Life-length Distribution Research Articles

Change of Life Distribution Via a Hazard Transformation: An Inequality with Application to Minimal Repair

We introduce a general transformation of hazard rates and discuss the corresponding change of a life length distribution. Minimal repair transformations are shown to be special cases of this framework which builds on general results concerning likelihood ratios for point processes, and in particular on the Girsanov theorem for point processes. We then study the role of the available information and the consequent definition of “state” in the change of distributions. By using the general notion of F-minimal repair, where F stands for the information which identifies the state of the considered device, we show that the “black box”-minimal repair modeling leads to a stochastically longer total life length than the more realistic one based on internal state information. Thus the former forms a potential source of bias in minimal repair modeling.

Inference for step-stress accelerated life tests under arbitrary right-censorship

A step-stress test is an accelerated life test in which the stress on an unfailed item is allowed to increase at a preassigned time. Step-stress testing is considered here from a non-parametric point of view when the observations have been arbitrarily right-censored. Thus, no assumptions are made about the particular form of the underlying distribution of life lengths. An estimator of the underlying life distribution under the use conditions stress is obtained. The estimator is shown to be strongly consistent.

On coupling of renewal processes with use of failure rates

We construct a coupling of renewal processes by using failure rates; it is particularly useful when the failure rate function of the lifelength distribution is monotone. For the case when that function is decreasing, the speed of convergence towards stationarity of the renewal processes is established, with ease, and a unifying treatment of monotonicity properties of functionals of such processes is given. A relaxed condition for the key renewal theorem is presented. The value of the coupling for the IFR case is discussed briefly.

Formulae for the average unavailability (MFDT) of a coherent system with periodically tested components

In this paper we give some formulae (bounds) for the average unavailability/mean fractional dead time of a coherent system with periodically tested components. The components are assumed to be independent (associated) with exponential lifelength distributions.

On the computation of certain measures of importance of system components

In this paper we establish some approximation-formulae (bounds) for two measures of importance of the components in a coherent system: Barlow-Proschan ( Stochastic Process. Appl. 3, 153–173, 1975) measure and Natvig ( Stochastic Process. Appl. 9, 319–330, 1979) measure. A finite time-interval [0, τ] is considered. The components are assumed to have Weibull lifelength distributions.

Robustness of Sequential Exponential Life-Testing Procedures

Abstract Sequential life-testing procedures for the exponential distribution are often used when the underlying distribution of life lengths is not exponential. In this article, we investigate the robustness of these procedures with respect to the risks and the expected sample sizes, when the underlying distribution has a monotone failure rate. We also describe the regions in which the true operating characteristic curves of the misused tests lie. A main conclusion is that a producer could penalize a consumer by overloading the test bench with unsatisfactory items, whenever the latter have an increasing failure rate.

Robustness of Sequential Exponential Life-Testing Procedures

Abstract Sequential life-testing procedures for the exponential distribution are often used when the underlying distribution of life lengths is not exponential. In this article, we investigate the robustness of these procedures with respect to the risks and the expected sample sizes, when the underlying distribution has a monotone failure rate. We also describe the regions in which the true operating characteristic curves of the misused tests lie. A main conclusion is that a producer could penalize a consumer by overloading the test bench with unsatisfactory items, whenever the latter have an increasing failure rate.

Systems weakend by failures

The ideas of a dynamic approach to the analysis of multivariate life length distributions, introduced in Arjas (1981a) and Arjas and Norros (1984), are developed further. Basic definitions are given in terms of prediction processes. Properties of martingales jumping downwards at failure times are studied. Finally, the spaecial case of a general multivariate exponential distribution is considered.

Inference for step-stress accelerated life tests

In this paper we consider the more realistic aspect of accelerated life testing wherein the stress on an unfailed item is allowed to increase at a preassigned test time. Such tests are known as step-stress tests. Our approach is nonparametric in that we do not make any assumptions about the underlying distribution of life lengths. We introduce a model for step-stress testing which is based on the ideas of shock models and of wear processes. This model unifies and generalizes two previously proposed models for step-stress testing. We propose an estimator for the life distribution under use conditions stress and show that this estimator is strongly consistent.

Cell elongation and division probability during the Escherichia coli growth cycle.

A new method is presented for determining the growth rate and the probability of cell division (separation) during the cell cycle, using size distributions of cell populations grown under steady-state conditions. The method utilizes the cell life-length distribution, i.e., the probability that a cell will have any specific size during its life history. This method was used to analyze cell length distributions of six cultures of Escherichia coli, for which doubling times varied from 19 to 125 min. The results for each culture are in good agreement with a single model of growth and division kinetics: exponential elongation of cells during growth phase of the cycle, and normal distributions of length at birth and at division. The average value of the coefficient of variation was 13.5% for all strains and growth rates. These results, based upon 5,955 observations, support and extend earlier proposals that growth and division patterns of E. coli are similar at all growth rates and, in addition, identify the general growth pattern of these cells to be exponential.

Hazard functions and life distributions in discrete time

The problem of studying lifelength distributions in discrete time is considered for certain forms of hazard functions. A class of life distributions that consists of the geometric, the Waring and the negative hypergeometric distributions is shown to result when the hazard function is inversely proportional to some linear function of time.

Almost sure limit results for the supercritical Bellman-Harris process

As a consequence of the convergence of the age distribution, several other limit results follow for supercritical Bellman-Harris processes. These concern the distribution of split times, the growth ‘rate' of the process, and the distribution of ancestral lifelengths.

Almost sure limit results for the supercritical Bellman-Harris process

As a consequence of the convergence of the age distribution, several other limit results follow for supercritical Bellman-Harris processes. These concern the distribution of split times, the growth ‘rate' of the process, and the distribution of ancestral lifelengths.

Bayesian non-parametric estimation for age-dependent branching processes

The purpose of this paper is to obtain Bayes estimators for both the offspring and life-length distribution in the context of a Bellman-Harris age-dependent branching process. We take a non-parametric approach by letting the prior random distributions, for the offspring and life-length distributions, be independent Dirichlet processes. Our primary results concern the derivation of Bayes estimators, under weighted squared error loss for each distribution. We also indicate some of their asymptotic properties and briefly discuss the modifications that become necessary when the initial information is such that the prior random distribution cannot be taken to be independent.

Exponential growth of a branching process usually implies stable age distribution

Start a Bellman–Harris branching process from one or several ancestors, whose ages are identically distributed random variables. Assume that the life-length distribution decays more quickly than exponentially and that the distribution of ages at start does not give too much mass to high ages (in a sense to be made precise). Then, if the expected population size is an exponential function of time, the ages must follow the stable age distribution of the process.

Exponential growth of a branching process usually implies stable age distribution

Start a Bellman–Harris branching process from one or several ancestors, whose ages are identically distributed random variables. Assume that the life-length distribution decays more quickly than exponentially and that the distribution of ages at start does not give too much mass to high ages (in a sense to be made precise). Then, if the expected population size is an exponential function of time, the ages must follow the stable age distribution of the process.

On possible rates of growth of age-dependent branching processes with immigration

It is shown that if ϕ is a given function out of a large class satisfying a certain regularity condition, then a supercritical age-dependent branching process {Z(t)} exists with deterministic immigration and given life-length and family-size distributions such that Z(t)/(eat ϕ(t)) converges in probability to a non-zero constant, a being the appropriate Malthusian parameter.As an easy corollary one discovers the asymptotic behaviour of some processes with random immigration.

On possible rates of growth of age-dependent branching processes with immigration

It is shown that if ϕ is a given function out of a large class satisfying a certain regularity condition, then a supercritical age-dependent branching process {Z(t)} exists with deterministic immigration and given life-length and family-size distributions such that Z(t)/(eat ϕ(t)) converges in probability to a non-zero constant, a being the appropriate Malthusian parameter. As an easy corollary one discovers the asymptotic behaviour of some processes with random immigration.

The application of stochastic processes in function theory

(v) A new method which promises to be very useful in the critical case is based on a comparison of generating functions. Consider two age-dependent processes with the same lifelength distribution G, but different particle production generating functions f(s) and h(s). Let F(s, t) and H(s, t) be the corresponding generating functions of the process. Then we have the following age-dependent analog of Spitzer's comparison lemma for the G-W process.

Asymptotic growth of a class of size-and-age-dependent birth processes

A class of binary fission stochastic population models is described, in which the fission probabilities may depend on the age of an individual and the total population size. Age-dependent binary branching processes with Erlangian lifelength distributions are a special case. An asymptotic expression for the growth of the population size is developed, which generalizes known theorems about the asymptotic exponential growth of a branching process.