Discrete Lifetime Distributions Research Articles

Evaluating the Discrete Generalized Rayleigh Distribution: Statistical Inferences and Applications to Real Data Analysis

Various discrete lifetime distributions have been observed in real data analysis. Numerous discrete models have been derived from a continuous distribution using the survival discretization method, owing to its simplicity and appealing formulation. This study focuses on the discrete analog of the newly generalized Rayleigh distribution. Both classical and Bayesian statistical inferences are performed to evaluate the efficacy of the new discrete model, particularly in terms of relative bias, mean square error, and coverage probability. Additionally, the study explores different important submodels and limiting behavior for the new discrete distribution. Various statistical functions have been examined, including moments, stress–strength, mean residual lifetime, mean past time, and order statistics. Finally, two real data examples are employed to evaluate the new discrete model. Simulations and numerical analyses play a pivotal role in facilitating statistical estimation and data modeling. The study concludes that the discrete generalized Rayleigh distribution presents a notably appealing alternative to other competing discrete distributions.

Residual and Past Discrete Tsallis and Renyi Extropy with an Application to Softmax Function.

In this paper, based on the discrete lifetime distribution, the residual and past of the Tsallis and Renyi extropy are introduced as new measures of information. Moreover, some of their properties and their relation to other measures are discussed. Furthermore, an example of a uniform distribution of the obtained models is given. Moreover, the softmax function can be used as a discrete probability distribution function with a unity sum. Thus, applying those measures to the softmax function for simulated and real data is demonstrated. Besides, for real data, the softmax data are fit to a convenient ARIMA model.

Applying User Surveys and Accelerated Tests Data to Estimate Reliability of New Consumer Products Using a Discrete Life Distribution Model

Assessing the reliability of consumer products that are subject to random discrete damage is critical during their design. Previous studies have approximated a continuous life distribution for consumer products by treating the occurrence (cycles) of accumulating damage as a continuous random variable. However, when the lifetime of a product is only a few damage cycles (e.g., ten drop cycles of a laptop), using a discrete lifetime distribution is more accurate. Using a discrete lifetime distribution is challenging because it contains a summation term with an unknown upper bound, which makes calculating its likelihood function cumbersome. This paper proposes a method to address this issue. First, the upper bound of the summation term is approximated through a gradient descent algorithm and the Maximum Likelihood Estimation method. Then, the upper bound is fixed, and the other parameters of the reliability model are estimated using Bayesian analysis. The paper presents a hypothetical case study that shows that using the discrete model leads to a more accurate estimation of product life when dealing with a small number of cycles.

New Discrete Lifetime Distribution with Applications to Count Data

In this paper, we present a new class of distribution called generalized Hermite–Genocchi distribution (GHGD). This model is obtained by compounding generalized Hermite–Genocchi polynomials given by Gould and Hopper with powers series distribution. Statistical properties and reliability characteristics are studied. The model has been applied to several real data. Finally, a simulation study is performed to assess the performance of the model.

Revisiting discrete time age replacement policy for phase-type lifetime distributions

For a system (or unit) whose lifetime is measured by the number cycles, according to the discrete time age replacement policy, it is replaced preventively after n cycles or correctively at failure, whichever occurs first. In this paper, discrete time age replacement policy is revisited when the lifetime of the system is modeled by a discrete phase-type distribution. In particular, the necessary conditions for the unique and finite replacement cycle which minimizes the expected cost per unit of time are obtained. The necessary conditions are mainly based on the behavior of the hazard rate. The results are illustrated for some special discrete phase-type lifetime distributions. Computational results are also presented for the optimal replacement cycle under specific real life setups.

Ageing Properties of Modified Proportional Hazard Rates Model for Discrete Lifetime Distributions

Ageing Properties of Modified Proportional Hazard Rates Model for Discrete Lifetime Distributions

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.

Odds Function and Odds Rate for Discrete Lifetime Distributions

There has been recent interest in studying the potential of odds function and odds rate in lifetime data analysis. In the present study, we define odds function and odds rate for discrete lifetime distributions. The relationships of these measures with other reliability concepts are discussed. We present the usual aging properties in terms of these measures. The comparison of various reliability concepts of two lifetimes by stochastic orders of their odds functions is investigated. Finally, we apply the proposed results to two real life data.

A Differential Equation for a Class of Discrete Lifetime Distributions with an Application in Reliability

It is shown that the probability generating function of a lifetime random variable T on a finite lattice with polynomial failure rate satisfies a certain differential equation. The interrelationship with Markov chain theory is highlighted. The differential equation gives rise to a system of differential equations which, when inverted, can be used in the limit to express the polynomial coefficients in terms of the factorial moments of T. This then can be used to estimate the polynomial coefficients. Some special cases are worked through symbolically using Computer Algebra. A simulation study is used to validate the approach and to explore its potential in the reliability context.

Some Discrete Lifetime Distributions with Bathtub-Shaped Hazard Rate Functions

ABSTRACT Unlike continuous distributions, discrete models capable of modeling failure times with bathtub-shaped hazard rate functions have not received deserving attention in the statistical reliability literature. In this article, we show that the shape of the hazard rate function of a continuous lifetime model is partially preserved under the usual discretization of the model. Then, two continuous distributions with bathtub-shaped hazard rates are discretized, and some properties of the obtained discrete versions are studied. Graphical methods are utilized to obtain initial estimates of the parameters that are necessary for calculation of the maximum likelihood estimator. Finally, we fit the studied models along with some other proper discrete models to a set of real data and show that these models may be good candidates for describing discrete lifetime data.

Reversed variance residual life function and its properties in discrete lifetime models

PurposeIn reliability studies, interests in discrete failure data came relatively late in comparison to its continuous analogue. Also, discrete failure data arise in several common situations. So, in this paper the authors try to study some reliability concepts such as reversed variance and reversed mean residual life functions based on discrete lifetime random variable.Design/methodology/approachSupposed T be a non‐negative discrete random variable, then based on reversed residual random variable Tk*=(k−T|T≤k), some useful and applicable relations and bounds are achieved.FindingsIn this paper, the authors study the reversed variance residual life in discrete lifetime distributions, the results of which are not similar to the continuous case. Its relationship with reversed mean residual life and reversed residual coefficient of variation are obtained. Also, its monotonicity and the associated ageing classes of distributions are discussed. Some characterization results of the class of increasing reversed variance residual life, which is denoted by IRVR, are presented and the upper bound for reversed variance residual life under some conditions is obtained.Practical implicationsThere are many situations where a continuous time is inappropriate for describing the lifetime of devices and other systems. For example, the lifetime of many devices in industry, such as switches and mechanical tools, depends essentially on the number of times that they are turned on and off or the number of shocks they receive. In such cases, the time to failure is often more appropriately represented by the number of times they are used before they fail, which is a discrete random variable.Originality/valueAll the results based on discrete reversed residual lifetime, such as the relationships among reversed mean, variance and coefficient of variation residual lifetime and also their monotonicity ageing classes, are new.

A Note on Closure Properties of Classes of Discrete Lifetime Distributions

The main purpose of this note is to provide further properties of discrete lifetime distributions based on variance residual lifetimes (VRL). New discrete aging classes are introduced in terms of discrete version of VRL. We demonstrate closure of discrete variance residual lifetime under convolution and mixing.

Probabilistic properties of discrete mean and variance reversed residual lifetime functions

The random variable () () t = where X is a discrete random variable. Besides similar results for discrete and continuous lifetime distributions, relationships with its mean, monotonicity and the associated ageing classes of distributions are obtained for discrete cases. Furthermore, some characterization results about the class of increasing variance reversed residual lifetime distributions based on the mean reversed residual lifetime and the reversed residual coefficient of variation, are presented and the lower and upper bound for them are achieved. Also, we will investigate the connection between discrete increasing variance reversed residual lifetimes and other classes of distributions. I will provide two simple characterizations of the increasing discrete variance reversed residual lifetime class of distributions based on discrete mean and variance reversed residual lifetime. Since Geometric, Weibull, Shift Geometric distributions belong to increasing variance reversed residual lifetime class distributions, therefore the

Variance residual life function in discrete random ageing

The random variable Xt= X − t¦X ≥ t, which is called the residual life random variable, has gathered the attention of most researchers in reliability. The mean and the variance of this variable in continuous distribution have been studied by several authors. But, in discrete case, only in recent years, some studies have been done for the mean of this variable. In this paper, we define and study the properties of variance of Tk= T − k¦T ≥ k where T is a discrete random variable. Besides similar results for discrete and continuous lifetime distributions, relationships with its mean, monotonicity and the associated ageing classes of distributions are obtained for discrete cases. Furthermore, some characterization results about the class of increasing (decreasing) variance residual life distributions based on mean residual life and residual coefficient of variation, are presented and the lower and upper bound for them are achieved.

Aging Properties of a Discrete-Time Failure and Repair Model

We introduce a failure and repair model that is a discrete-time version of the pure birth shock model. Its discrete lifetime is a random sum with support on N, and with geometric summands, that represent the number of successful tasks between consecutive shocks performed by an engineering system. We provide distributional properties and conditions that lead to the discrete aging properties of such random sums. Our results provide qualitative insight into the discrete lifetime distribution of our model.

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.

Time-Resolved Fluorescence Studies on Bovine Serum Albumin Denaturation Process

The denaturation of Bovine Serum Albumin (BSA) by a chaotropic agent, guanidinium hydrochloride (GuH+Cl-) was studied by fluorescence lifetime analysis. The BSA was labelled with 1-anilino-8-naphthalene sulfonate (ANS) at two different molar ratios (1:1) and (1:10). The non-exponential fluorescence kinetics of the BSA-ANS complex at different stages of denaturation is analysed using three different models: a discrete tri-exponential sum, stretched exponential, and Gaussian lifetime distribution. In all cases, the fluorescence decay times decreased with protein denaturation. The results from the models show that there are at least two different binding sites located in the BSA protein with different water accessibility.

Classes of Discrete Lifetime Distributions

Several results that characterize the distribution of a lifetime variable, T, with probability mass function (pmf) p t , where t = 0, 1, 2,…, by its survivor function, S t = ∑ j≥t p j , its hazard function, h t = p t /S t , its cumulative hazard function, Λ t = − lnS t , its accumulated hazard function, , and its mean residual life function, L t = E[(T − t)|T ≥ t] (an initially faulty item is deemed to have a zero lifetime), are presented. These include results that have previously appeared in the literature as well as some new results. Differences in the terminology used by engineers, actuaries, and biostatisticians are pointed out and clarified. Attention is focussed on the relationships between the IFR/DFR, IFRA/DFRA, NBU/NWU, NBUE/NWUE, and IMRL/DMLR classes to which a discrete lifetime distribution and its current age distribution belong.

Optimal maintenance decisions on the basis of uncertain failure probabilities

Due to a lack of data, many maintenance optimisation models have to be initialised on the basis of expert judgment. Rather than eliciting the parameters of a continuous lifetime distribution, experts give more reliable answers when assessing a discrete lifetime distribution. If the prior uncertainty in the probabilities of failure per unit time is expressed in terms of a Dirichlet distribution, Bayes estimates can be obtained of three cost‐based criteria to compare maintenance decisions over unbounded time‐horizons: first, the expected average costs per unit time; second, the expected discounted costs over an unbounded horizon; and third, the expected equivalent average costs per unit time. Illustrates the maintenance model by determining optimal age replacement and lifecycle costing policies, which optimally balance both the failure cost against the preventive repair cost, and the initial cost against the future cost.

Mean residual life of lifetime distributions

This paper characterizes the general behaviors of the MRL (mean residual lives) for both continuous and discrete lifetime distributions, with respect to their failure rates. For the continuous lifetime distribution with failure rates with only one or two change-points, the characteristic of the MRL depends only on its mean and failure rate at time zero. For failure rates with "roller coaster" behavior, the subsequent behavior of the MRL depends on its MRL and failure-rates at the change points. Using the characterization, their behaviors for the: Weibull; lognormal; Birnbaum-Saunders; inverse Gaussian; and bathtub failure rate distributions are tabulated in terms of their shape parameters. For discrete lifetime distributions, for upside-down bathtub failure rate with only one change point, the characteristic of the MRL depends only on its mean and the probability mass function at time zero.