Discrete Lifetimes Research Articles

Nonparametric Predictive Inference for Discrete Lifetime Data

This paper presents nonparametric predictive inference for discrete lifetime data. While lifetimes are mostly treated as continuous random variables in statistics, there are scenarios where time observations are recorded as discrete values, for example, in actuary, where lifetimes are often recorded as integers in years. The presented method provides lower and upper probabilities for a variety of events of interest involving discrete lifetimes, with examples provided for illustration. Furthermore, the discrete-time situation is considered for inference of the reliability of systems, with discrete-time data for components of different types and using the survival signature to combine inference on components’ reliability to quantify the overall system reliability.

Repair alert model when the lifetimes are discretely distributed

This paper deals with the repair alert models. They are used for analyzing lifetime data coming from engineering devices under maintenance management. Repair alert models have been proposed and investigated for continuous component lifetimes. Existing studies are concerned with the lifetimes of items described by continuous distributions. However, discrete lifetimes are also frequently encountered in practice. Examples include operating a piece of equipment in cycles, reporting field failures that are gathered weekly, and the number of pages printed by a device completed before failure. Here, the repair alert models are developed when device lifetimes are discrete. A wide class of discrete distributions, called the telescopic family, is considered for the component lifetimes, and the proposed repair alert model is explained in detail. Furthermore, the problem of estimating parameters is investigated and illustrated by analyzing a real data set.

Reliability and performance evaluation of weighted [formula omitted]-out-of-[formula omitted]:G system consisting of components with discrete lifetimes

For the k-out-of-n system consisting of components that have different weights, the system is in a good state if the total weight of working components is at least k. Such a system is known to be weighted k-out-of- n:G system. Although the weighted k-out-of-n system that has continuously distributed components’ lifetimes has been extensively studied, the discrete weighted k-out-of-n:G system has not been considered yet. The present paper fills this gap by modeling and analyzing the weighted k -out-of-n:G system that consists of discretely distributed components’ lifetimes. In particular, the behavior of the total capacity/weight of the system with respect to the component failures is evaluated. An optimization problem that is concerned with the determination of optimal number of spare components is also formulated by utilizing the mean lost capacity of the system.

On conditional residual lifetimes of coherent systems consisting of components with discrete lifetimes

On conditional residual lifetimes of coherent systems consisting of components with discrete lifetimes

Acceptance reliability sampling plan for discrete lifetime models

Various types of acceptance reliability sampling plans have been reported in the literature. Most of the existing studies deal with the lifetimes of items described by continuous distributions. However, discrete lifetimes are also frequently encountered in practice. For example, field failures are often collected and reported daily, weekly, and so forth. Items often operate in cycles and the experimenter observes the number of cycles successfully completed prior to failure. In this study, we develop a variables acceptance reliability sampling plan for items having discrete lifetimes. Furthermore, the reliability improvement in the population reliability characteristics after the testing procedure, as compared to those before the test, is discussed in detail. The reliability sampling acceptance test is applied for heterogeneous populations using the alternative discrete hazard rate for the corresponding proportional hazards model. Some numerical examples illustrate our findings.

Some conditional reliability properties of k-out-of-n system composed of different types of components with discrete independent lifetimes

In this paper, we study reliability properties of k-out-of-n system consisting of l(1le lle n) different types of components with discrete, independent lifetimes. We obtain some conditional survival functions of lifetime of a used system. Next, we use them to calculate two conditional failure probabilities of k-out-of-n systems and show that they are equal to unconditional failure probability of a k-out-of-(n-r) system, r<n-k+1. These results are extended versions of the respective ones existing in the literature.

Maximum likelihood estimators based on discrete component lifetimes of a k-out-of-n system

This paper deals with parametric inference about the independent and identically distributed discrete lifetimes of components of a k-out-of-n system. We consider the maximum likelihood estimation assuming that the available data consists of component failure times observed up to and including the moment of the breakdown of the system. First, we provide general conditions for the almost sure existence of a strongly consistent sequence of maximum likelihood estimators (MLE’s). Then, we focus on three typical discrete failure distributions—the Poisson, binomial and negative binomial distributions—and prove that in these cases the MLE’s are unique, provided they exist, and that they are strongly consistent. Finally, we complete our results by Monte Carlo simulation study. Interestingly, the inference considered in the paper can be viewed as equivalent to one based on Type-II right censored discrete data. Therefore, our results can as well be applied to the case when Type-II right censored sample from a discrete distribution is observed.

Moments of order statistics from DNID discrete random variables with application in reliability

In this paper, we present methods of obtaining single moments of order statistics arising from possibly dependent and non-identically distributed discrete random variables. We derive exact and approximate formulas convenient for numerical evaluation of such moments. To demonstrate their use, we tabulate means and second moments of order statistics from random vectors having some typical discrete distributions with selected parameter values. Next, we apply our results in reliability theory. We establish moments of discrete lifetimes of coherent systems consisting of heterogeneous and not necessarily independently working components. In particular, we obtain expression for expectations and variances of lifetimes of such systems. We give some illustrative examples.

Mixture and nonmixture cure fraction models assuming discrete lifetimes: Application to a pelvic sarcoma dataset.

Different cure fraction models have been used in the analysis of lifetime data in presence of curedpatients. This paper considers mixture and nonmixture models based on discrete Weibull distribution to model recurrent eventdata in presence of a cure fraction. The novelty of this study is the use of a discrete lifetime distribution in place ofusual existing continuous lifetime distributions for lifetime data in presence of cured fraction, censored data, andcovariates. In the verification of the fit of the proposed model it is proposed the use of randomized quantile residuals. Anextensive simulation study is considered to evaluate the properties of the estimates of the parameters related to the proposedmodel. As an illustration of the proposed methodology, it is considered an application considering a medical dataset related tolifetimes in a retrospective cohort study conducted by Puchner etal. (2017) that consists of 147 consecutive cases withsurgical treatment of a sarcoma of the pelvis between the years of 1980 and2012.

On Reliability Analysis of $k$ -Out-of-$n$ Systems Consisting of Heterogeneous Components With Discrete Lifetimes

In this paper, we study reliability of $k$ -out-of- $n$ systems with possibly heterogeneous components. We focus on the case when the lifetimes of components have a joint discrete distribution. The basic tool we use is a new closed-form formula describing the joint probability mass function of any subset of order statistics arising from not necessarily independent nor identically distributed discrete variates. This formula and some its modifications enable us to establish explicit expressions for unconditional and some conditional probabilities of a failure of the $k$ -out-of- $n$ system as well as expressions describing survival functions of various types of residual lifetimes of a used system.

On Methods of Estimation for the Type II Discrete Weibull Distribution

In this paper, we describe and analyze several methods of estimation for the type II discrete Weibull distribution, outlining their applicability and properties, assessing and comparing their performance via intensive Monte Carlo simulation experiments. We consider the standard maximum likelihood method, a method of proportion, and two variants of the least-squares method. The type II discrete Weibull distribution can be used in reliability engineering for modeling count data or discrete lifetimes and its use is theoretically motivated by its capability of modeling either bounded or unbounded support, and either increasing or decreasing failure rate. Statistical analyses of real datasets are presented to show the capability of the distribution in fitting reliability data and illustrate the application of the proposed inferential techniques.

Discrete Burr XII-Gamma Distributions: Properties and Parameter Estimations

In this paper, we propose three discrete analogues of the Burr XII-gamma distribution to meet the need of modelling discrete lifetimes in reliability. It is shown that hazard rate functions of discrete analogues are increasing, U-shaped and modified unimodal. Then we investigate estimation of their parameters using three methods (maximum likelihood, moments and empirical). From the results of extensive simulation runs, their accuracies are compared. Finally, we fit three proposed discrete distributions and discrete Burr XII and discrete gamma distributions to a discrete real data set and use AIC to compare them.

A maximum likelihood estimator for left-truncated lifetimes based on probabilistic prior information about time of occurrence

ABSTRACTIn forestry, many processes of interest are binary and they can be modeled using lifetime analysis. However, available data are often incomplete, being interval- and right-censored as well as left-truncated, which may lead to biased parameter estimates. While censoring can be easily considered in lifetime analysis, left truncation is more complicated when individual age at selection is unknown. In this study, we designed and tested a maximum likelihood estimator that deals with left truncation by taking advantage of prior knowledge about the time when the individuals enter the experiment. Whenever a model is available for predicting the time of selection, the distribution of the delayed entries can be obtained using Bayes' theorem. It is then possible to marginalize the likelihood function over the distribution of the delayed entries in the experiment to assess the joint distribution of time of selection and time to event. This estimator was tested with continuous and discrete Gompertz-distributed lifetimes. It was then compared with two other estimators: a standard one in which left truncation was not considered and a second estimator that implemented an analytical correction. Our new estimator yielded unbiased parameter estimates with empirical coverage of confidence intervals close to their nominal value. The standard estimator leaded to an overestimation of the long-term probability of survival.

The Arabidopsis exocyst subunit SEC3A is essential for embryo development and accumulates in transient puncta at the plasma membrane

The exocyst is a protein complex that is essential for polarized secretion in mammals and fungi. Although the exocyst is essential for plant development, its precise function has not been elucidated. We studied the role of exocyst subunit SEC3A in plant development and its subcellular localization. T-DNA insertional mutants were identified and complemented with a SEC3A-green fluorescent protein (GFP) fusion construct. SEC3A-GFP localization was determined using confocal microscopy. sec3a mutants are defective in the globular to heart stage transition in embryogenesis. SEC3A-GFP has similar cell plate localization to the other plant exocyst subunits. In interphase cells, SEC3A-GFP localizes to the cytoplasm and to the plasma membrane, where it forms immobile, punctate structures with discrete lifetimes of 2-40s. These puncta are equally distributed over the cell surface of root epidermal cells and tip growing root hairs. The density of puncta does not decrease after growth termination of these cells, but decreases strongly when exocytosis is inhibited by treatment with brefeldin A. SEC3A does not appear to be involved in polarized secretion for cell expansion in tip growing root hairs. The landmark function performed by SEC3 in mammals and yeast is likely to be conserved in plants.

The application of artificial neural networks to the inversion of the positron lifetime spectrum

A new method of processing positron annihilation lifetime spectra is proposed. It is based on an artificial neural network (ANN)-back propagation network (BPN). By using data from simulated positron lifetime spectra which are generated by a simulation program and tested by other analysis programs, the BPN can be trained to extract lifetime and intensity from a positron annihilation lifetime spectrum as an input. In principle, the method has the potential to unfold an unknown number of lifetimes and their intensities from a measured spectrum. So far, only a proof-of-principle type preliminary investigation was made by unfolding three or four discrete lifetimes. The present study aims to design the network. Besides, the performance of this method requires both the accurate design of the BPN structure and a long training time. In addition, the performance of the method in practical applications is dependent on the quality of the simulation model. However, the chances of satisfying the above criteria appear to be high. When appropriately developed, a trained network could be a very efficient alternative to the existing methods, with a very short identification time. We have used the artificial neural network codes to analyze data such as the positron lifetime spectra for single crystal materials and monocrystalline silicon. Some meaningful results are obtained.

Limit of hazard rate function of coherent system with discrete life

In this paper we study the limiting behavior of the hazard rate function of a coherent system &#x1d4ae; whose components are independent and have discrete lifetimes. The lifetime of the system &#x1d4ae; can be viewed as the lifetime of a parallel system consisting of the series systems obtained from the minimal paths of &#x1d4ae;. If we appropriately interpret the ‘strongest’ minimal path(s) of the system &#x1d4ae;, it will be shown that the limit of the hazard rate functions of &#x1d4ae; is the same as that of the ‘strongest’ path(s) of &#x1d4ae;. The application of the main results are also provided. Copyright © 2010 John Wiley & Sons, Ltd.

Signatures of carrier multiplication in the frequency resolved fluorescence spectra from polaritons

Using the micro-cavity coupled to weakly confined excitons in quantum dots we investigated the signatures of carrier multiplication in the single and two-photon emitted frequency resolved fluorescence. Those are compared to the spectra provided by the conventional multi-photon induced fluorescence. The two processes are distinguished by the statistics of the initial polariton distribution. We argue that exciton emission spectra turned out to be CM insensitive, while the polaritons emission demonstrates strong spectral signatures of the CM. When the polaritonic effect dominates the confinement-boosted excitons' nonlinearities, increasing quantum yield leads to consecutive appearance of Rabi multiplets. The correlation spectra reveal the formation of more multiplets as compared to the single photon emission. When the confinement increases the multiplets start to overlap and their classification requires transient measurements of the spectral snapshots. These are based on the discrete Auger lifetimes of the multi-excitons.

Excited‐state lifetime studies of the three tryptophan residues in the N‐lobe of human serum transferrin

The energy transfer from the three Trp residues at positions 8, 128, and 264 within the human serum transferrin (hTF) N-lobe to the ligand to metal charge transfer band has been investigated by monitoring changes in Trp fluorescence emission and lifetimes. The fluorescence emission from hTF N-lobe is dominated by Trp264, as revealed by an 82% decrease in the quantum yield when this Trp residue is absent. Fluorescence lifetimes were determined by multifrequency phase fluorometry of mutants containing one or two Trp residues. Decays of these samples are best described by two or three discrete lifetimes or by a unimodal Lorentzian distribution. The discrete lifetimes and the center of the lifetime distribution for samples containing Trp128 and Trp264 are affected by iron. The distribution width narrows on iron removal and is consistent with a decrease in dynamic mobility of the dominant fluorophore, Trp264. Both the quantum yield and the lifetimes are lower when iron is present, however, not proportionally. The greater effect of iron on quantum yields is indicative of nonexcited state quenching, i.e., static quenching. The results of these experiments provide quantitative data strongly suggesting that Förster resonance energy transfer is not the sole source of Trp quenching in the N-lobe of hTF.

Confidence intervals of the hazard rate function for discrete distributions using mixtures

The statistical models and methods for lifetime data mainly deal with continuous nonnegative lifetime distributions. However, discrete lifetimes arise in various common situations where either the clock time is not the best scale for measuring lifetime or the lifetime is measured discretely. In most settings involving lifetime data, the population under study is not homogenous. Mixture models, in particular mixtures of discrete distributions, provide a natural answer to this problem. Nonparametric mixtures of power series distributions are considered, as for instance nonparametric mixtures of Poisson laws or nonparametric mixtures of geometric laws. The mixing distribution is estimated by nonparametric maximum likelihood (NPML). Next, the NPML estimator is used to build estimates and confidence intervals for the hazard rate function of the discrete lifetime distribution. To improve the performance of the confidence intervals, a bootstrap procedure is considered where the estimated mixture is used for resampling. Various bootstrap confidence intervals are investigated and compared to the confidence intervals obtained directly from the NPML estimates.

A SURVEY ON DISCRETE LIFETIME DISTRIBUTIONS

This paper presents a comprehensive survey of discrete probability distributions used in reliability for modeling discrete lifetimes of nonrepairable systems. The basic properties of each model are given. A classification into two families is proposed, highlighting the interest of using a Pólya urn scheme. The quality of the estimation of models parameters is numerically assessed. Some criteria are given in order to select among the presented distributions the most useful for applications.