Reversed Failure Rate Research Articles

A Novel Modified Alpha Power Transformed Weibull Distribution and Its Engineering Applications

This paper suggests a new modified version of the traditional Weibull distribution by adding a new shape parameter utilising the modified alpha power transformed technique. We refer to the new model as modified alpha power transformed Weibull distribution. The attractiveness and significance of the new distribution lie in its power to model monotone and non-monotone failure rate functions, which are quite familiar in environmental investigations. Its hazard rate function can be decreasing, increasing, bathtub and upside-down then bathtub shaped. Diverse structural properties of the proposed model are acquired including quantile function, moments, entropies, order statistics, residual life and reversed failure rate function. The parameters of the distribution were estimated using the maximum likelihood function. The maximum likelihood method is employed to estimate the model parameters and the approximate confidence intervals are also computed. Via a simulation study, the performance of the point and interval estimates are compared using different criteria. Employing real lifetime data sets, we verify that the offered model furnishes a better fit than some other lifetime models including Weibull, gamma and alpha power Weibull models.

Stochastic Ordering Inequalities in Parallel and Series Systems with Gumble Distributed Components

The stochastic comparisons of parallel and series systems are worthy of study. In this article, we present some stochastic comparisons of parallel and series systems having independent components from Gumble distribution which is also known as an extreme value type-I distribution with two parameters (one location and one shape). This study only deals with the location parameter. Here, we first put a condition for the likelihood ratio ordering of the parallel systems and second, we use the concept of vector majorization technique to compare the systems by the reversed failure rate ordering, the failure rate ordering, the dispersive ordering, and the less uncertainty ordering with respect to the location parameter. AMS 2020 subject classifications: 62G30, 60E15, 60K10

Modeling Reliability Engineering Data Using Scale-Invariant Quasi-Inverse Lindley Model

An important property that any lifetime model should satisfy is scale invariance. In this paper, a new scale-invariant quasi-inverse Lindley (QIL) model is presented and studied. Its basic properties, including moments, quantiles, skewness, kurtosis, and Lorenz curve, have been investigated. In addition, the well-known dynamic reliability measures, such as failure rate (FR), reversed failure rate (RFR), mean residual life (MRL), mean inactivity time (MIT), quantile residual life (QRL), and quantile inactivity time (QIT) are discussed. The FR function considers the decreasing or upside-down bathtub-shaped, and the MRL and median residual lifetime may have a bathtub-shaped form. The parameters of the model are estimated by applying the maximum likelihood method and the expectation-maximization (EM) algorithm. The EM algorithm is an iterative method suitable for models with a latent variable, for example, when we have mixture or competing risk models. A simulation study is then conducted to examine the consistency and efficiency of the estimators and compare them. The simulation study shows that the EM approach provides a better estimation of the parameters. Finally, the proposed model is fitted to a reliability engineering data set along with some alternatives. The Akaike information criterion (AIC), Kolmogorov-Smirnov (K-S), Cramer-von Mises (CVM), and Anderson Darling (AD) statistics are used to compare the considered models.

On a new flexible Lomax distribution: statistical properties and estimation procedures with applications to engineering and medical data

<abstract><p>This paper introduces a new extension of the traditional Lomax distribution. The current distribution, which has one scale and two shape parameters, is known as logarithmic transformed Lomax distribution. The hazard rate function of the new distribution based on parameter values, making monotonically decreasing, upside-down and bathtub upside-down shaped. Some properties of the proposed distribution including mixture expansion, moments, entropies, probability weighted moments, order statistics, reversed failure rate and mean past life time are discussed. The distribution parameters are estimated using four frequentist methods: namely, maximum likelihood, least squares, weighted least squares, and maximum product of spacing. The efficiency of the different estimators is compared using a simulation analysis using mean square error. Finally, two real data sets for service times of Aircraft Windshield and survival times of a group of patients given chemotherapy medication are analyzed and compared to some other competing distributions. Based on the results of data analysis, the estimates of minimum and maximum values for the service times of Aircraft Windshield and the survival times of patients given chemotherapy medication are obtained.</p></abstract>

SOME INFERENCES ON OFFSET NORMAL DISTRIBUTION: PROPERTIES AND CHARACTERIZATIONS

The offset normal distribution, defined as the induced shape parameter of a Gaussian random configuration in the plane, is one of the most important probability distributions with tremendous applications both in pure and applied mathematics. In this paper some basic properties of the offset normal distribution are discussed. Based on the basic properties, several characterizations are presented by the left and right truncated moments. The first characterization result is based on a relation between left truncated moment and failure rate function. The second characterization result is based on a relation between right truncated moment and reversed failure rate function.

On the behavior of the failure rate and reversed failure rate in engineering systems

AbstractIn this paper the behaviour of the failure rate and reversed failure rate of an n-component coherent system is studied, where it is assumed that the lifetimes of the components are independent and have a common cumulative distribution function F. Sufficient conditions are provided under which the system failure rate is increasing and the corresponding reversed failure rate is decreasing. We also study the stochastic and ageing properties of doubly truncated random variables for coherent systems.

Connection of Generalized Failure Rate and Generalized Reversed Failure Rate with Inequality Curves

The purpose of this paper is to introduce some properties of generalized failure rate (GFR) and generalized reversed failure rate (GRFR), study their connection with inequality measures and then apply it to make comparisons of some reliability models. In the second part of our paper, we obtain the properties for the GRFR based on the properties of the GFR and weighted distributions. The importance of such a study lies in the usefulness of using. Finally, we present two examples to illustrate the applications of the relationships outlined in this paper. A real dataset is used for clarifying presented concepts.

THE TRANSMUTED GENERALIZED LINDLEY DISTRIBUTION: PROPERTIES AND AN APPLICATION TO A DATA SET ON TIME-UP-TO-CURE OF PATIENTS TREATED WITH A TRIAZOLE ANTIFUNGAL DRUG IN AN INTENSIVE CARE UNIT

In this paper, we consider transmuted generalized Lindley distribution, obtained via the quadratic rank transmutation map under the Lindley distribution. This distribution exhibits, in addition to decreasing, increasing and bathtub hazard rates, depending on its parameters, also unimodal hazard rate shape. A comprehensive mathematical treatment of this distribution is provided. Expressions for the moment generating function, moments, order statistics, residual life and reversed failure rate function are derived. The model parameters are estimated by the maximum likelihood method. A simulation study is performed to verify the behavior of the estimation procedure in terms of mean square errors and coverage probability. Global and local influence diagnostic procedures are provided. We then analyse a real data set on time-up-to-cure of patients treated with a triazole antifungal drug in an intensive care unit in Brazil.

A New Five - Parameter Lifetime Model: Theory and Applications

In this paper we defined a new lifetime model called the the Exponentiated additive Weibull (EAW) distribution. The proposed distribution has a number of well-known lifetime distributions as special sub-models, such as the additive Weibull, exponentiated modified Weibull, exponentiated Weibull and generalized linear failure rate distributions among others. We obtain quantile, moments, moment generating functions, incomplete moment, residual life and reversed Failure Rate Functions, mean deviations, Bonferroni and Lorenz curves. The method of maximum likelihood is used for estimating the model parameters. Applications illustrate the potentiality of the proposed distribution.

Weighted Lomax distribution.

The Lomax distribution (Pareto Type-II) is widely applicable in reliability and life testing problems in engineering as well as in survival analysis as an alternative distribution. In this paper, Weighted Lomax distribution is proposed and studied. The density function and its behavior, moments, hazard and survival functions, mean residual life and reversed failure rate, extreme values distributions and order statistics are derived and studied. The parameters of this distribution are estimated by the method of moments and the maximum likelihood estimation method and the observed information matrix is derived. Moreover, simulation schemes are derived. Finally, an application of the model to a real data set is presented and compared with some other well-known distributions.

SOME SUFFICIENT CONDITIONS FOR RELATIVE AGING OF LIFE DISTRIBUTIONS

In reliability theory, Cox's proportional hazard model is quite popular and widely used. In many situations, it is observed that failure rates under consideration are not proportional, rather they cross each other. In such situations, an alternative to Cox's proportional hazard model may be monotone hazard ratio model (provided the ratio exists). A notion of relative aging based on increasing hazard ratio was introduced by Kalashnikov and Rachev [19]. Sengupta and Deshpande [40] further explored this model and posited two other notions of relative aging based on increasing reversed failure rate ratio and increasing mean residual life ratio. In this study, for two life distributions, we derive sufficient conditions under which a life distribution ages faster than the other with respect to notions of relative aging described above. These sufficient conditions are easy to verify and can be used in practical applications where one is interested in studying relative aging of two life distributions. Applications of these results to relative aging of weighted distributions have also been illustrated. We also introduce a new relative aging ordering in terms of mean inactivity time order and study its fundamental properties.

Characterization of Lindley distribution by truncated moments

ABSTRACTIn the past few years, the Lindley distribution has gained popularity for modeling lifetime data as an alternative to the exponential distribution. This paper provides two new characterizations of the Lindley distribution. The first characterization is based on a relation between left truncated moments and failure rate function. The second characterization is based on a relation between right truncated moments and reversed failure rate function.

Stochastic comparisons in systems with Frèchet distributed components

In this paper, we study stochastic comparisons of parallel/series systems having Frèchet distributed components. These comparisons are with respect to the usual stochastic ordering, the reversed failure rate ordering and the failure rate ordering.

Relative ageing of ([formula omitted])-out-of-[formula omitted] systems

Various partial orders describing relative ageing of two life distributions have been proposed in the literature. These partial orders are based on various reliability measures such as failure rate, reversed failure rate and mean residual life. In this paper we study the relative ageing of (n−k+1)-out-of-n systems.

Further results on closure properties of LPQE order

Di Crescenzo and Longobardi (2002) introduced the past entropy, Sunoj et al. (2013) gave a quantile version for the past entropy, termed as the past quantile entropy (PQE). Based on the PQE function, they defined a new stochastic order called as less PQE (LPQE) order and studied some properties of this order. In the present paper, we focus our interests on further closure properties of this new order. Some characterizations of the LPQE order are investigated, closure and reversed closure properties are obtained. The preservation of the LPQE order in the proportional failure rate and reversed failure rate models is discussed.

Characterizing Discrete Life Distributions by Properties of Reversed Variance Residual life

The present article discusses the characterization of non negative integer-valued random variable using reversed variance residual life. A special attention is given to the characterizations by relationship between conditional variance and the reversed failure rate. A lower bound to the conditional variance is also established. Our bound is compared to the Cramer-Rao and Chapman-Robbins lower bounds so that construction of minimum variance unbiased estimators of relevant parametric functions in truncated distributions can be possible.

On proportional reversed failure rate class

Motivated by the recent use of the proportional reversed failure rate in economics (rates of increase and elasticity, see Veres-Ferrer and Pavia (Stat Pap 55:275–284, 2014) and in reliability (stochastic comparisons among systems, see Khaledi et al. (J Stat Plan Inference 141:276–286, 2011), in this work, we investigate characterizations and closure properties of the decreasing proportional reversed failure rate (DPRFR) classes for continuous, nonnegative random variables. Among others, we prove that DPRFR distributions are closed under convolutions. In addition, we relate this class of distributions with the class of monotone failure rate, proportional failure rate and likelihood ratio distributions.

Stochastic comparisons of residual lifetimes and inactivity times of coherent systems with dependent identically distributed components

In this paper we compare the residual lifetime of a used coherent system of age t>0 with the lifetime of the similar coherent system made up of used components of age t. Here ‘similar’ means that the system has the same structure and the component lifetimes have the same dependence (joint reliability copula). Some comparison results are obtained for the likelihood ratio order, failure rate order, reversed failure rate order and the usual stochastic order. Similar results are reported for comparing inactivity time of a coherent system with lifetime of similar coherent system having component lifetimes same as inactivity times of failed components.

Stochastic comparisons of component and system redundancies with dependent components

Under the assumption of dependent identically distributed components and redundant (spares) components, the problem of stochastic comparison of component and system redundancies have been considered. This study has been carried out under the criteria of the likelihood ratio ordering, the reversed failure rate ordering, the failure rate ordering and the usual stochastic ordering.

A new Characterization of some distributions Based on the r th conditional moment of doubly truncated mean function

New characterizations of doubly truncated exponential and geometric distributions are presented by using the s th conditional expectation in terms of their failure rate and reversed failure rate .these results may serve as generalization of several other results in the literature. This characterization generalizes a result of Nofal (2010).