Some results on generalized ageing classes

In this paper we provide new results about generalized ageing classes on the excess lifetime of a renewal process. We also obtain some characterizations of generalized ageing classes by means of the residual life at random time.     

The percentile residual life up to time t0: Ordering and aging properties

Motivated by practical issues, a new stochastic order for random variables is introduced by comparing all their percentile residual life functions until a certain instant. Some interpretations of these stochastic orders are given, and various properties of them are derived. The relationships to other stochastic orders are studied and also an application in reliability theory is described. Finally, we present some characterization results of the decreasing percentile residual life up to time t₀ aging notion.     

Identifying the Pareto and Yule distributions by properties of their reliability measures

This paper provides characterizations of a family of distributions in the context of reliability theory placing emphasis on a particular family member, the Yule distribution. In particular, it is shown that the distribution of a non-negative, integer-valued random variable X with E(X)<+∞ is uniquely identified to belong to the class of distributions consisting of the geometric, the Waring and the negative hypergeometric distributions if and only if anyone of the following conditions is satisfied:

• a.The mean residual life is a linear function of time.
• b.The vitality function is a linear function of time.
• c.The product of the hazard rate by the mean residual life is constant.

Characterizations of the Yule distribution based on reliability measures of its size-biased version are also provided. Continuous analogues of the results are considered. These include characterizations of the exponential and the beta of the first and second kind distributions.     

A note on the mean residual life function of a coherent system with exchangeable or nonidentical components

This article investigates some properties of the mean residual life function of (n−k+1)-out-of-n systems, when the lifetimes of the system components are independent random variables but not necessarily identically distributed and when the joint distribution of the component lifetimes is exchangeable, extending the results of Asadi and Goliforushani (2008) [On the mean residual life function of coherent systems. IEEE Transactions on Reliability 57 (4) 574-580] for the case of independent and identically distributed components. The extension to a coherent system with exchangeable components is also given.     

Efficient estimates in linear and nonlinear regression with heteroscedastic errors

In this paper we consider the heteroscedastic regression model defined by the structural relation Y = r(V, β) + s(W)ε, where V is a p-dimensional random vector, W is a q-dimensional random vector, β is an unknown vector in some open subset B of R^m, r is a known function from R^p × B into R, s is an unknown function on R^q, and ε is an unobservable random variable that is independent of the pair (V, W). We construct asymptotically efficient estimates of the regression parameter β under mild assumptions on the functions r and s and on the distributions of ε and (V, W).     

On Upshifted Reversed Mean Residual Life Order

Recently, the concept of reversed mean residual life order based on the mean of the random variable X _t  = (t ? X | X ≤ t), t > 0, called the reversed residual life, defined for the nonnegative random variable X, has been introduced in the literature. In this paper, a stochastic order based on the shifted version of the reversed mean residual life is proposed, based on the reversed mean residual life function for a random variable X with support (l _X , ∞), where l _X may be negative infinity, and its properties are studied. Closure under the Poisson shock model and properties for spare allocation are also discussed.     

Some results on residual life and inactivity time at random time

In this article, we enhance the study of residual life at random time (RLRT) and inactivity time at random time (ITRT). To this aim, first we provide some stochastic orderings results among ITRT in two-sample problems when they failed at two different random times. Then, we develop some sufficient conditions which lead to the stochastic comparisons of RLRT and ITRT based on variance residual life order. The results are expected to be useful in reliability theory, forensic science, queue theory, and actuarial science.     

Inequalities involving expectations of selected functions in reliability theory to characterize distributions

Recently, authors have studied inequalities involving expectations of selected functions, viz. failure rate, mean residual life, aging intensity function, and log-odds rate which are defined for left truncated random variables in reliability theory to characterize some well-known distributions. However, there has been growing interest in the study of these functions in reversed time (X ? x, instead of X > x) and their applications. In the present work we consider reversed hazard rate, expected inactivity time, and reversed aging intensity function to deal with right truncated random variables and characterize a few statistical distributions.     

Ordering conditional lifetimes of coherent systems

Consider a system of n components that has the property that there exists a number r  (r<n)$(r<n)$, such that if it is known that at most r components have failed, the system is still functioning with probability 1. Suppose that such a system is equipped with a warning light that comes up at the time of the failure of the rth component. The system is still working then, and we are interested in its residual life. In this paper we obtain some results which stochastically compare the residual lives of such systems with the same type, or with different types, of components. Some applications are given. In particular, we derive upper and lower bounds on the expected residual lives of such systems given that the warning light has not come up yet, and given that the component hazard rate functions are bounded from below or from above by a known constant.     

Semiparametric distribution forecasting

Given m time series regression models, linear or not, with additive noise components, it is shown how to estimate semiparametrically the predictive probability distribution of one of the time series conditional on past random covariate data. This is done by assuming that the distributions of the residual components associated with the regression models are tilted versions of a reference distribution.     

CARMA(p,q) generalized random processes

There is now a vast literature on the theory and applications of generalized random processes, pioneered by Itô (1953), Gel’fand (1955) and Yaglom (1957). In this note we make use of the theory of generalized random processes as defined in the book of Gel’fand and Vilenkin (1964) to extend the definition of continuous-time ARMA(p,q  ) processes to allow q≥p$q\ge p$, in which case the processes do not exist in the classical sense. The resulting CARMA generalized random processes provide a framework within which it is possible to study derivatives of CARMA processes of arbitrarily high order.     

On residual lifetime of coherent systems after the rth failure

In this article we study the residual lifetime of a coherent system after the rth failure, i.e. the time elapsed from the rth failure until the system failure given that the system operates at the time of the rth failure. We provide a mixture representation for the corresponding residual lifetime distribution in terms of signature. We also obtain some stochastic ordering results for the residual lifetimes.     

On tail parameter estimation in certain point process models

Let μ(ds, dx) denote Poisson random measure with intensity dsG(dx) on (0, ∞) × (0, ∞), for a measure G(dx) with tails varying regularly at ∞. We deal with estimation of index of regular variation α and weight parameter ξ if the point process is observed in certain windows K_n = [0, T_n] × [Y_n, ∞), where Y_n → ∞ as n → ∞. In particular, we look at asymptotic behaviour of the Hill estimator for α. In certain submodels, better estimators are available; they converge at higher speed and have a strong optimality property. This is deduced from the parametric case G(dx) = ξαx^−α−1 dx via a neighbourhood argument in terms of Hellinger distances.     

A new test for decreasing mean residual lifetimes

The mean residual life of a non negative random variable X with a finite mean is defined by M(t) = E[X ? t|X > t] for t ? 0. One model of aging is the decreasing mean residual life (DMRL): M is decreasing (non increasing) in time. It vastly generalizes the more stringent model of increasing failure rate (IFR). The exponential distribution lies at the boundary of both of these classes. There is a large literature on testing exponentiality against DMRL alternatives which are all of the integral type. Because most parametric families of DMRL distributions are IFR, their relative merits have been compared only at some IFR alternatives. We introduce a new Kolmogorov–Smirnov type sup-test and derive its asymptotic properties. We compare the powers of this test with some integral tests by simulations using a class of DMRL, but not IFR alternatives, as well as some popular IFR alternatives. The results show that the sup-test is much more powerful than the integral tests in all cases.     

La méthode de vraisemblance pour les processus linéaires spatiaux définis sur un treillis triangulaire

Spatial linear processes {X_s, s ? T} where T is a triangular lattice in R² are considered. Special attention is given to the class of spatial moving-average processes. Precisely, for each site s T, the variable X_s is defined as a linear combination of real-valued random shocks located at the vertices of regular concentric hexagons centered at s. For Gaussian random shocks, the process is also Gaussian, and estimates of its parameters are obtained by maximizing the exact likelihood. For non-Gaussian random shocks, the exact likelihood is difficult to obtain; however, the Gaussian likelihood is still used giving the pseudo-Gaussian likelihood estimates. The behaviour of these estimates is analyzed through the study of asymptotic properties and some simulation experiments based on an isotropic model defined with one coefficient.     

On the mean residual life of a generalized k-out-of-n system

A generalized k-out-of-n system consists of N modules in which the i th module is composed of n_i components in parallel. The system failswhen at least f components in the whole system or at least k consecutive modules have failed. In this article, we obtain the mean residual life function of such a generalized k-out-of-n system under different conditions, namely, when the number of components in each module is equal or unequal and when the components of the system are independent or exchangeable.     

Optimal fractional factorial designs and their construction

Optimal symmetrical fractional factorial designs with n$n$ runs and m$m$ factors of s$s$ levels each are constructed. We consider only designs such that no two factors are aliases. The minimum moment aberration criterion proposed by Xu (2003) is used to judge the optimality of the designs. The minimum moment aberration criterion is equivalent to the popular generalized minimum aberration criterion proposed by Xu and Wu (2001), but the minimum moment criterion is simpler to formulate and employ computationally. Some optimal designs are constructed by using generalized Hadamard matrices.     

On the mean residual life of records

Suppose that a technical system is subject to shocks, e.g. peaks of voltages from a sequence of identically independent voltages having a lower limit value v>0$v>0$. We propose a new definition for the mean residual life of the records of the sequence and study its various properties.     

Bounds for Expectations of Differences of Generalized Order Statistics Based on General and Life Distributions

Let X _? ^(r), r ≥ 1, denote generalized order statistics based on an arbitrary distribution function F with finite pth absolute moment for some 1 ≤ p ≤ ∞. We present sharp upper bounds on E(X _? ^(s) ? X _? ^(r)), 1 ≤ r < s, for F being either general or life distribution. The bounds are expressed in various scale units generated by pth central absolute or raw moments of F, respectively. The distributions achieving the bounds are specified.     

On the dynamic cumulative residual entropy

Recently, Rao et al. [(2004) Cumulative residual entropy: a new measure of information. IEEE Trans. Inform. Theory 50(6), 1220–1228] have proposed a new measure of uncertainty, called cumulative residual entropy (CRE), in a distribution function F and obtained some properties and applications of that. In the present paper, we propose a dynamic form of CRE and obtain some of its properties. We show how CRE (and its dynamic version) is connected with well-known reliability measures such as the mean residual life time.