a random sample with density

f(x; θ) = θ(1 − x)θ−1 for 0 < x < 1

where θ > 0.

(a) Show that  (1 − Xi) is a sufficient statistic for θ.

(b) Find the MLE of 1/θ.

(c) Given that  (1 − Xi) is complete statistic, find the best unbiased estimator of 1/θ.

(d) Find the best unbiased estimator of θ. (Hint: you may start with the MLE of θ; and show that Y1 = − log(1 − X1) follows an exponential distribution).

3. Let X1 and X2 be two independent and identically distributed variables with probability mass function

f(x; θ) =  for x = θ, θ + 1, . . .

where θ ∈ {0, 1, 2, . . .}.

(a) Show that min(X1, X2) is a complete sufficient statistic for θ.

(b) Find P(min(X1, X2) = k) for k = θ, θ + 1, . . . and E(min(X1, X2)).

(c) Find the best unbiased estimator of θ.

4. Let X1, . . . , Xn be a random sample from the Gamma(α, θ) distribution with density

f(x; θ) =  for x > 0,

where α > 0 is known and θ > 0. Consider the hypothesis test H0 : θ = 1 versus H1 : θ = 3. Show that the rejection region of the most powerful test takes the form. of R = {x :  xi ≤ c} for some constant c.

5. Let X1 and X2 be two independent and identically distributed random variables whose probability mass function under H0 and H1 is given by

Use the Neyman–Pearson lemma to find the most powerful test for H0 versus H1 with size α = 0.09. Also, compute the type II error probability for this test.