AMA529 STATISTICAL INFERENCE 2

最新推荐文章于 2025-12-19 12:03:37 发布

原创最新推荐文章于 2025-12-19 12:03:37 发布 · 750 阅读

11 ·

CC 4.0 BY-SA版权

文章标签：

#开发语言

Java Python 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.

(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. Letdai 写AMA529 STATISTICAL INFERENCE Assignment 2 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)).

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