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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