Reference no: EM132889738

STAT6110 Statistical Inference - Macquarie University

In answering this assignment questions, you may use standard properties of common proba- bility distributions, such as those given in the document Common Probability Distributions available on iLearn.

1. Let Y₁ . . . , Y_n denote a random sample from the probability density function

f (y; θ) = 1/y√2πθ. exp {-(logy)²/2θ}   for y > 0 and θ > 0

and f (y; θ) = 0 otherwise.

You may assume that for this distribution E(log Y_i) = 0 and Var(log Y_i) = θ. (a)Find the maximum likelihood estimator (MLE) of θ.

(b) Find the expected information and provide an approximate standard error for the MLE in (a).

(c) Consider testing the hypothesis that the parameter θ is equal to a particular value:
H₀ : θ = 1 versus H₁ : θ ≠ 1.

Write down the test statistics corresponding to each of the following tests and evaluate them under the assumption that n = 100 and the observed value of the MLE is θˆ = 1.2:

i. likelihood ratio test

ii.score test

iii.Wald test

iv.approximate Wald test.

2. Two independent random samples X₁, . . . , X_n and Y₁, . . . , Y_m follow Poisson distribu- tions: X_i Poi(λ) and Y_i Poi(γλ), where λ ≥ 0 and γ ≥ 0 are two unknown parameters. Suppose we wish to test the hypothesis

H₀ : λ = λ₀, γ = γ₀ v.s. H₁ : not H₀,
where λ₀ and γ₉ are given values. Let l(λ, γ) denote the log-likelihood function from these two samples.

(a) Show that the log-likelihood function is given by

l(λ, γ) = -nλ -mγλ + logλ (Σ_i=1ⁿ X_i + Σ_j=1^m Yj) + logYΣ_j=1^mY_j

(b) Hence show that the maximum likelihood estimators of λ and γ are

λ^ = Σ_i=1ⁿ X_i/n

γ^ = Σ_j=1^m Y_j/m/Σ_i=1ⁿ X_i/n

(c) Derive the likelihood ratio test statistic for testing this hypothesis.

(d) Specify the rejection region given by the likelihood ratio test.