Monotone Failure Rate Research Articles

A proof of simple insensitive bounds for a pure overflow system

In [1], [4] and [6] simple and intuitively obvious bounds were suggested for various overflow systems and claimed to be insensitive, that is to be valid also in the non-exponential case. This insensitivity was formalized in [4] and [6] when the service distributions have a monotone failure rate. This technical note provides another proof which for the case of a pure overflow system with one customer class extends the insensitivity to the general case.

Testing for exponentiality against monotone failure rate average alternatives

We propose a test for exponentiality against the class of non-exponential distributions having monotone failure rate averages. The test statistic, which is a U-statistic and hence asymptotically normally distributed, is much simpler than its competitors yet compares favorably with them in efficiency and power comparisons.

Bathtub distributions: a review

Bathtub distributions are characterized by bathtub failure rate functions . These are possibly more realisitic models than the monotone failure rate models . A systematic account of such distributions is not available and this review aims to give such an account . We give some easily verifiable conditions to check the bathtub property of a distribution along with methods to construct such distributions . We also discuss some stochastic and reliablity mechanisms which lead to bathtub distributions. These include mixtures ( stochastic failure rate models ) , series system , stochastic differential equation models and so on. We also review inference on bathtub distributions. The paper concludes with a rather exhaustive list of bathtub distributions.

Survey of Soviet Work in Reliability

Abstract : We review the research of Soviet school in reliability theory for the last twenty years. The main subjects under review are complex systems and their reliability, families with monotone failure rate, optimal maintenance and control, and statistical problems of reliability theory (121 defs.). Keywords: Coherent systems, Families with increasing failure rate, Optimal control, Standby items, Confidence intervals, and Exponential distribution.

A note on dispersive ordering defined by hazard functions

Recently many authors (e.g.Shaked (1982), Deshpande and Kochar (1983), Sathe (1984), Bartoszewicz (1985), Ahmed et al. (1986)) have studied relations between the dispersive ordering and some other partial orderings of distributions and monotone failure rate distributions. This note presents an approach to the studyong these relations, which is based on properties of hazard functions. A connection between NBU and NWU distribution and the dispersive ordering is established.

A class of tests for exponentiality against monotone failure rate alternatives with incomplete data

A class of tests is proposed for exponentiality against monotone failure rate alternatives when randomly right censored data are available. The class is based on an expected value inequality characterizing monotone versus constant failure rate. The inequality motivates also consistency of derived statistics. Attention is subsequently restricted to a wide subclass of tests whose asymptotic distribution is found. Some evaluations are given on the basis of Monte Carlo trials and — in the complete sample case — through Pitman asymptotic realative efficiency.

Testing constant failure rate against some partially monotone alternatives

In many practical situations, it is necessary to know whether the failure rate of a unit is partially monotone, e. g. monotone under a part of its life time. Some interesting and useful classes of life distributions with partially monotone failure rate is defined. The total time on test (TTT) concept which is useful for model identification and for tests against aging can also be used for testing exponentiality against these partially monotone alternatives. Some numerical results will also be given.

Two tests for monotone failure rate with incomplete data

Two statistics are proposed for testing for the exponential distribution against monotone failure rate alternatives when ran-domly right censored data are available. One of them is a general-ization of the Billmann, Antle and Bain test based on the MLE of the shape parameter of the Weibull distribution. The second has the advantage of being given in closed form. For this test the asymptotic null distribution is given. Consistency of the two tests is proved starting from an expected value inequality characterizing monotone failure rate.

A Test of Exponentiality Vs. Monotone-Hazard Alternatives Derived from the Empirical Characteristic Function

SUMMARY A test of exponentiality is proposed which is consistent for alternatives with monotone failure rate. Derived as a weighted integral of the difference between the characteristic functions of the sample and of the exponential distribution, the scale-invariant statistic depends on the mean value of the exponential density function, evaluated at the sample observations. The power of the test is similar to that of the Gini test of Gail and Gastwirth (1978b), and both compare favourably with other leading tests of exponentiality.

Robustness of Sequential Exponential Life-Testing Procedures

Abstract Sequential life-testing procedures for the exponential distribution are often used when the underlying distribution of life lengths is not exponential. In this article, we investigate the robustness of these procedures with respect to the risks and the expected sample sizes, when the underlying distribution has a monotone failure rate. We also describe the regions in which the true operating characteristic curves of the misused tests lie. A main conclusion is that a producer could penalize a consumer by overloading the test bench with unsatisfactory items, whenever the latter have an increasing failure rate.

Robustness of Sequential Exponential Life-Testing Procedures

Abstract Sequential life-testing procedures for the exponential distribution are often used when the underlying distribution of life lengths is not exponential. In this article, we investigate the robustness of these procedures with respect to the risks and the expected sample sizes, when the underlying distribution has a monotone failure rate. We also describe the regions in which the true operating characteristic curves of the misused tests lie. A main conclusion is that a producer could penalize a consumer by overloading the test bench with unsatisfactory items, whenever the latter have an increasing failure rate.

A bayesian approach to inference for monotone failure rates

In reliability theory, the notion of monotone failure rates plays a central role. When prior information indicates that such monotonicity is meaningful, it must be incorporated into the prior distribution whenever inference about the failure rates needs to be made. In this paper we show how this can be done in a straightforward and intuitively pleasing manner. The time interval is partitioned into subintervals of equal width and the number of failures and censoring in each interval is recorded. By defining a Dirichlet as the joint prior distribution for the forward or the backward differences of the conditional probabilities of survival in each interval, we find that the monotonicity is presenved in the posterior estimate of the failure rates. A posterior estimate of the survival function can also be obtained. We illustrate our method by applying it to some real life medical data.

Dispersive ordering and monotone failure rate distributions

Recently many authors (e.g. Shaked (1982), Deshpande and Kochar (1983), Sathe (1984)) have established relations between the dispersive ordering and some other partial orderings of distributions. This note presents connections which the dispersive ordering has with monotone failure rate distributions.

Dispersive ordering and monotone failure rate distributions

Recently many authors (e.g. Shaked (1982), Deshpande and Kochar (1983), Sathe (1984)) have established relations between the dispersive ordering and some other partial orderings of distributions. This note presents connections which the dispersive ordering has with monotone failure rate distributions.

Optimal Replacement of One-Unit Systems under Periodic Inspection

In this paper the optimal replacement problem of a one-unit system under periodic inspection is analyzed utilizing Markov decision theory. The problem is formulated in a general setting and it is shown that an age replacement policy is in fact optimal if the component lifetime has monotone failure rate. A necessary and sufficient condition of optimality together with several characterizations of the optimal policy are presented.

Testing exponenttality against monotone failure rate average

Let X1 … Xn be a random sample from an absol-utely continuous life distribution with distribution function F. We consider the problem of testing the null hypothesis H0: F is negative exponential (with unknown scale parameter) against the alternative H1 F is IFRA and not exponential. The test statistic is . The test has good ARE properties and is consistent for testing Ho against H1The exact critical values have been tabulated for n = 2(1)20 and power comparisons have been made on the basis of Monte Carlo studies. This test has been applied to one real example.

Probability of ruin with variable premium rate

Abstract The probability of ruin is investigated under the influence of a premium rate which varies with the level of free reserves. Section 4 develops a number of inequalities for the ruin probability, establishing upper and lower bounds for it in Theorem 4. Theorem 5 gives an expression for the ruin probability, and it is seen in Section 5 that this amounts to a generalization of the ruin probability given by Gerber for the special case of a negative exponential claim size distribution. In that same section it is shown the Lundberg's inequality is not derivable from the generalized theory of Section 4, and this is seen as a drawback of the methods used there. Sections 6 and 7 deal with some special cases, including claim size distributions with monotone failure rates. Section 8 shows that, in contrast with the result for a constant premium that the probability of ruin for zero initial reserve is independent of the claim size distribution, the same result does not hold when the premium rate is allowed to...

Maximum likelihood estimation of a distribution function with increasing failure rate based on censored observations

Abstract : The maximum likelihood estimator of a distribution function with monotone failure rate is derived based on a set of observations subject to arbitrary right censorship. This estimator is defined everywhere on the positive real line while the Kaplan-Meier estimator may not be. The small sample properties of this estimator are indicated by results of a Monte Carlo study for the Weibull distribution. (Author)

Some Inequalities for Failure Distributions

Inequalities for continuous and discrete failure distributions with monotone failure rates are given. Their use and some physical meanings are shown.

On inequalities for functionals of bivariate distributions with monotone failure rates

Estimates are obtained for functionals of convolutions of distributions functions with monotone failure rate.These estimates are expressed in terms of the corresponding functionals for convolutions of bivariate exponential distributions with dependent components.The obtained inequalities cannot be improved in a special sense in the class of all bivariate distributions with the monotone failure rate.The mentioned inequalities are used for the estimation of some characteristics of bivariate cumulative processes and to obtain conservative confidence bands.