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LANCASTER UNIVERSITY
2007 EXAMINATIONS
Part II (Third or Fourth Year)
MATHEMATICS & STATISTICS 2 hours
Math 350: Likelihood Inference
You should answer ALL Section A questions and TWO Section B questions. In Section A there are questions worth a total of 50 marks, but the maximum mark that you can gain there is capped at 40. Please note, MLE will denote maximum likelihood estimator or maximum likelihood estimate.
If the random variable W has a χ^21 distribution, i.e. a chi-squared distribution with 1 degree of freedom, then P (W > 3 .84) = 0. 05 and P (W > 2 .71) = 0. 10. If the random variable W has a χ^22 , i.e. a chi-squared distribution with 2 degrees of freedom, then P (W > 5 .99) = 0. 05. If X ∼ N (0, 1), then P (X < 1 .96) = 0. 975 and P (X < 1 .64) = 0. 95. The inverse of a 2 × 2 matrix is given by
( a b c d
= (^) ad −^1 bc
d −b −c a
SECTION A
A1. (a) Define the Deviance Function for a statistical model with^ d-dimensional parameter^ θ, D(θ). [3] (b) State carefully the asymptotic distribution of D(θ), for different values of θ. [You need not state precisely the regularity conditions under which your result holds.] [4] (c) Consider the linear regression model: Yi ∼ N (α + βzi, 1) where the Yi’s are assumed independent and the {zi} are known covariates. Calculate the MLE ˆα, βˆ and D(α, β) and write down its asymptotic distribution. [11] (d) Give one reason why deviance based confidence intervals might be preferred to those based on the asymptotic distribution of the MLE. [3] please turn over
SECTION A continued
A2. (a) X 1... Xn are IID from f (x|θ). Define the Fisher’s Information matrix IE (θ) for this problem. [3] (b) State the asymptotic distribution of the Maximum Likelihood Estimator, θˆ. [2] (c) Suppose now that θ = (θ 1 , θ 2 )T^. From the result you’ve just stated, write down the asymptotic distribution of θˆ 1. [3] (d) Suppose X takes values 1, 2, 3 written respective probabilities θ 1 , θ 2 , 1 − θ 1 − θ 2. State constraints that apply to (θ 1 , θ 2 ) to make this a sensible model. [2] (e) Given data x 1... xn from the model given in (d), calculate the MLE ˆθ. [4] (f) Given that in this case
IE (θ) =
θ 1 +^1 −θ^11 −θ 2 1 −θ^11 −θ 2 1 −θ^1 1 −θ 2 θ^1 2 +^ 1 −θ^1 1 −θ 2
calculate the asymptotic distribution of θˆ 1. [3]
A3. (a) What is meant by the parameters θ 1 and θ 2 being orthogonal? [4] (b) State an advantage of using an orthogonal parameterisation. [3] (c) Consider the Gamma model where data x 1... xn are IID from Gamma(α, β). Calculate the log likelihood for this model and by computing a suitable second derivative, show that α and β are not orthogonal. [5]
please turn over
SECTION B continued
B3. Consider IID data x 1... xn from the Gamma model, Gamma(α, β); α > 0, β > 0.
(a) Compute the profile log-likelihood of α, P `(α). Simplify this expression for α = 1 and for α = 2. [10] (b) Suppose now that a factory making Christmas tree lights produces two types. Type A fail completely after one individual bulb fails, while type B continue to function until two bulbs have failed. For each set of light sets, the time until the first light bulb fails, and time between first and second bulb failures are assumed to be IID exponential random variables with unknown parameter. Find appropriate models for type A and type B complete failure times. [5] (c) There has been a mix up in the labelling department, and a large batch of light sets are of unknown (but identical) origin. An experiment is carried out from light sets in this batch, giving life times x 1... xn. We are required to choose the maximum likelihood type from these data. Give a condition on x 1 ,... , xn under which type A is chosen. [5]
end of exam
