Increasing Mean Residual Life Research Articles

An exact test for exponentiality against renewal increasing mean residual life class

In this paper, we develop an exact test for testing exponentiality against renewal increasing mean residual life class. The critical values are presented for different sample sizes. Asymptotic properties of the test statistic are also studied. Some numerical results are presented for evaluating the performance of the proposed test. The test procedure is illustrated by using two real data sets. The proposed method is then modified to incorporate right censored observations and the efficiency loss due to censoring is evaluated using Pitman's asymptotic efficacy value.

Some new results on the preservation of the NBUE and NWUE aging classes under the formation of coherent systems

AbstractWe consider the classical problem of whether certain classes of lifetime distributions are preserved under the formation of coherent systems. Under the assumption of independent and identically distributed (i.i.d.) component lifetimes, we consider the NBUE (new better than used in expectation) and NWUE (new worse than used in expectation) classes. First, a necessary condition for a coherent system to preserve the NBUE class is given. Sufficient conditions are then obtained for systems satisfying this necessary condition. The sufficient conditions are satisfied for a collection of systems which includes all parallel systems, but the collection is shown to be strictly larger. We also prove that no coherent system preserves the NWUE class. As byproducts of our study, we obtain the following results for the case of i.i.d. component lifetimes: (a) the DFR (decreasing failure rate) class is preserved by no coherent systems other than series systems, and (b) the IMRL (increasing mean residual life) class is not preserved by any coherent systems. Generalizations to the case of dependent component lifetimes are briefly discussed.

SHARP BOUNDS FOR SURVIVAL PROBABILITY WHEN AGEING IS NOT MONOTONE

We exploit a novel bounding argument to obtain sharp bounds for survival functions belonging to the Increasing initially then Decreasing Mean Residual Life (IDMRL) class introduced by Guess, Hollander and Proschan (1986) [8]. The bounds obtained are in terms of the mean, change point and pinnacle of the mean residual life function. The bounds for the monotonic ageing classes Decreasing Mean Residual Life (DMRL) and Increasing Mean Residual Life (IMRL) are obtained as special cases. Discussions on the bounds as well as two concrete illustrative examples are included.

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.

On renewal increasing mean residual life distributions: An age replacement model with hypothesis testing application

We study the life distribution of an operating device through the notion of mean residual life. The device is experiencing a random number of shocks governed by a homogeneous Poisson process, and a new U-statistic test procedure is introduced to test the hypothesis of that the life is exponentially distributed against the alternative that the life distribution is the life distribution has renewal increasing mean residual property.

Reversed Preservation Properties for Series and Parallel Systems

Recently Li and Yam (2005) studied which ageing properties for series and parallel systems are inherited for the components. In this paper we provide new results for the increasing convex and concave orders, the increasing mean residual life (IMRL), decreasing failure rate (DFR), the new worse than used in expectation (NWUE), the increasing failure rate in average (IFRA), the decreasing failure rate in average (DFRA), and the new better than used in the convex order (NBUC) ageing classes.

Reversed Preservation Properties for Series and Parallel Systems

Recently Li and Yam (2005) studied which ageing properties for series and parallel systems are inherited for the components. In this paper we provide new results for the increasing convex and concave orders, the increasing mean residual life (IMRL), decreasing failure rate (DFR), the new worse than used in expectation (NWUE), the increasing failure rate in average (IFRA), the decreasing failure rate in average (DFRA), and the new better than used in the convex order (NBUC) ageing classes.

Connections among various variability orderings

Recently, a new variability ordering, called right spread ordering or excess wealth ordering has been introduced. This new ordering is weaker than dispersive ordering. We show in this note that if X is less variable than Y in the sense of right spread ordering or convex ordering, then it implies that | X 1 − X 2| is less variable than | Y 1 − Y 2| according to increasing convex ordering. Here X 1 and X 2 ( Y 1 and Y 2) are two independent copies of X ( Y). An application of the right spread ordering in the study of spacings from an increasing mean residual life distribution is given.

DFR Property of First-Passage Times and its Preservation Under Geometric Compounding

It is shown that if a discrete-time Markov chain on the state space $\{0, 1,\ldots\}$ has a transition probability matrix $\mathbf{P}$ and a transition survival probability matrix $\mathbf{Q}$ which is totally positive of order two $(TP_2)$, where $Q(i, j) = \sum_{k\geq j}P(i, k)$, then the first-passage time from state 1 to state 0 has decreasing failure rate (DFR). This result is used to show that (i) the sum of a geometrically distributed number (i.e., geometric compound) of i.i.d. DFR random variables is DFR, and (ii) the number of customers served during a busy period in an M/G/1 queue with increasing failure-rate service times is DFR. Recent results of Szekli (1986) and the closure property of i.i.d. DFR random variables under geometric compounding are combined to show that the stationary waiting time in a GI/G/1 (M/G/1) queue with DFR (increasing mean residual life) service times is DFR. We also provide sufficient conditions on the inter-renewal times under which the renewal function is concave. These results shed some light on a conjecture of Brown (1981).

Testing Exponentiality Versus a Trend Change in Mean Residual Life

Given that an item is of age $t$, the expected value of the random remaining life is called the mean residual life (MRL) at age $t$. We propose two new nonparametric classes of life distributions for modeling aging based on MRL. The first class of life distributions consists of those with "increasing initially, then decreasing mean residual life" (IDMRL). The IDMRL class models aging that is initially beneficial, then adverse. The second class, "decreasing, then increasing mean residual life" (DIMRL), models aging that is initially adverse, then beneficial. We propose two testing procedures for $H_0$: constant MRL (i.e., exponentiality) versus $H_1$: IDMRL, but not constant MRL (or $H'_1$: DIMRL, but not constant MRL). The first testing procedure assumes the turning point, $\tau$, from IMRL to DMRL is specified by the user or is known. The second procedure assumes knowledge of the proportion, $\rho$, of the population that "dies" at or before the turning point (knowledge of $\tau$ itself is not assumed).

Approximating IMRL Distributions by Exponential Distributions, with Applications to First Passage Times

It is shown that if $F$ is an IMRL (increasing mean residual life) distribution on $\lbrack 0, \infty)$ then: $\max\{\sup_t |\bar F(t) - \bar G(t)|, \sup_t|\bar F(t) - e^{-t/\mu}|, \sup_t|\bar G(t) - e^{-t/\mu}|, \\ \sup_t |\bar G(t) - e^{-t/\mu_G}|\} = \frac{\rho}{\rho + 1} = 1 - \frac{\mu}{\mu_G}$ where $\bar F(t) = 1 - F(t), \mu = E_FX, \mu_2 = E_FX^2, G(t) = \mu^{-1} \int^t_0 \bar F(x) dx, \mu_G = E_GX = \mu_2/2\mu$, and $\rho = \mu_2/2\mu^2 - 1 = \mu_G/\mu - 1$. Thus if $F$ is IMRL and $\rho$ is small then $F$ and $G$ are approximately equal and exponentially distributed. IMRL distributions with small $\rho$ arise naturally in a class of first passage time distributions for Markov processes, as first illuminated by Keilson. The current results thus provide error bounds for exponential approximations of these distributions.

The marshall and stoyan bounds for IMRL/G/1 queues are tight

Previously established upper and lower bounds for the mean waiting time in a GI/G/1 queue given an interarrival-time distribution with increasing mean residual life are shown to be tight. Distributions for which the inequalities become equalities are displayed. The corresponding bounds for DMRL distributions are not tight.

Further Monotonicity Properties for Specialized Renewal Processes

Abstract : Define Z(t) to be the forward recurrence time at t for a renewal process with interarrival time distribution, F, which is assumed to be IMRL (increasing mean residual life). It is shown that E phi (z(t)) is increasing in t or = 0 for all increasing convex phi. An example demonstrates that Z(t) is not necessarily stochastically increasing nor is the renewal function necessarily concave. Both of these properties are known to hold for F DFR (decreasing failure rate). (Author)

On a characterization of gamma distributions via exponential mixtures

A new and simpler proof of Morrison's result that within exponential mixtures only IFR gamma mixing produces linearly increasing mean residual life functions is given. A parallel and new characterization of the DFR gamma laws follows as a consequence. The method of proof used suggests a general result on the infinite divisibility of the mixing distributions in exponential mixtures.

On a characterization of gamma distributions via exponential mixtures

A new and simpler proof of Morrison's result that within exponential mixtures only IFR gamma mixing produces linearly increasing mean residual life functions is given. A parallel and new characterization of the DFR gamma laws follows as a consequence. The method of proof used suggests a general result on the infinite divisibility of the mixing distributions in exponential mixtures.

Bounds, Inequalities, and Monotonicity Properties for Some Specialized Renewal Processes

Renewal processes with increasing mean residual life and decreasing failure rate interarrival time distributions are investigated. Various two-sided bounds are obtained for $M(t)$, the expected number of renewals in $\lbrack 0, t\rbrack$. It is shown that if the interarrival time distribution has increasing mean residual life with mean $\mu$, then the expected forward recurrence time is increasing in $t \geqslant 0$, as is $M(t) - t/\mu$. If the interarrival time distribution has decreasing failure rate then $M(t)$ is concave, and the forward and backward recurrence time distributions are stochastically increasing in $t \geqslant 0$.