ABSTRACT
We designate the hazard function as h(i;) and the integral function as A(i;). Thus,
h-h(i:.)- g(i;) i - "'i - 1 - G(i;)' (7.1.15)
JS! ag Ai = A(i;) = - dz. -3/<xj aa3
af..L 2J..L 2J..L<X3i=1 2a3i=1 (7.1.16)
Maximum likelihood estimates(')!, ,1, &3) and subsequently 6-= ,1 &3/3, can be obtained by simultaneously solving the three estimating equations of (7 .1.16). This can be accomplished by employing various iterative procedures, but the evaluation of the integral function A(i;) is likely to be troublesome. This problem can be avoided if we bypass the third equation of (7 .1.16) and proceed directly to the Joglikelihood function (7 .1.11 ). Accordingly, we select a first approximation a~n and solve the first two equations of (7 .1.16) simultaneously for conditional estimates -y1 = -y (a~l)) and J..L 1 = J..L (a~1 l). We then substitute the three approximations a~!), -y ~> J..L 1 into (7 .1.11) to calculate ln L( a~l)). We repeat these calculations for several values in the vicinity of the maximum and a plot of In L as a function of a 3 can enable us to determine the estimate &3 that is
required. With &3 thus calculated, the remaining estimates can be obtained from the first two equations of (7 .1.16) or by interpolation.
