Question 1
A generalised linear model has independent normal responses Yi , with mean µi and common variance σ2.
(i) Show that N(µ, σ2) is a member of the exponential family
(ii) Write down an expression for the variance function.
(iii) Identify the canonical link function
(iv) Consider a linear predictor of the form ?? = α + βx used in conjunction with the canonical link for a sample of data points (xi , yi), i-1,2,3...,n. Obtain the maximum likelihood estimates of α and β.
Question 2
The following study was carried out into the mortality rate of leukaemia sufferers. A white blood cell count was taken from each of 17 patients and their survival times were recorded.
Suppose that Yi represents the survival time (in weeks) of the ith patient and xi represent the logarithm of the ith patient's initial white blood cell count (i = 1, 2, .......,17).
The response variables yi are assumed to be exponentially distributed with mean 1/µ. A possible specification for E(Yi) is E(Yi) = exp(α + βxi). This will ensure that the E(Yi) is non-negative for values of xi.
(i) Write down the natural link function associated with the linear predictor ηi = α + βxi)
(ii) Use this link function and linear predictor to derive the equations that must be solved in order to obtain the maximum likelihood estimates of α and β.
(iii) The following two models are now to be compared:
Model 1: E(Yi) = α
Model 2: E(Yi) = exp(α + βxi)
The deviance for Model 1 is found to be 26.282 and the deviance for Model 2 is 19.457. Using the x2 distribution, test the null hypothesis that β = 0 against the alternative hypothesis that β ≠ 0 . What can you conclude?
Question 3
Consider the AR(2) process Zt = Zt-1 +αZt-2 + et with first and second lag ACF's ρ1 and ρ2.
(i) Find the range of α such that the above process is stationary.
(ii) Show that if the above process is stationary then ρ12 < (ρ2 + 1)/2.
Question 4
An ARMA(2,1) process defined by:
where Zt ~ N(0, 1) has been running for a very long time.
(i) Show that the process is weakly stationary.
(ii) (a) Find the autocorrelation function for lags 0, 1, 2.
(b) Show that the autocorrelation function at lag k ( k= 0, 1, 2..... ) can be expressed in the form:
where A and B are constants that you should specify.
(iii) What is the probability distribution of Xt?