Download Tutorial 3 - Maximum Likelihood Estimation

Tutorial 3 - Maximum Likelihood Estimation & Canonical Link
(last updated January 30, 2009)
1. Find the canonical link for
(a) Normal distribution with unknown mean and known variance
(b) Poisson distribution
(c) Binomial distribution
2. A sample of 100 plants were classified into three groups according to their status for a particular gene,
and the resulting frequencies are reported as follow:
Group
1
2
3
Frequency
18
55
27
T
T
Let Y = (Y1 , Y2 , Y3 ) denote the frequencies where Y ∼ M ult(π) and π = (π1 , π2 , π3 ) the vector of
probabilities of the multinomial model.
(a) There are only 2 degree of freedom in this model and it is common to reparameterize this by
α2
α3
letting π1 = 1+eα21+eα3 , π2 = 1+eαe2 +eα3 , π3 = 1+eαe2 +eα3 . Write down the likelihood function in
T
terms of α = (α2 , α3 ) . Derive the score and information function in terms of α.
T
2
2
(b) Genetic theory says that the hypothesis H0 : π = θ2 , 2θ (1 − θ) , (1 − θ)
. Derive the MLE
of θ under the hypothesis based on the data above. (We will assess the evidence against this
hypothesis in next week tutorial.)
1