5.39 (Examination question, University of Nottingham.)

The EM algorithm is to be used as an alternative to the Newton-Raphson method for maximising the likelihood of Exercise 3.21 by imputing the values of the censored observations.

(i) Show that the E step of the algorithm involves the term E(Ti|Ti > ti, λ(r)

) for m + 1 ≤ i ≤ m + n, where λ(r) is the current estimate of λ.

(ii) Prove that E(Ti|Ti > ti, λ(r)

) = 2

λ(r) +

λ(r)t 2

i 1 + λ(r)ti

. (∗)

[You may assume that for θ > 0

 ∞

s

θnyn−1e−θy

(n − 1)! dy =

n

−1 j=0 sj θj e−θs j! . ]

(iii) Show how the M step is used to produce the next estimate of λ in the iteration, denoted by λ(r+1). Give an intuitive explanation of your result.

(iv) Suppose that you had been unable to derive the formula for the expectation in (∗) above. How else could you have calculated E(Ti|Ti > ti, λ(r)

)?