3.21 (Examination question, University of Nottingham.) An electrical component has failure time given by the gamma, (2,

3.21 (Examination question, University of Nottingham.) An electrical component has failure time given by the gamma, Γ(2, λ), pdf:

f(t) = λ2te−λt for t > 0, f(t) = 0 for t ≤ 0, where λ > 0.

The corresponding cdf is F(t)=1 − e−λt(1 + λt) for t > 0,

= 0 for t ≤ 0.

For m components the failure times are known exactly, and given by {ti, 1 ≤

i ≤ m}. The failure times of n components are all right-truncated, with truncation times, {ti, m + 1 ≤ i ≤ m + n}.

(i) Show that apart from an additive constant, the log-likelihood may be expressed as

(λ)=2m log λ − λ

m



+n i=1 ti +

m



+n i=m+1 log(1 + λti).

(ii) The Newton-Raphson iterative method is to be used to find the maximum likelihood estimate of λ. Let λ(r) be the value of λ at the rth iteration. Show that

λ(r+1) = λ(r)

, 1 +

2m − λ(r) m+n i=1 ti + λ(r) m+n i=m+1 z(r)

i 2m + (λ(r))2 m+n i=m+1(z(r)

i )2

-

, where z

(r)

i = ti 1 + λ(r)ti

.