Life length

About: Life length is a research topic. Over the lifetime, 379 publications have been published within this topic receiving 7956 citations.

The asymptotic theory of extreme order statistics

Abstract: . Let X j denote the life length of the j th component of a machine. In reliability theory, one is interested in the life length Z n of the machine where n signifies its number of components. Evidently, Z n = min (X j : 1 ≤ j ≤ n). Another important problem, which is extensively discussed in the literature, is the service time W n of a machine with n components. If Y j is the time period required for servicing the j th component, then W n = max (Y j : 1 ≤ j ≤ n). In the early investigations, it was usually assumed that the X's or Y's are stochastically independent and identically distributed random variables. If n is large, then asymptotic theory is used for describing Z n or W n . Classical theory thus gives that the (asymptotic) distribution of these extremes (Z n or W n ) is of Weibull type. While the independence assumptions are practically never satisfied, data usually fits well the assumed Weibull distribution. This contradictory situation leads to the following mathematical problems: (i) What type of dependence property of the X's (or the Y's) will result in a Weibull distribution as the asymptotic law of Z n (or W n )? (ii) given the dependence structure of the X's (or Y's), what type of new asymptotic laws can be obtained for Z n (or W n )? The aim of the present paper is to analyze the recent development of the (mathematical) theory of the asymptotic distribution of extremes in the light of the questions (i) and (ii). Several dependence concepts will be introduced, each of which leads to a solution of (i). In regard to (ii), the following result holds: the class of limit laws of extremes for exchangeable variables is identical to the class of limit laws of extremes for arbitrary random variables. One can therefore limit attention to exchangeable variables. The basic references to this paper are the author's recent papers in Duke Math. J. 40 (1973), 581–586, J. Appl. Probability 10 (1973, 122–129 and 11 (1974), 219–222 and Zeitschrift fur Wahrscheinlichkeitstheorie 32 (1975), 197–207. For multivariate extensions see H. A. David and the author, J. Appl. Probability 11 (1974), 762–770 and the author's paper in J. Amer. Statist. Assoc. 70 (1975), 674–680. Finally, we shall point out the difficulty of distinguishing between several distributions based on data. Hence, only a combination of theoretical results and experimentations can be used as conclusive evidence on the laws governing the behavior of extremes.

Mathematics in Population Biology

Abstract: The formulation, analysis, and re-evaluation of mathematical models in population biology has become a valuable source of insight to mathematicians and biologists alike This book presents an overview and selected sample of these results and ideas, organized by biological theme rather than mathematical concept, with an emphasis on helping the reader develop appropriate modeling skills through use of well-chosen and varied examples Part I starts with unstructured single species population models, particularly in the framework of continuous time models, then adding the most rudimentary stage structure with variable stage duration The theme of stage structure in an age-dependent context is developed in Part II, covering demographic concepts, such as life expectation and variance of life length, and their dynamic consequences In Part III, the author considers the dynamic interplay of host and parasite populations, ie, the epidemics and endemics of infectious diseases The theme of stage structure continues here in the analysis of different stages of infection and of age-structure that is instrumental in optimizing vaccination strategies Each section concludes with exercises, some with solutions, and suggestions for further study The level of mathematics is relatively modest; a "toolbox" provides a summary of required results in differential equations, integration, and integral equations In addition, a selection of Maple worksheets is provided The book provides an authoritative tour through a dazzling ensemble of topics and is both an ideal introduction to the subject and reference for researchers

A new family of life distributions

Abstract: : A new two parameter family of life length distributions is presented which is derived from a model for fatigue. This derivation follows from considerations of renewal theory for the number of cycles needed to force a fatigue crack extension to exceed a critical value. Some closure properties of this family are given and some comparisons made with other families such as the lognormal which have been previously used in fatigue studies.

Estimation for a family of life distributions with applications to fatigue

Abstract: : The estimation problem is studied for a new two-parameter family of life length distributions which has been previously derived from a model of fatigue crack growth. Maximum likelihood estimates of both parameters are obtained and iterative computing procedures are given and examined. A simple estimate of the median life is exhibited, shown to be consistent and then compared, favorably, with the maximum likelihood estimate. More importantly the asymptotic distribution of this estimate is shown to be within the same class of distributions as the observations themselves. This model, and these estimation procedures, are tried by fitting this distribution to several extensive sets of fatigue data and then some comparisons of practical significance are made.

Joint life annuities and annuity demand by married couples

Abstract: This article summarizes the range of joint-life annuity products that are currently available to married couples, and it explores the potential utility gain that such couples receive from access to actuarially fair annuity markets. It is more difficult to estimate this utility gain for couples than for individuals because a couple's value of annuitization will depend in part on the survivor benefits that are available after one spouse has died but while the other is still alive. Valuing joint and survivor annuities also requires recognition of the potentially important interactions between the members of a married couple, such as joint consumption, interdependent utilities, and correlated mortality rates. This article considers each of these issues. It develops an annuity valuation model for married couples and it estimates their "annuity equivalent wealth," the amount of wealth that couples would need in the absence of actuarially fair annuity markets in order to achieve the same utility level that they rece ive when such markets are available. The utility gain from annuitization is smaller for couples than for single individuals. Since most potential annuity buyers are married, this finding may help to explain the limited size of the market for single premium annuities in the United States. INTRODUCTION Annuities play an important role in the standard theory of consumer choice when life length is uncertain. Yaari (1965) showed that an individual with a fixed stock of resources and an uncertain lifetime should purchase an annuity contract to insure against the risk of outliving his resources. More recent work, such as Mitchell, Poterba, Warshawsky, and Brown (1999) (hereafter MPWB), finds substantial gains to annuitization for individual life-cycle consumers. Typical results suggest that a 65-year-old man who does not have access to an actuarially fair annuity market would be willing to forgo roughly one-third of his wealth if by doing so he could purchase an actuarially fair annuity with his remaining wealth. In spite of Yaari's (1965) insight, a number of studies have observed that the market for individual annuity contracts in the United States is very small. Standard explanations for the limited flow of new annuity purchases are adverse selection in the annuity market, individual bequest motives, and the presence of other annuitized resources. Several empirical studies, including Friedman and Warshawsky (1988, 1990) and MPWB (1999), have explored the extent of adverse selection. Numerous other studies surveyed in Laitner (1997) and (more briefly) in Altonji, Hayashi, and Kotlikoff (1997) and Laitner and Juster (1996) have focused on intergenerational altruism as a potential explanation for limited annuity demand. Simple altruistic models do not appear to provide a satisfactory explanation for observed patterns of intergenerational transfers. Auerbach, Kotlikoff, and Well (1992) show that when Social Security benefits, private defined benefit pension plan payments, and Medicare benefits are added together, more than half of the resources of the current elderly in the United States take the form of life-contingent payouts. Thus many households are already "annuitized." This may also explain the limited demand for additional private annuities. The functioning of annuity markets has recently attracted attention as part of the global policy debate on individual accounts Social Security systems. A central design issue in such systems concerns the way a retiree would spread accumulated resources over his or her remaining lifetime, or his or her lifetime and that of his or her spouse. Private annuities offer one way of spreading such resources; mandatory government-provided annuities are another. Although the treatment of married couples is an important issue in retirement income security, virtually all of the previous research on annuities has focused on individuals rather than couples as decision-making units. …