## Description

1. (40%) Show that the following distributions belong to the exponential family. Find

the natural parameter θ, scale parameter φ, and convex function b(θ). Also find the

E[Y ] and Var[Y ] as functions of the natural parameter. Specify the canonical link

functions.

(a) Binomial distribution Bin(n, π), f(y; π) =

n

y

π

y

(1 − π)

n−y

, where n is known;

(b) Poisson distribution P ois(λ), f(y; λ) = 1

y!

λ

y

e

−λ

;

(c) The Gamma distribution Gamma(α, β), f(y; β) = β

α

Γ(α)

y

α−1

e

−βy, where the

shape parameter α is known.

(d) Negative binomial distribution NB(m, β), f(y; β) =

y+m−1

m−1

β

m(1−β)

y

, where

m is known;

(e) Beta distribution Beta(α, β), f(y; α, β) = Γ(α+β)

Γ(α)Γ(β)

x

α−1

(1−x)

β−1

, where α > 0,

β > 0, and x ∈ [0, 1].

2. (30%) Assume Y1, Y2, …, Yn are independent and follow a binomial distribution where

Yi ∼ Bin(m, πi) and m is known. Furthermore, assume log πi

1−πi

= Xiβ. What are

the expressions of deviance residuals and Pearson residuals respectively (use βˆ to

represent the MLE)? What are the expressions of the deviance and Pearson’s χ

2

statistic?

3. (30%) Consider the binary response variable Y ∼ Bernoulli with P(Y = 1) = π and

P(Y = 0) = 1 − π. Observations Yi

, i = 1, . . . , n, are independent and identically

distributed as Y .

(a) Find the Wald test statistic, the score test statistic, and the likelihood ratio

test statistic to test hypothesis H0 : π = π0.

(b) With large samples, the Wald test statistic, score test statistic and the likelihood ratio test statistic approximately have the χ

2

(1) distribution. For n = 10

and data (0, 1, 0, 0, 1, 0, 0, 0, 1, 0), use these statistics to test null hypotheses

on for (i) π0 = 0.1, (ii) π0 = 0.3, (iii) π0 = 0.5.

(c) Do the Wald test, score test, and the likelihood ratio test lead to the same

conclusions in (b)?

4. (+10%) Assume Yi ∼ P ois(λ), i = 1, …, n. We are interested in testing H0 : log λ =

log λ0. What are the Wald test statistic, the score test statistic, and the likelihood

ratio test statistic?