STAT 3023/3923
Semester 2 Statistical Inference 2021
Week 4 Tutorial
1. Let X = (X,Y) be bivariate normal with X and Y being standard normals with coefficient of correlation ρ. Show that U = X/Y is distributed with frequency function
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√1−ρ2 fU(u)= π(1−2ρu+u2).
2. Suppose that X1, . . . , Xn is a random sample.
(a) If Xi′s are iid Bernoulli with parameter θ, 0 < θ < 1, show that the likelihood can
be written as the canonical one-parameter exponential family
l(x;θ) = eηT(x)−ψ(η)h(x) for x = (x1,...,xn), xi ∈ {0,1}.
Identify the natural sufficient statistic T for θ, and find E(T) and Var(T) using ψ.
(b) Using the same argument, show that if Xi’s are iid N(θ,1), then T = ni=1 Xi is a sufficient statistic for θ. Find E(T) and Var(T).
3. Let X1, . . . , Xn be iid from a distribution with density f(x; θ) = (2πθ)−1/2e−x2/(2θ).
Find a sufficient statistic for θ.
4. Let X1, . . . , Xn be iid from a distribution with density
f(x;θ)= βα , x∈(0,β), α>0.
Show that (X(n), ni=1 log Xi) is sufficient for θ = (α, β), where X(n) = max(Xi).
5. Recall the density of Gamma(α, β),
f(x;α,β) = 1 xα−1e−x/β for x > 0. βαΓ(α)
(a) Show that if β is fixed, this belongs to the one-parameter exponential family. (b) Show that if β is unknown, this belongs to the two-parameter exponential family.
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