Can We Express ēxxx…(n) and I/μx by the Ordinates of Frequency Curves?

Abstract

Many years ago it was suggested—I cannot remember by whom—that instead of graduating lx or qx or some such function it might be preferable to graduate annuity values. When I was fitting frequency curves to many kinds of statistics some fifty years ago, I noticed that the expectation of life from birth onwards looked something like the ordinates of a very skew frequency curve with a mode at age 4 or thereabouts. Partly because it seemed difficult to bring expectations of life within a definition of frequency distributions and partly because the example was neither a straightforward graduation nor helpful to the work on which I was then engaged, the notes I made were put aside and have long since been destroyed. From time to time, however, I have wondered whether life annuity values and expectations of life could be expressed roughly by one of the algebraic expressions used for frequency curves, and whether there would be advantages in working from joint expectations of life for n lives of the same age rather than for one life. Such a device, or the use of annuity values, may be taken to mean that we are using a series of values that could represent an expectation of life from a mortality table with heavier but related mortality. Approximations even if they are as rough as I had to anticipate in this case may be useful in unexpected ways, and that thought supplied me with some sort of justification for taking up the subject again and making some fresh calculations.