13.5 Maximum likelihood estimation of decrement probabilities
In Section 12.4 methods were introduced for estimating mortality probabilities with large data sets. One of the methods was a seriatim method using exact exposure. In this section that estimator will be shown to be maximum likelihood under a particular assumption. To do this, we need to develop some notation. Suppose we are interested in estimating the probability an individual alive at age a dies prior to age b where a > b. This is denoted q = [S(a) − S(b)]/S(a). Let X be the random variable with survival function S(x), the probability of surviving from birth to age x. Now let Y be the random variable X conditioned on X > a. Its survival function is SY (y) = Pr(X > y|X > a) = S(y)/S(a).
We now introduce a critical assumption about the shape of the survival function within the interval under consideration. Assume that SY(y) = exp[-(y − a)λ] for a < y ≤ b. This means that the survival function decreases exponentially within the interval. Equivalently, the hazard rate (called the force of mortality in life insurance mathematics) is assumed to be constant within the interval. Beyond b a different hazard rate will be used. Our objective is to estimate the conditional probability q, Thus we can perform the estimation using only data from and a functional form for this interval. Values of the survival function beyond b will not be needed.
Now consider data collected on n individuals, all of whom were observed during the age ...