Three observations were made of a random variable Y , namely y1 = 51, y2 = 35, y3 = 31.

(a) Assume that Y ∼ N(40, σ2 ), i.e. the mean is known but the variance must be estimated. Obtain the maximum likelihood estimate of σ2 from this sample. Is this estimate unbiased?

(b) Now assume that Y ∼ N(µ, σ2 ), i.e. both parameters must be estimated. Obtain the maximum likelihood estimates of µ and σ2. Are these estimates unbiased? Obtain the residual maximum likelihood (REML) estimate of σ2. Is this estimate unbiased?

(c) Make a sketch of the data space defined by this sample, corresponding to that presented in Figure 10.3 for a sample of two observations. Show the Y1, Y2 and Y3 axes, the µ-effects axis and the observed values.

(d) Describe briefly how this geometrical representation can be used to obtain estimates of µ and σ2, designated µˆ and σˆ 2. What is the distance from the point y = (y1, y2, y3) to the point µˆ = (µ,ˆ µ,ˆ µ)ˆ  ? When this graphical approach is used to represent a sample of two observations, and when any particular values, µˆ and σˆ 2, are postulated for µ and σ2, the contours of the resulting probability distribution are circles, as shown in Figure 10.5.

(e) What is the shape of the corresponding contours for this sample of three observations?

(f) Sketch the contour at a distance √3σˆ from the point (µ,ˆ µ,ˆ µ)ˆ  : (i) when the postulated values µˆ and σˆ 2 are the maximum likelihood estimates, (ii) when the postulated value µˆ is above the maximum likelihood estimate and the postulated value σˆ 2 is below the maximum likelihood estimate. Now suppose that µˆ is restricted to the maximum likelihood estimate. 354 Why is the criterion for fitting mixed models called REML? (g) In what subspace of the data space must the point y then lie? (h) Within this subspace, what will be the shape of the contours of the probability distribution?