Classes Of Life Distributions Research Articles

Tests of non-monotonic stochastic aging notions in reliability theory

Testing of various classes of life distributions has been a subject of investigation for more than four decades. In this study we restrict ourselves to the problem of testing exponentiality against non-monotonic aging notions. We model non-monotonic aging using the notions of bathtub failure rate, increasing and then decreasing mean residual life and new worse then better than used in expectation classes. The different tests of exponentiality against these alternatives are discussed in detail.

On HNBUE class after specific age

In this article we introduce new classes of life distributions namely harmonic new better (worse) than used in expectation after specific age t 0 HNBUE ∗ t 0 ( HNWUE ∗ t 0 ). The closure properties under various reliability operations such as convolution, mixture, mixing and the homogeneous Poisson shock model of these classes are studied. Furthermore, nonparametric test is proposed to test exponentiality vs. the HNBUE ∗ t 0 class. The critical values and the powers of this test are calculated to assess the performance of the test. It is shown that the proposed test have high efficiencies for LFR and Weibull distributions. Sets of real data are used as examples to elucidate the use of the proposed test for practical problems.

New classes at Specific Age: Properties and Testing hypotheses

In this article we introduce new classes of life distributions namely new better (worse) than used in expectation at specific age t0 NBUE-t0 (NWUE-t0) and harmonic new better (worse) than used in expectation at specific age t0 HNBUE-t0 (HNWUE-t0). The closure properties under various reliability operations such as convolution, mixture, mixing and the homogeneous Poisson shock model of these classes are studied. Furthermore, nonparametric tests are proposed to test exponentiality versus the NBUE-t0 and HNBUE-t0 classes. The critical values and the powers of this tests are calculated to assess the performance of the tests. It is shown that the proposed tests have high efficiencies for some commonly used distributions in reliability. Sets of real data are used as examples to elucidate the use of the proposed tests for practical problems.

On quantile reversed residual lifetime and its aging properties

If the random variable T denotes the lifetime (T > 0, with probability one) of a unit, then the random variable T (t) = [t − T|T ⩽ t],, for a fixed t, is known as the inactivity time or the reversed residual lifetime. In this paper, we define a new class of life distributions based on the quantile of the random variable T( t ) and study its reliability properties. Some new results of the proposed class are demonstrated including some closure properties. In particular, the quantile reversed residual lifetime functions of some well-known life distributions are illustrated. Finally, we define a new stochastic ordering based on the quantile reversed residual lifetime and study its reliability properties and closure properties.

Tests for exponentiality against NBUE alternatives: a Monte Carlo comparison

Testing of various classes of life distributions has been addressed in the literature for more than 45 years. In this paper, we consider the problem of testing exponentiality (which essentially implies no ageing) against positive ageing which is captured by the fairly large class of new better than used in expectation (NBUE) distributions. These tests of exponentiality against NBUE alternatives are discussed and compared. The empirical size of the tests is obtained by simulations. Power comparisons for different popular alternatives are done using Monte Carlo simulations. These comparisons are made for both small and large sample sizes. The paper concludes with a discussion in which suggestions are made regarding the choices of the test when a particular alternative is suspected.

On some properties of the IDMRL class of life distributions

In this paper we consider some properties of the IDMRL class of life distributions, which have not been addressed in the literature thus far. Specifically we show that the IDMRL class of life distributions are closed under weak convergence. The second result concerns preservation under the Poisson shock model. We know that if shocks arrive according to a homogeneous Poisson process and P¯k is the probability of surviving the first k shocks, then H¯(t)=∑k=0∞P¯kP[N(t)=k] is the probability that the device survives beyond time. We show that if P¯k has the discrete IDMRL property, then H¯(t) is continuous IDMRL. We also investigate some closure properties of IDMRL distributions.

A Goodness of Fit Approach to the Class of Life Distributions with Unknown Age

Based on the goodness of fit approach, a new test is presented for testing exponentiality against the unknown age (used better than aged in convex ordering (UBAC)) Class of life distributions. The percentiles of this test are tabulated for sample sizes n=5(5)40. It is shown that the proposed test is simple, has high relative efficiency for some commonly used alternatives and enjoys a good power. An example in medical science is considered as a practical application of the proposed test. Copyright © 2011 John Wiley & Sons, Ltd.

Characterization and preservations of the variance inactivity time ordering and the increasing variance inactivity time class

If the random variable X denotes the lifetime of a unit, then the random variable X ( t ) = [ t - X X ⩽ t ] for a fixed t > 0 is known as the inactivity time. In this paper, based on the random variable X ( t ), a new class of life distributions, namely increasing variance inactivity time ( IVIT) and the concept of inactivity coefficient of variation ( ICV), are introduced. The closure properties of the IVIT class under some reliability operations, such as mixing, convolution and formation of coherent systems, are obtained.

Starshaped Ordering of Life Distributions and Its Aging Properties

In this paper, we introduce a new aging class of life distributions based upon the starshaped ordering. This class can be useful in situations where some new better than used aging classes fail. Some properties are presented including the relationship of this class to other aging classes, and the preservation properties under some reliability operations. Finally, we discuss testing exponentially against this class, and apply this test to some data sets in the context of reliability.

A new three-parameter ageing distribution

We propose a new three-parameter ageing distribution called the Weibull–Poisson (WP) distribution, which generalizes the exponential-Poisson (EP) distribution introduced by Kus (2007). This new distribution has a more general form of failure rate (hazard rate) function. With appropriate choice of parameter values, it is able to model three ageing classes of life distributions including decreasing failure rate (DFR), increasing failure rate (IFR), and modified upside-down-bathtub (MUBT)-shaped failure rate. It thus provides an alternative to many existing life distributions. Various properties of this distribution are discussed and the estimation of the parameters is considered by the expectation maximization (EM) algorithm. Also, the asymptotic variance–covariance matrices of these estimates are obtained. Furthermore, some expressions for the Rènyi and Shannon entropies are given. Simulation studies are performed and experimental results are illustrated based on a real data set.

Bayesian multivariate imperfect repair model

This paper generalizes a multivariate imperfect repair model by adapting a Bayesian approach. We concern with modeling a system whose components have specific multivariate distribution of dependent lifelengths. Upon failure, each components undergoes imperfect repair. Different sources of failure which affect the probabilities of perfect and minimal repair are distinguished. Each probability of perfect (minimal) repair is assumed unknown and assigned a prior distribution. Based on the proposed model, we derive the distribution of waiting time between two successive perfect repairs and its corresponding failure rates. We also study some preservation properties for certain nonparametric classes of life distributions and describe the singularity for the resulting distribution.

Testing NBU(2) class of life distribution based on goodness of fit approach

In this paper, new testing procedures for exponentiality against the NBU(2) class is addressed based on the goodness of fit approach. It is shown that the proposed test has high relative efficiency for some commonly used alternative and enjoys a good power. The critical values of the proposed statistic are calculated and some applications are given to elucidate the use of the proposed test in reliability analysis.

Median Inactivity Time Function and its Reliability Properties

Let the non-negative random variable X denote the life time of a unit, then the random variable X(t) = t – X for X ≤ t for a fixed t ∈ {x : FX (x) > 0}, is known as inactivity time or reversed residual life time. In this paper, we define a new non-parametric class of life distributions based on the median of the random variable X(t) and study its reliability properties. Some new results concerning the proposed class are given including some closure properties and characterizations. We also introduce and study a new stochastic ordering based on the median of X(t) and find its relationship with other well-known order relations. Finally, we provide the median inactivity time function of some well-known life time distributions.

Recent tests of exponentiality against IFR alternatives: a survey

Testing of various classes of life distributions has been a subject of investigation for more than four decades. In this study, we restrict ourselves to the problem of testing exponentiality (which essentially means no aging) against positive aging, which is captured by the class of increasing failure rate alternatives. Recent tests are discussed and compared. The empirical size of the tests is obtained by simulation. Power computations, using simulations, are done for each test procedure. These comparisons are done both for small and large sample sizes. Suggestions are made regarding the choice of the test when a particular alternative is suspected.

Some New Result Involving the NBUCA Class of Life Distributions

Some New Result Involving the NBUCA Class of Life Distributions

On a representation of weighted distributions

The representation of weighted distributions given by Błażej [Błażej, P., 2008. Preservation of classes of life distributions under weighting with a general weight function. Statist. Probab. Lett. 78, 3056–3061] is developed. New relations between weighted distributions and classes of life distributions and stochastic orders are established.

New results on the NBUFR and NBUE classes of life distributions

Some properties of the “new better than used in failure rate” (NBUFR) and the “new better than used in expectation” (NBUE) classes of life distributions are given. These properties include moment inequalities and moment generating

On The New Better Than Used Renewal Failure Rate at Specified Time

In this paper we introduce a new class of life distribution called the new better (worse) than used renewal failure rate at specified time t 0 NBURFR-t 0 (NWURFR-t 0). Some closure properties, such as closure under convolution, arbitrary and general mixture, and Shock model are discussed. Furthermore, a nonparametric test is proposed to test exponentiality against the NBURFR-t 0 class. The critical values, the powers and Pitman efficiencies are calculated for t 0 = 0:25.

Stochastic comparison and monotonicity of inactive record values

In this paper, we study the inactive record values and carry out a stochastic comparison of these quantities from two independent samples. The results established here strengthen those of Amiripour et al. [Amiripour, F., Khaledi, B.-E., Hu, T., Shojaei, R., 2009. Some new results on stochastic comparisons of record values, Communications in Statistics — Theory and Methods (in press)]. We also describe the behavior of the IFR (DFR) class of life distributions through the stochastic monotonicity of the inactive record values.

On testing alternative classes of life distribution with guaranteed survival times

Nonparametric procedures are proposed for testing exponentiality against several new aging classes of life distributions. The main idea is to test if the lifetime of a system, whose failure time has occurred in between visits, belongs to some alternative new aging classes of life distribution. This knowledge enables us to better estimate premium amounts for clients by more accurately estimating value when encountering left or interval-censored data. One possible approach to perform these tests is based on the empirical distribution function. The limiting distributions of the presented test statistics are given for a well known alternative when the null distribution is exponential. For small sample sizes, the power of the tests are calculated. The results derived by a Monte Carlo method show an excellent power of these procedures for some common alternative distributions.