“Comparison of future lifetime distribution and its approximations,” / Reply

Jones, Bruce L

BRUCE L. JONES* AND JOHN A. MEREU^

We found this paper to be of interest because the topic is one we recently did some work on (Jones and Mereu 2000, 2002). In our papers, we examined families of fractional age assumptions including the three traditional ones analyzed by Frostig. To help in choosing specific fractional age assumptions, we presented an optimality criterion based on the length of the probability density function over the range of the mortality table.

Our findings are consistent with those obtained by Frostig. They show that UDD results are better than those found with the constant-force and the hyperbolic (Balducci) assumptions. We postulated assumptions that were better than UDD, but not as simple. These could be used if the error with UDD was not acceptable.

REFERENCES

JONES, B. L., AND J. A. MEREU. 2000. “A Family of Fractional Age Assumptions,” Insurance: Mathematics & Economics 27: 261-76.

–.2002. “A Critique of Fractional Age Assumptions,” Insurance: Mathematics & Economics 30: 363-70.

AUTHOR’S REPLY^^

I appreciate the discussion of Professors John Mereu and Bruce Jones, and thank them for bringing to my attention their interesting papers, and the relevance of their work to my contribution.

In my paper I argued that the UDD (uniform distribution of deaths) approximation is better than the constant force of mortality and the Balducci approximations. Hurlimann (1990) proved that the graph of survival function of the UDD approximation lies above the corresponding graphs for the constant force of mortality and the Balducci approximations. In their papers Mereu and Jones (Jones and Mereu 2000, 2002) present a very rich family of fractional age assumption approximations. In the first paper, their approximation is based on linear interpolation of ^sub t^P^sub k^^sup alpha^ between ^sub o^p^sub k^^sup alpha^ = 1 and p^sub k^^sup alpha^. By choosing appropriate alpha, probably alpha > 1, one might derive approximation, which overestimates the survival function under the UDD assumption.

My paper shows that, when the life survival function has increasing force of mortality, then it is underestimated by the constant force of mortality and the Balducci approximations, while under the UDD approximation it might be either underestimated or overestimated. By choosing appropriate alpha, probably alpha > 1, one might derive approximations, which, in this case overestimate the life survival function. This might occur also with linear force of mortality (LFM) and the quadratic survival function (QSF) approximations presented by Mereu and Professor Jones in their second paper (Jones and Mereu 2002). It would be worthwhile to examine the Jones and Mereu approximations by relating them to the unknown survival function as I did in my paper.

REFERENCES

HURLIMANN, W. 1990. “On Life Table Applications of Ordering among Risks,” Insurance: Mathematics & Economics 9: 277-79.

JONES, B. L., AND J. A. MEu. 2000. “A Family of Fractional Age Assumptions,” Insurance: Mathematics & Economics 27: 261-76.

–.2002. “A Critique of Fractional Age Assumptions,” Insurance: Mathematics & Economics 30: 363-70.

* Bruce L. Jones, F.S.A., F.C.I.A., is an Associate Professor at the University of Western Ontario, Department of Statistical and Actuarial Sciences, Western Science Centre, 1151 Richmond Street, London, ON, Canada, N6A 5157, e-mail: jones@stats.uwo.ca.

^ John A. Mereu, F.S.A., F.C.I.A., is a lecturer at the University of Western Ontario, Western Science Centre, London, ON, Canada, N6A 5137, e-mail: mereu@stats.uwo.ca.

^^ Esther Frostig is an Associate Professor at the Department of Statistics, University of Haifa, Haifa, Israel, 31905, e-mail: frostig@stat.haifa.ac.il

Copyright Society of Actuaries Jan 2003

Provided by ProQuest Information and Learning Company. All rights Reserved