A. The likelihood of a time-continuous Markov process. Consider now the general set-up, whereby the development of an insurance policy is represented by a continuous time Markov process Z on a finite state space Z = {0, 1, . . . , J}. As usual, let Ig(t) and Ngh(t) denote, respectively, the indicator of the event that the process is staying in state g at time t ≥ 0, and the number of transitions from state g to state h in the time interval (0, t]. The transition intensities µghare assumed to exist, and to be piecewise continuous.
Suppose the policy is observed continuously throughout the time period [t, ¯t ], commencing in state g0 at time t. One then speaks of left-censoring and right-censoring at times t and
¯t, respectively, and the triplet z = (t, ¯t, g0) will be referred to as observational design or censoring schemeof the policy.
Consider a specific realization of the observed part of the process:
X(τ ) =
By the given censoring, the probability of this realization is as follows, where t0= t , tq= ¯t, and µg=P
h6=gµghdenotes the total intensity of transition out of state g:
exp
= exp
It follows that the likelihood of the observables is
Λ = exp
B. ML estimation of parametric intensities. Now consider a parametric model where the intensities are of the form µgh(t, θ), with θ = (θ1, . . . , θs)0 varying in an open set in the s-dimensional euclidean space, s < ∞. We assume they are twice continuously differentiable functions of θ.
Suppose that inference is to be made about the intensities or, equivalently, the parameter θ on the basis of data from a sample of n similar policies. Equip all quantities related to the m-th policy by topscript (m). The processes X(m)are assumed to be stochastically independent replicates of the process Z described above, but their censoring schemes z(m)may be different.
By independence, the likelihood of the whole data set is the product of the individual likelihoods: Λ =Qn
The censoring schemes are not visualized in (11.34), and they need not be if, as a matter of definition, dNgh(m)(t) and Ig(m)(t) are taken as 0 for t /∈ [t(m), ¯t(m)]. Likewise, introduce
p(m)g (t) = pg(m)
0 g(t(m), t)1[t(m),¯t(m)](t) ,
the probability that the censored process Z(m) stays in g at time t, by definition taken as 0 for t /∈ [t(m), ¯t(m)].
In the MLE construction we need the derivatives of (11.34), of first order (an s-vector),
∂ A comment on the form of the likelihood (9.31): For each type of transition g → h introduce Ngh, the number of transitions of that type, and (if Ngh> 0) Tgh(1), . . . , Tgh(Ngh), the
times when such transitions occurred. In terms of these quantities the log likelihood in (9.32)
and the ML equations (9.35) become
X comes to numerical computation of the MLE: The good thing about the form (9.32) is that it, by use of the counting processes, writes the sum on the left as a sum of contributions from all small time intervals. This is particularly useful in the derivation of the statisti-cal properties of the MLE. (A similar remark could be made about the benefit of using the counting processes to define the payment stream for a general insurance policy in Section 7.5.) Referring to Appendix D, the large sample distribution properties of the MLE are given by
θ ∼ˆ asN(θ, Σ(θ)) , (11.39)
where Σ(θ) is given by its inverse, the so-called information matrix, Σ(θ)−1= −E
∂
∂θ∂θ0ln Λ
. (11.40)
Taking expectation in (11.36), noting that the terms dNgh(τ ) − µgh(τ, θ)Ig(τ )dτ have zero means, we obtain
Σ(θ)−1=X The expression in parentheses under the integral sign is an s×s matrix and all other quantities are scalar.
It is seen that the information matrix tends to infinity, hence the variance matrix of the MLE tends to 0, if the termsPn
m=1p(m)g (τ, θ) grow to infinity as n increases, roughly speaking, which means that the expected number of individuals exposed to risk in different states gets unlimited.
C. Estimating the parameters of a Gompertz-Makeham mortality law.
The actuarial office in a life insurance company is to estimate the mortality law governing the company’s portfolio of term insurance policies. (The mortality law for the portfolio of life annuities may be different since people who (believe they) are in good health would probably find a pure survival benefit more useful and profitable than a pure death benefit. Thus, it seems appropriate to perform a separate mortality investigation for each line of life insurance business. Moreover, since mortality also depends on sex, the study would typically include only males or only females.)
Suppose the statistical data comprises n individuals who have been insured under the scheme during a certain period of time. For each individual No. m (= 1, . . . , n) there is a policy record with the following pieces of information:
– xm, the age on entry into the study;
– ym, the age on exit from the study;
– Nm, the number of deaths during the study (0 or 1).
Here xmwould typically be the age at issue of the policy. If Nm= 1, then ymis the age at death, and if Nm= 0, then ymis the age at the time of censoring, either at the term of the contract or upon termination of the study, whichever occurred first. In any case ym− xmis the time spent under observation as alive during the study. With these definitions xmtakes the role of t(m) in the general set-up and, for Nm= 0, ymtakes the role of ¯t(m).
The state space is now just Z = {0, 1} (”alive” and ”dead”). Assume the mortality law is Gompertz-Makeham so that the mortality intensity at age t is
µ(τ, θ) = α + βeγτ,
We need the derivatives of the intensity w.r.t. all three parameters,
∂ and their integrals with respect to time,
Z y The MLE equations (9.35) specialize to
X
To find the information matrix (9.38) we need the matrix with the products of the deriva-tives,
(symmetric) and the probabilities p(m)0 (τ, θ), which are p(m)0 (τ, θ) = exp transition, from 0 to 1, and the summation over g, h in the information matrix (9.38) can be dropped.)
We see that all ingredients in the asymptotic variance matrix are given by explicit formu-las, and it remains only to perform a numerical integration to find its value for given θ.