Question 1

A generalised linear model has independent normal responses Y_i , with mean µ_i 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 (x_i , y_i), 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 Y_i represents the survival time (in weeks) of the ith patient and x_i represent the logarithm of the ith patient's initial white blood cell count (i = 1, 2, .......,17).

The response variables y_i are assumed to be exponentially distributed with mean 1/µ.  A possible specification for E(Y_i) is E(Y_i) = exp(α + βx_i).  This will ensure that the E(Y_i) is non-negative for values of x_i.

(i) Write down the natural link function associated with the linear predictor η_i = α + βx_i)

(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(Y_i)    = α

Model 2:     E(Y_i) = exp(α + βx_i)

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  Z_t = Z_t-1  +αZ_t-2 + e_t  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 Z_t ~ 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 X_t?