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P8131 Homework 1 solved

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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?