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For a given pair of random lifetimes whose dependence is described by a time-transformed exponential model, we provide analytical expressions for the distribution of their sum. These expressions are obtained by using a representation of the joint distribution in terms of bivariate distortions, which is an alternative approach to the classical copula representation. Since this approach allows one to obtain conditional distributions and their inverses in simple form, then it is also shown how it can be used to predict the value of the sum from the value of one of the variables (or vice versa) by using quantile regression techniques. Open Access funding provided thanks to the CRUE-CSIC agreement with Springer Nature. JN and JM are supported by Ministerio de Ciencia e Innovación of Spain under Grant PID2019-103971GB-I00/AEI/10.13039/501100011033. FP is partially supported by the Grant Progetto di Eccellenza, CUP: E11G18000350001 and by the Italian GNAMPA research group of INdAM (Istituto Nazionale di Alta Matematica).

Given a finite set of independent random variables, assume one can observe their sum, and denote with s its value. Efron in 1965, and Lehmann in 1966, described conditions on the involved variables such that each of them stochastically increases in the value s, i.e., such that the expected value of any non-decreasing function of the variable increases as s increases. In this paper, we investigate conditions such that this stochastic monotonicity property is satisfied when the assumption of independence is removed. Comparisons in the stronger likelihood ratio order are considered as well.

Consider a vector ( X , Y ) that describes the failure times of two non-independent components of a system. Assuming that the first component has survived up to a given time t > 0 , i.e., assuming X > t , one can define the corresponding residual lifetime under different assumptions on the failure of the second component, having lifetime Y . In particular, one can observe that the second component has not failed before a time s ≥ 0 (maybe different from t ), thus the conditioned residual lifetime X ˜ t = [ X − t | X > t , Y > s ] can be considered, or one cannot observe Y , and in this case the conditioned residual lifetime X t = [ X − t | X > t ] has to be studied. This note deals with conditions on the survival copula of ( X , Y ) such that X ˜ t and X t are comparable according to the main reliability stochastic orders. Similar conditions, based on the connecting copula of ( X , Y ) , are described also for the conditioned inactivity times X ˜ t = [ t − X | X ≤ t , Y ≤ s ] and X t = [ t − X | X ≤ t ] .

New weak notions of positive dependence between the components X and Y of a random pair (X,Y) have been considered in recent papers that deal with the effects of dependence on conditional residual lifetimes and conditional inactivity times. The purpose of this paper is to provide a structured framework for the definition and description of these notions, and other new ones, and to describe their mutual relationships. An exhaustive review of some well-know notions of dependence, with a complete description of the equivalent definitions and reciprocal relationships, some of them expressed in terms of the properties of the copula or survival copula of (X,Y), is also provided.

Among Mixed Poisson processes, counting processes having geometrically distributed increments can be obtained when the mixing random intensity is exponentially distributed. Dealing with shock models and compound counting models whose shocks and claims occur according to such counting processes, we provide various comparison results and aging properties concerning total claim amounts and random lifetimes. Furthermore, the main characteristic distributions and properties of these processes are recalled and proved through a direct approach, as an alternative to those available in the literature. We also provide closed-form expressions for the first-crossing-time problem through monotone nonincreasing boundaries, and numerical estimates of first-crossing-time densities through other suitable boundaries. Finally, we present several applications in seismology, software reliability and other fields.

Recent literature reports that a large percentage of teachers complain that teaching has an adverse effect on their voice status. Thus, more needs to be done to study their vocal behavior. The objective of this longitudinal study was twofold: to determine changes in the voice use of teachers over a school year, and to study the relationships between voice use and classroom acoustic parameters. Thirty-one teachers from two secondary schools in Turin (Italy) were involved at the beginning of the 2014-2015 school year, and 22 of them also participated at the end of the same school year. The results show that teachers adjust their voices with noise and reverberation. A minimum value of the sound pressure level of voice (SPL) was found at a mid-frequency reverberation time of 0.8 s in both periods. Moreover, the teachers who worked in the worst classroom acoustic conditions showed an increase of 2.3 dB in the mean SPL and a decrease of 10% in the voicing time percentage at the end of the school year. A predictive model that can be used to estimate the mean SPL from the background noise level and the reverberation time, based on collected data, is here proposed.

Several studies have shown so far that poor acoustics inside classrooms negatively affects the teaching and learning processes, especially at the lowest grades of education. However, the extent to which noise exposure or excessive reverberation affect well-being of children at school in their early childhood is still unanswered, as well as their awareness of noise disturbance. This work is a pilot study to investigate to which extent classroom acoustics affects the perceived well-being and noise disturbance in first graders. About 330 pupils aged from 6 to 7 years participated in the study. They belonged to 20 classes of 10 primary schools located in Torino (Italy), where room acoustic measurements were performed and where noise level was monitored during classes. The school buildings and the classrooms were balanced between socioeconomic status and acoustic conditions. Trained experimenters administered questionnaires in each class, where pupils answered all together during the last month of the school year (May). Questions included the happiness scale, subscales assessing self-esteem, emotional health, relationship at home and with friends, enjoyment of school, intensity and noise disturbance due to different sound sources, and quality of voice. The findings of the study suggest that long reverberation times, which are associated with poor classroom acoustics as they generate higher noise levels and degraded speech intelligibility, bring pupils to a reduced perception of having fun and being happy with themselves. Furthermore, bad classroom acoustics is also related to an increased perception of noise intensity and disturbance, particularly in the case of traffic noise and noise from adjacent school environments. Finally, happy pupils reported a higher perception of noise disturbance under bad classroom acoustic conditions, whereas unhappy pupils only reported complaints in bad classroom acoustics with respect to the perception of pleasances with himself or herself and of fitting in at school. Being a mother tongue speaker is a characteristic of children that brings more chances of attending classes in good acoustics, of being less disturbed, and of having more well-being, and richer districts presented better acoustic conditions, in turn resulting in richer districts also revealing a greater perception of well-being.

A new property defined on the class of symmetric copulas is introduced and studied along this note. It is shown here that such a property can define a family of bivariate distribution functions satisfying all the characteristics listed in Kimeldorf and Sampson (1989) to be considered as a positive dependence notion. Applications, relationships with other positive dependence notions, further properties and the corresponding negative dependence notion are discussed as well.

An innovative integrated self-learning health monitoring system has been developed and implemented on a fleet of helicopters in actual service. This system improves significantly the efficiency of a previous accelerometric vibrational monitoring tool by means of multivariate third level processing of the accelerometric features. The paper describes the way in which several problems, typical for the monitoring process of a mechanical system, have been treated in this specific case. The applied techniques could be of much more general interest.

In order to take into account any possible dependence between alternatives in optimization problems, bivariate characterizations of some well-know univariate stochastic orders have been defined and studied by Shanthikumar and Yao (Adv Appl Probab 23:642–659, 1991). These characterizations gave rise to new stochastic comparisons, commonly called joint stochastic orders, which are equivalent to the original ones under assumption of independence, but are different whenever the variables to be compared are dependent. In this note we provide sufficient conditions on the survival copula describing the dependence among the compared variables such that the standard stochastic orders imply the corresponding joint stochastic orders, and viceversa. Also, simple conditions for joint stochastic orders between the components of random vectors defined through multivariate frailty models are provided.

Consider random vectors formed by a finite number of independent groups of independent and identically distributed random variables, where those of the last group are stochastically smaller than those of the other groups. Conditions are given such that certain functions, defined as suitable means of supermodular functions of the random variables of the vectors, are supermodular or increasing directionally convex. Comparisons based on the increasing convex order of supermodular functions of such random vectors are also investigated. Applications of the above results are then provided in risk theory, queueing theory, and reliability theory, with reference to (i) net stop-loss reinsurance premiums of portfolios from different groups of insureds, (ii) closed cyclic multiclass Gordon-Newell queueing networks, and (iii) reliability of series systems formed by units selected from different batches.

A set of logistic mathematical models to estimate the daylight amount and the energy demand for lighting of a room is presented. The models were built upon a database of results obtained for a sample room through Daysim simulations: features such as site, orientation, external obstructing angle, window size, glazing visible transmittance and room depth were parametrically changed, resulting in 102 cases. For each case, the target workplane illuminance and the lighting power density were also changed, producing a final database of 408 cases. Two groups of models were built, for two different external obstructions: a building ahead and an overhang used as shading system. Each group contains a set of models for different daylighting metrics (daylight autonomy, continuous daylight autonomy and spatial daylight autonomy) and for the energy demand for lighting, considering a manual on/off switch or automated daylight responsive lighting control. The estimates that were obtained using the models showed, compared to the corresponding simulations results, a coefficient of variation CV lower than 16% for all the models, with one exception, having a CV up to 30%. The aim of the study was to elaborate models that could be used to incorporate daylighting strategies since the earliest stages of the building design process. Using the models, it is possible to predict the annual daylight amount in a room and the corresponding energy consumption of the lighting systems starting from some given room features (for instance the room depth or the window area) or, the other way around, estimate the suitable values of room features (glazing area, visible transmittance or room depth) to guarantee a target value of energy demand for lighting or of a daylighting metric.

This longitudinal work explores the relationships between three analyses used for assessing teachers’ voice use: the voice monitoring during lessons that describes the teachers’ Vocal Behavior (VB), the perceptual assessment of voice by speech-language pathologists and the estimation of objective parameters from vocalizations to define teachers’ Vocal Performance (VP). About 30 Italian teachers from secondary schools were involved at the beginning and at the end of a school year. In each period, teachers’ vocal activity was monitored using the Voice Care device, which acquires the voice signal through a contact microphone fixed at the neck to estimate sound pressure level, fundamental frequency, and voicing time percentage. Once in each period, two speech-language pathologists performed a perceptual assessment of teachers’ voice using the GIRBAS-scale. On that occasion, teachers vocalized a sustained vowel standing in front of a sound level meter in a quiet room. Jitter, Shimmer and other parameters were ...

The accurate prediction of windblown sand drift events approaching human infrastructures and activities is fundamental in arid lands. In both scientific literature and technical practice sand drift estimation is carried out in mean terms. Typically, sand drift net direction and intensity are assessed by means of the resultant drift potential. However, windblown sand suffers a number of epistemic and aleatory uncertainties, related to both the wind and the sand fields. The windblown sand drift estimation in probabilistic terms is useful in the infrastructure design perspective and allows to obtain characteristic values of windblown sand transport. In this study windblown sand is considered as an environmental action in analogy to wind action. Several uncertainties involved in the phenomenon are considered: threshold shear velocity and 10-min average wind velocity are assumed as random variables. Monte Carlo approach is adopted within a bootstrapping technique in order to assess sand drift in probabilistic terms. The proposed approach is applied to five sites in the Arabian Peninsula. Directional statistics of the sand drift are given for each site.

A weak version of the joint hazard rate order, useful to stochastically compare not independent random variables, has been recently defined and studied in Belzunce et al. (2016). Comparison of hazard rates for dependent random variables, Statistics 50, 630-648. In the present paper, further results on this order are proved and discussed. In particular, some statements dealing with the relationships between the joint weak hazard rate order and other stochastic orders are generalized to the case of non symmetric copulas, and its relations with some multivariate aging notions (studied in Bassan and Spizzichino (2005), Relations among univariate aging, bivariate aging and dependence for exchangeable lifetimes. J. Multivariate Anal. 93, 313-339) are presented. For this purpose, the new notions of Generalized Supermigrative and Generalized Submigrative copulas are defined. Other new results, examples and discussions are provided as well

In this note we compare bivariate additive models with respect to their Pearson correlation coefficients, Kendall’s τ concordance coefficients, and Blomqvist β medial correlation coefficients. The conditions that enable the comparisons involve variability stochastic orders such as the dispersive and the peakedness orders. Specifically we show that we can compare the Kendall’s τ concordance coefficients of Cheriyan and Ramabhadran’s bivariate gamma distributions, in spite of the fact that it is hard (and not necessary) to compute them.

In this paper, we introduce a new multivariate stochastic order that compares random vectors in a direction which is determined by a unit vector, generalizing the previous upper and lower orthant orders. The main properties of this new order, together with its relationships with other multivariate stochastic orders, are investigated and, we present some examples of application in the determination of optimal allocations of wealth among risks in single period portfolio problems.

Since multivariate lifetime data frequently occur in applications, various properties of multivariate distributions have been previously considered to model and describe the main concepts of aging commonly considered in the univariate setting. The generalization of univariate aging notions to the multivariate case involves, among other factors, appropriate definitions of multivariate quantiles and related notions, which are able to correctly describe the intrinsic characteristics of the concepts of aging that should be generalized, and which provide useful tools in the applications. A new multivariate version of the well-known New Better than Used in Expectation univariate aging notion is provided, by means of the concepts of the upper corrected orthant and multivariate excess-wealth function. Some of its properties are described, with particular attention paid to those that can be useful in the analysis of real data sets. Finally, through an example it is illustrated how the new multivariate aging notion influences the final results in the analysis of data on tumor growth from the Comprehensive Cohort Study performed by the German Breast Cancer Study Group.

Customer satisfaction data collected by a large cellular phone service provider are to be used to evaluate and improve the quality of their service. For this purpose, we propose a Bayesian treatment of a joint-response chain graph relating partial assessments of specific aspects of quality to an overall assessment of the service quality. The resulting Bayesian model can be used to render basic geographical and temporal differentiation, allowing the company to undertake direct corrective actions. Both normal and binary models are considered for our customer satisfaction data and are compared with other currently used methods in the study of customer satisfaction. Copyright © 2011 John Wiley & Sons, Ltd.

In this paper, some new properties of the upper-corrected orthant of a random vector are proved. The univariate right-spread or excess wealth function, introduced by Fernandez-Ponce et al. (1996) , is extended to multivariate random vectors, and some properties of this multivariate function are studied. Later, this function was used to define the excess wealth ordering by Shaked and Shanthikumar (1998) and Fernandez-Ponce et al. (1998) . The multivariate excess wealth function enable us to define a new stochastic comparison which is weaker than the multivariate dispersion orderings. Also, some properties relating the multivariate excess wealth order with stochastic dependence are described.

A class of generalized bivariate Marshall–Olkin distributions, which includes as special cases the Marshall–Olkin bivariate exponential distribution and the Marshall–Olkin type distribution due to Muliere and Scarsini (1987) [19] are examined in this paper. Stochastic comparison results are derived, and bivariate aging properties, together with properties related to evolution of dependence along time, are investigated for this class of distributions. Extensions of results previously presented in the literature are provided as well.

Consider a series or parallel system of independent components and assume that the components are randomly chosen from two different batches, with the components of the first batch being more reliable than those of the second. In this note it is shown that the reliability of the system increases, in usual stochastic order sense, as the random number of components chosen from the first batch increases in increasing convex order. As a consequence, we establish a result analogous to the Parrondo’s paradox, which shows that randomness in the number of components extracted from the two batches improves the reliability of the series system.

We discuss suitable conditions such that the lifetime of a series or of a parallel system formed by two components having nonindependent lifetimes may be stochastically improved by replacing the lifetimes of each of the components by an independent mixture of the individual components' lifetimes. We also characterize the classes of bivariate distributions where this phenomenon arises through a new weak dependence notion. © 2011 Wiley Periodicals, Inc. Naval Research Logistics, 2011

Let X = (X, Y) be a pair of lifetimes whose dependence structure is described by an Archimedean survival copula, and let X t = [(X − t, Y − t) | X > t, Y > t] denotes the corresponding pair of residual lifetimes after time t ≥ 0. Multivariate aging notions, defined by means of stochastic comparisons between X and X t , with t ≥ 0, were studied in Pellerey (2008), who considered pairs of lifetimes having the same marginal distribution. Here, we present the generalizations of his results, considering both stochastic comparisons between X t and X t+s for all t, s ≥ 0 and the case of dependent lifetimes having different distributions. Comparisons between two different pairs of residual lifetimes, at any time t ≥ 0, are discussed as well.

In this paper, we define and study a new notion for the comparison of the hazard rates of two random variables taking into account their mutual dependence. Properties, applications and the comparison for a data set are given.

The frailty approach is commonly used in reliability theory and survival analysis to model the dependence between lifetimes of individuals or components subject to common risk factors; according to this model the frailty (an unobservable random vector that describes environmental conditions) acts simultaneously on the hazard functions of the lifetimes. Some interesting conditions for stochastic comparisons between random vectors defined in accordance with these models have been described in the literature; in particular, comparisons between frailty models have been studied by assuming independence for the baseline survival functions and the corresponding environmental parameters. In this paper, a generalization of these models is developed, which assumes conditional dependence between the components of the random vector, and some conditions for stochastic comparisons are provided. Some examples of frailty models satisfying these conditions are also described.

The reliable prediction in probabilistic terms of the consequences of aeolian events related to sand transport phenomena is a key element for human activities in arid regions. Threshold shear velocity generating sand lifting is a key component of such a prediction. It suffers from the effect of uncertainties of different origin, such as those related to the physical phenomena, measurement procedures, and modelling. Semi empirical models are often fitted to a small amount of data, while recent probabilistic models needs the probability distribution of several random variables. Triggered by this motivation, this paper proposes a purely statistical approach to fluid threshold shear velocity for sand saltation, treated as a single comprehensive random variable. A data set is derived from previously published studies. Estimates of conditional probability distributions of threshold shear velocity for given grain diameters are given. The obtained statistical moments are critically compared to some deterministic semi empirical models refitted to the same collected data. The proposed statistical approach allows to obtain high order statistics useful for practical purposes.

In this note we consider branching processes whose behavior depends on a dynamic random environment, in the sense that we assume that the offspring distributions of individuals are parametrized, over time, by the realizations of a process describing the environmental evolution. We study how the variability in time of the environment modifies the variability of total population by considering two branching processes of this kind (but subjected to different environments). We also provide conditions on the random environments in order to stochastically compare their marginal distributions in the increasing convex sense. Weaker conditions are also provided for comparisons at every fixed time of the expected values of the two populations.

Different aggregate preference orders based on rankings and top choices have been defined in the literature to describe preferences among items in a fixed set of alternatives. A useful tool in this framework is constituted by random utility models, where the utility of each alternative, or object, is represented by a random variable, indexed by the object, which, for example, can capture the variability of preferences over a population. Applications are derived in diverse research fields, including computer science, management science and reliability. Recently, some stochastic ordering conditions have been provided for comparing alternatives by means of some aggregate preference orders in the case of independent random utility variables by Joe (Math Soc Sci 43:391–404, 2002). In this paper we provide new conditions, based on some joint stochastic orderings, for aggregate preference orders among the alternatives in the case of dependent random utilities. We also provide some examples of application in different research fields.

Dependencies among random numbers of summands in random sums or among summands have been widely studied in recent literature, and several applications have been developed in actuarial sciences and reliability, where dependencies among risks or lifetimes are quite usual in common problems. In this note we derive stochastic comparisons for random sums when the summands and number of summands depend on a common random environment. The main results presented here describe how such dependence among the random environmental parameters influences the variability of random vectors of sums. As a corollary, we provide stochastic bounds for total claim amounts. Other applications and a discussion of a related result are included.

In this note we provide new results of interest in the portfolio choice problem when the risky opportunities are correlated: for a general vector (X1;X2;:::;Xn) of risky opportunities we give new conditions for stochastic comparison among dierent portfolios choices and new necessary and sucient conditions to characterize the portfolio which gives the maximal expected utility return.

Consider a general coherent system with independent or dependent components, and assume that the components are randomly chosen from two different stocks, with the components of the first stock having better reliability than the others. Then here we provide sufficient conditions on the component’s lifetimes and on the random numbers of components chosen from the two stocks in order to improve the reliability of the whole system according to different stochastic orders. We also discuss several examples in which such conditions are satisfied and an application to the study of the optimal random allocation of components in series and parallel systems. As a novelty, our study includes the case of coherent systems with dependent components by using basic mathematical tools (and copula theory).

In many applicative fields, the behavior of a process \(\mathbf {Z}\) is assumed to be subjected to an underlying process \(\varTheta \) that describes evolutions of environmental conditions. A common way to define the environmental process is by letting the marginal values of \(\varTheta \) subjected to specific environmental factors (constant along time) and factors describing the conditions of the environment at the specified time. In this paper we describe some recent results that can be used to compare two of such environmental discrete-time processes \(\varTheta \) and \(\widetilde{\varTheta }\) in dependence. A sample of applications of the effects of these comparison results on the corresponding processes \(\mathbf {Z}\) and \(\widetilde{\mathbf {Z}}\) in some different applicative contexts are provided.

Consider m counting processes @C"i,i=1,2,...,m, and a random vector of positive integers M=(M"1,M"2,...,M"m). Denote by X"i"j the jth interpoint interval in the process @C"i,i=1,2,...,m,j=1,2,..., and define Z"[email protected]?"j"="1^M^"^iX"i"j,i=1,2,...,m. It is assumed that M is independent of the @C"i's, however, the processes @C"i's are not necessarily mutually independent. In this paper we identify some situations in which the positive (negative) association of the M"i's implies the positive (negative) association of the Z"i's. Some applications in reliability theory and in insurance are indicated.

Ebrahimi (2001) proposes an interesting model for assessing a system's reliability, expressing the failure time of the system in terms of a deterioration process and covariates. He also provides illustrative examples and gives some properties of the model. In this note, we give conditions for negative ageing of the system's lifetime, and we correct one of his statements. Moreover, for the lifetimes of two systems of the same kind, some stochastic comparisons are presented.

The geometric telegrapher's process is proposed as a model to describe the dynamics of the price of risky assets. When the underlying random inter-times have Erlang distribution we express the probability law of such process in terms of a suitable two-index pseudo-Bessel function. Stochastic comparisons of two geometric telegrapher's processes based on the usual stochastic order (FSD comparison) and on the stop-loss order are also performed. Various examples of application of such comparisons are then provided. Copyright © 2002 John Wiley & Sons, Ltd.

We consider the problem of stochastic comparison of general Garch-like processes, for different parameters and different distributions of the innovations. We identify several stochastic orders that are propagated from the innovations to the Garch process itself, and discuss their interpretations. We focus on the convex order and show that in the case of symmetric innovations it is also propagated to the cumulated sums of the Garch process. More generally, we discuss multivariate comparison results related to the multivariate convex and supermodular order. Finally we discuss ordering with respect to the parameters in the Garch (1,1) case. Key words: Garch, Convex Order, Peakedness, Kurtosis, Supermodularity.

In this article, we identify conditions under which the epoch times and the interepoch intervals of a nonhomogeneous Poisson process have logconcave densities. The results are extended to relevation counting processes. We also study discrete-time counting processes and find conditions under which the epoch times and the interepoch intervals of these discrete-time processes have logconcave discrete probability densities. The results are interpreted in terms of minimal repair and record values. Several examples illustrate the theory.

LetX andY be two random variables with finite expectationsE X andE Y, respectively. ThenX is said to be smaller thanY in the dilation order ifE[ϕ(X-E X)]≤E[ϕ(Y-E Y)] for any convex functionϕ for which the expectations exist. In this paper we obtain a new characterization of the dilation order. This characterization enables us to give new interpretations to the dilation order, and using them we identify conditions which imply the dilation order. A sample of applications of the new characterization is given.

In this note we consider a vector (X1, X2) of lifetimes whose dependenceis described by an Archimedean copula, and we provide conditions useful to describe the relationships between ageing properties of X1 and X2and ageing properties of their minimum. Examples are also provided.

The paper presents a set of logistic mathematical models to quickly estimate, since the earliest design stages, the electric lighting energy demand for a generic room with different architectural features (site, orientation, external obstructing angle, window size, glazing visible transmittance and room depth), lighting system characteristics or users’ lighting requirements (working plane illuminance, lighting power density, type of lighting control). The models were built upon the data obtained from a parametric study, based on simulations carried out through Daysim of 828 case-studies. Two dataset were obtained: one for rooms with a manual on-off switch lighting control and one for rooms with an automatic daylight responsive lighting control. Consistently, two specific models were derived, one for each control system. The average error of the mathematical models in estimating the lighting energy demand with respect to the results of Daysim simulations was quantified in terms of the normalized Mean Bias Error value (0.66% – for the manual control system model and 0.29% – for the automatic daylight responsive control system model) and of the Coefficient of Variation of the Root Mean Squared Error (3.45% and 5.34% for the two models, respectively).

Some moment inequalities are known to be valid for non-parametric lifetime distribution classes. Here we consider one set of these inequalities, which hold for random variables that are DMRL (decreasing in mean residual life). We prove that such inequalities are satisfied by variables which are sums of DMRL random variables too, though these sums are not necessarily DMRL. Related results are shown, together with similar results valid for the stochastic comparison in mean residual life.

Consider an n–component coherent system having random lifetime \(T_{\boldsymbol{X}}\), where \(\boldsymbol{X} = (X_{1},\ldots,X_{n})\) is the vector of the non-independent components’ lifetimes. Stochastic comparisons of the residual life of \(T_{\boldsymbol{X}}\) at a fixed time t ≥ 0, conditioned on \(\{T_{\boldsymbol{X}} > t\}\) or on \(\{X_{i} > t,\forall i = 1,\ldots,n\}\), are investigated. Sufficient conditions on the vector \(\boldsymbol{X}\) that imply this comparison in the usual stochastic order are provided, together with sufficient conditions under which the lifetime \(T_{\boldsymbol{X}}\) satisfies the NBU aging property.

A new partial ordering among life distributions in terms of their uncertainties is introduced. Our measure of uncertainty is Shannon information applied to the residual lifetime. The relationship between this ordering and various existing orderings of life distributions are discussed. Various properties of our proposed concept are examined. Based on our proposed ordering and various existing orderings, the notion of a ‘better system' is introduced.