Stat613: Intermediate Theory of Statistics
Homework 0 (8/27/07)
Instructor: Jianhua Huang Students: Student A
TA: Seokho Lee Student B
Welcome to Statistics 613! This homework helps you warm up. Please submit your solution in class on
Friday, August 31. Please also create a web page that links to your solutions.
Thanks!
–Jianhua Huang
Problem 1.
Acceptance/Rejection Method
Consider the generation of a x∼Beta(α, β ) quantity with density f(x|α, β) = kα,β xα−1(1 −x)β−1I(x∈
[0,1]), α, β > 0. When α, β ≤1, xcan be generated by the following algorithm: (i) generate u1and u2
independently from a U[0,1] distribution; (ii) set v1=u1/α
1,v2=u1/β
2and w=v1+v2; (iii) if w > 1, then
go back to step i, otherwise set x=v1/w.
(a) Certify yourself of the correctness of the algorithm by writing a R program, drawing a sample of size
10,000 and plotting the resulting histogram along with the Beta(α, β) density function for several choices
of α,β.
(b) If x∼G(α, 1), y∼G(β, 1) and xand yare independent variables, show that x/(x+y)∼Beta(α, β).
Here we use G(α, β) to denote the Gamma distribution with density f(x;α, β) = βαxα−1e−βx /Γ(α), x > 0.
(c) Prove that the algorithm indeed generates a Beta(α, β) random number.
Solution
Give you solution here.
Problem 2.
Identifiability of Parametrization
A statistical model X∼P(x, θ) is called identifiable if and only if P(x, θ) = P(x, θ0) implies that θ=θ0
or, equivalently, if and only if θ6=θ0implies that P(x, θ)6=P(x, θ0). Identifiability means that the model
parameters are totally determined by the observed data. Only when the model is identifiable, interpretation
of the model parameters is meaningful. If a model is not identified, it can be made identifiable by imposing
constraints on the model parameter θ.
The notation used above is generic. The Xcould represent a random variable, a vector, or even more
complicated object. Similarly, the θcould be a number, a vector, or a function, or more complicated object.
Discuss if the following models are identifiable. If not, suggest a re-parameterization to make the model
identifiable.
(a) Single index model. Suppose we have data (Xi, Yi) independent identically distributed as (X, Y ). The
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