Increasing Failure Rate Average Research Articles

A new weighted means of failure rate and associated quantile versions

Abstract In this paper, we define weighted failure rate and their means from the stand point of an application. We begin by emphasizing that the formation of n independent component series system having weighted failure rates with sum of weight functions being unity is same as a mixture of n distributions. We derive some parametric and non-parametric characterization results. We discuss on the form invariance property of baseline failure rate for a specific choice of weight function. Some bounds on means of aging functions are obtained. Here, we establish that weighted increasing failure rate average (IFRA) class is not closed under formation of coherent systems unlike the IFRA class. An interesting application of the present work is credited to the fact that the quantile version of means of failure rate is obtained as a special case of weighted means of failure rate.

Estimation of a distribution function with increasing failure rate average

A life distribution function F is said to have an increasing failure rate average if H(x)∕x is nondecreasing where H(x) is the corresponding cumulative hazard function. In this paper we provide a uniformly strongly consistent estimator of F and derive the convergence in distribution of the estimator at a point where H(x)∕x is increasing using the arg max theorem. We also show using simulations that, unlike other estimators of the past, this new estimator outperforms the empirical distribution in terms of mean square error at all quantiles. An example is also discussed to illustrate the theoretical results

On testing more IFRA ordering-II

ABSTRACTSuppose F and G are two life distribution functions. It is said that F is more IFRA (increasing failure rate average) than G (written by F ⩽ *G) if G− 1F(x) is star-shaped on (0, ∞). In this paper, the problem of testing H0: F = *G against H1: F ⩽ *G and F ≠ *G is considered in both cases when G is known and when G is unknown. We propose a new test based on U-statistics and obtain the asymptotic distribution of the test statistics. The new test is compared with some well-known tests in the literature. In addition, we apply our test to a real data set in the context of reliability.

A Selection Procedure for Selecting the Least Increasing Failure Rate Average Distribution

For k independent absolutely continuous increasing failure rate average (IFRA) life distributions Fi, i = 1, 2, …, k, Link (1989) considered a measure of departure of against monotone failure rate average alternatives. In this paper, we use the measure defined by Link for detection of IFRA-ness of life distribution Fi. A two-stage selection procedure to select the least IFRA distribution is proposed. This selection procedure is based on a U-statistic which is an estimator of the measure and can be implemented even when the IFRA life distributions belong to different families. The applications of this procedure are discussed for some well known distributions.

On testing exponentiality against IFRA alternatives

In a recent paper, Ahmad [J. Statist. Plann. Inference 92 (2001) 121] proposes a family of tests for new better than used, new better than used in expectation and harmonic new better than used in expectation. His family of tests is based on some moment inequalities of these classes. In this paper we present a family of tests to test exponentiality against increasing failure rate average (IFRA) alternatives. Our proposed family is based on a moment inequality of the IFRA class. The results of a Pitman asymptotic relatives efficiency comparison indicate the superiority of our tests.

LIMIT THEOREMS FOR ASYMPTOTICALLY MINIMAX ESTIMATION OF A DISTRIBUTION WITH INCREASING FAILURE RATE UNDER A RANDOM MIXED CENSORSHIP/TRUNCATION MODEL

ABSTRACT The search for optimal non-parametric estimates of the cumulative distribution and hazard functions under order constraints inspired at least two earlier classic papers in mathematical statistics: those of Kiefer and Wolfowitz[1] and Grenander[2] respectively. In both cases, either the greatest convex minorant or the least concave majorant played a fundamental role. Based on Kiefer and Wolfowitz's work, Wang3-4 found asymptotically minimax estimates of the distribution function F and its cumulative hazard function Λ in the class of all increasing failure rate (IFR) and all increasing failure rate average (IFRA) distributions. In this paper, we will prove limit theorems which extend Wang's asymptotic results to the mixed censorship/truncation model as well as provide some other relevant results. The methods are illustrated on the Channing House data, originally analysed by Hyde.5-6

CLASSIFICATION OF MULTIVARIATE LIFE DISTRIBUTIONS BASED ON PARTIAL ORDERING

Barlow and Proschan presented some interesting connections between univariate classifications of life distributions and partial orderings where equivalent definitions for increasing failure rate (IFR), increasing failure rate average (IFRA), and new better than used (NBU) classes were given in terms of convex, star-shaped, and superadditive orderings. Some related results are given by Ross and Shaked and Shanthikumar. The introduction of a multivariate generalization of partial orderings is the object of the present article. Based on that concept of multivariate partial orderings, we also propose multivariate classifications of life distributions and present a study on more IFR-ness.

Exponential distributions on semigroups

Three fundamental characterizations of the standard exponential distribution on [0, ∞) are the remaining life, memoryless and constant failure properties. Analogs of these properties are studied for distributions on a class of semigroups in which the semigroup operation replaces addition, a compatible partial order replaces the ordinary order, and a left-invariant measure replaces Lebesgue measure. Partial characterizations of exponential distributions on such semigroups are obtained and the semigroup formulation provides new characterizations of certain aging properties studied in reliability-increasing failure rate, new better than used, and increasing failure rate average.

A class of efficient tests for increasing-failure-rate-average distribution under random censoring

A class of tests for the increasing failure rate average (IFRA) alternatives under random censoring is proposed. The tests are based on a function of the Kaplan-Meier estimator. Most of the IFRA tests in the literature depend on the (nuisance) parameter that appears in the definition of IFRA, and the choice of this parameter is crucial in performing the tests. The proposed class of tests does not have this disadvantage. Under some regularity conditions, the asymptotic normality of the tests is established and asymptotically distribution-free tests are obtained by using estimators for the null standard deviations. The efficacies of the tests under the proportional hazard censoring model are studied. The proposed test is most efficient for the Weibull family of IFRA alternatives among the existing tests available for the censored data. The test is applied to published appliance data. >

Some bounds on reliability of coherent systems of IFRA components

In this paper we find new bounds for the reliability of coherent systems of independent components with increasing failure rate average (IFRA) lifetimes. These bounds are based on certain bounds available for the survival functions of IFRA random variables and the fact that the IFRA class of life distributions is closed under the formation of coherent systems. These bounds are compared with other applicable bounds in this case. An illustration of explicit computations of the bounds is provided for the bridge structure with components having independent gamma life distributions.

Some bounds on reliability of coherent systems of IFRA components

In this paper we find new bounds for the reliability of coherent systems of independent components with increasing failure rate average (IFRA) lifetimes. These bounds are based on certain bounds available for the survival functions of IFRA random variables and the fact that the IFRA class of life distributions is closed under the formation of coherent systems. These bounds are compared with other applicable bounds in this case. An illustration of explicit computations of the bounds is provided for the bridge structure with components having independent gamma life distributions.

A class of tests for testing an increasing failure-rate-average distribution with randomly right-censored data

For testing exponentiality versus (nonexponential) increasing failure rate average (IFRA) alternatives (which are nonexponential) using the randomly right censored data, a class of test statistics based on a functional of the Kaplan-Meier estimator is proposed. The asymptotic relative efficiencies of tests from this class with respect to other test statistics are derived. The efficiency loss due to censoring is studied. The proposed tests are applied to published survival data. >

A new measure of ifra-ness and its application to two sample test

A continuous life distribution function f with f(x)=0 for x ≶ is said to be increasing failure rate average (IFRA) if and only if and all x For testing the null hypothesis that f is an exponentia distribution versus the alternative hypothesis that f is a nonexponential IFRA distribution DESHPANDE [BIometrika 70 (1983): 514:518] proposed a class of testing which depends on b the choice of b has been crucial since than In this paper we propose a new measure of IFRA ness that is independent of b and we use it to test whether one destribution is more IFRA than other. The Properties of the test such as unbiasedness, consistency and asymptotic normality are dscussed.Monte Carolo estimates of power are obtained for small sample sizes. Large sample performance is measured in terms of Asymptotic relative efficiency with respect to Tiwri and ZALKIAr,s (1988) test.

Testing an increasing failure rate average distribution with censored data

A life distribution F is an increasing failure rate average (IFRA) if For testing H0: F is exponential, versus H1 F is IFRA, but not exponential based on randomly censored data, we propose a test statistics where is the KAPLAN–MIEER Product limit estimator of F. The asymptotic normality of Jn c(b) is established and an asymptotically distribution – free test is obtained. /The efficiency loss due to ocensoring is studied compared to DESHPANDE'S (1983) test uncensored case. The asymptotic relati ve efficiency with respect to CHEN,HOLLANDER AND LANGBERG'S (1983) test is shoen to be reasonably high

A class of Large Sample Tests for Bivariate Exponentiality Versus Bifra and Bnbu (to, to) Alternatives

Two related problems are considered here. First, a class of tosts is proposed for testing bivariate exponentiality (BVE) against the class of bivariate increasing failure rate average (IFRA) distributions. Secondly, we propose a test of BVE versus bivariate new better than used of ago (to to)),a new class of bivariate distributions introduced here. The PITMAN asymptotic relative efficiencies of these tests with respect to BASU and EBRAHIMI's (1984) bivariate new better than used (BNBU) test, compared

Tests for Increasing Failure Rate Average with Randomly Right Censored Data

Tests for Increasing Failure Rate Average with Randomly Right Censored Data

On the first-passage times of pure jump processes

Let Tx be the time it takes for a pure jump process, which starts at 0, to cross a threshold x > 0. Sufficient conditions on the parameters of this process under which Tx has increasing failure rate average (IFRA), increasing failure rate (IFR) or logconcave density (PF2) are identified. The conditions for IFRA are weaker than those of Drosen (1986). Sufficient conditions on the parameter of a pure jump process for Tx to the IFR or PF2 are not available in the literature.

On the first-passage times of pure jump processes

Let Tx be the time it takes for a pure jump process, which starts at 0, to cross a threshold x > 0. Sufficient conditions on the parameters of this process under which Tx has increasing failure rate average (IFRA), increasing failure rate (IFR) or logconcave density (PF2) are identified. The conditions for IFRA are weaker than those of Drosen (1986). Sufficient conditions on the parameter of a pure jump process for Tx to the IFR or PF2 are not available in the literature.

A New MIFRA Class of Life Distributions

A new multivariate generalization of the increasing failure rate average (IFRA) class which can be intutively supported in terms of some averaging of the component failure rates is proposed. A piecewise approximability of the class in the bivariate case and an inequality characterizing the proposed class have been established. It has been further shown that the proposed class possesses some desirable properties which should hold for any multivariate ageing class.

Estimators of a distribution function with increasing failure rate average

A nonparametric estimator G n of a distribution function F with increasing failure rate average (IFRA) is constructed. This estimator G n is so constructed that its hazard function C n is the greatest star-shaped minorant of the sample hazard function. The estimator G n is itself IFRA. Denoting the sample distribution function by F n and the sample hazard function by H n , it follows from Millar (1979) that F n is an asymptotically minimax estimator of F if F is IFRA. We prove, under suitable restrictions of F, and for any fixed λ with F( λ) < 1, that n 1 2 sup x⩽λ||G n(x)−F n(x)|| and n 1 2 sup x⩽λ||C n(x)−H n(x)|| both tend to zero in probability. That is, G n and F n are asymptotically n 1 2 - equivalent as estimators of the true distribution functions. Similar results can be obtained for a DFRA distribution.