Discussion specifying prior distributions in reliability applications

Especially when facing reliability data with limited information (e.g., a small number of failures), there are strong motivations for using Bayesian inference methods. These include the option to use information from physics-of-failure or previous experience with a failure mode in a particular material to specify an informative prior distribution. Another advantage is the ability to make statistical inferences without having to rely on specious (when the number of failures is small) asymptotic theory needed to justify non-Bayesian methods. Users of non-Bayesian methods are faced with multiple methods of constructing uncertainty intervals (Wald, likelihood, and various bootstrap methods) that can give substantially different answers when there is little information in the data. For Bayesian inference, there is only one method of constructing equal-tail credible intervals-but it is necessary to provide a prior distribution to fully specify the model. Much work has been done to find default prior distributions that will provide inference methods with good (and in some cases exact) frequentist coverage properties. This paper reviews some of this work and provides, evaluates, and illustrates principled extensions and adaptations of these methods to the practical realities of reliability data (e.g., non-trivial censoring).3 página

Discussion of signature-based models of preventive maintenance

First of all, I congratulate the authors in Reference 1 for the pleasant and pedagogical tone in which they exhaustively describe most of the maintenance strategies found in the literature. This fact is a value in itself because it allows both novel researchers and experts having a conceptual map ofmany types ofmaintenance policies which go fromthe seminal work of Reference 2 to the most recent techniques based on the notion of the signature of a coherent system. Specifically, this article is highly recommended for researchers who are interested inmaintenance management strategies for the first time because the authors sequentially list the most relevant results and approaches in the literature, as well as provide a large number of bibliographic citations

Central regions for bivariate distributions

For a one-dimensional probability distribution, the classical concept of central region as a real interquantile interval arises in all applied sciences. We can find applications, for instance, with dispersion, skewness and detection of outliers. All authors agree with the main problem in a multivariate generalization: there does not exist a natural ordering in n-dimensions, n > 1. Because of this reason, the great majority of these generalizations depend on their use. We can say that is common to generalize the concept of central region under the definition of the well known concept of spatial median. In our work, we develop an intuitive concept which can be interpreted as level curves for distribution functions and this one provides a trimmed region. Properties referred to dispersion and probability are also studied and some considerations on more than two dimensions are also considered. Furthermore, several estimations for bivariate data based on conditional quantiles are discussed

Stochastic comparisons, differential entropy and varentropy for distributions induced by probability density functions

Stimulated by the need of describing useful notions related to information measures, we introduce the ‘pdf-related distributions’. These are defined in terms of transformation of absolutely continuous random variables through their own probability density functions. We investigate their main characteristics, with reference to the general form of the distribution, the quantiles, and some related notions of reliability theory. This allows us to obtain a characterization of the pdf-related distribution being uniform for distributions of exponential and Laplace type as well. We also face the problem of stochastic comparing the pdf-related distributions by resorting to suitable stochastic orders. Finally, the given results are used to analyse properties and to compare some useful information measures, such as the differential entropy and the varentropy

Stochastic orders and co-risk measures under positive dependence

Conditional risk measures (or co-risk measures) and risk contribution measures are increasingly used in actuarial portfolio analysis to evaluate the systemic risk, which is related to the risk that the failure or loss of a component spreads to another component or even to the whole portfolio: while co-risk measures are risk-adjusted versions of measures usually employed to assess isolate risks, risk contribution measures quantify how a stress situation for a component affects another one. In this paper, we provide sufficient conditions under which two random vectors could be compared in terms of CoVaR (conditional value-at- risk), CoES (conditional expected shortfall) and different risk contribution measures. Conditions are given in terms of the increasing convex order, the dispersive order and the excess wealth order of the marginals under some assumptions of positive dependence

Stochastic Bounds for Conditional Distributions Under Positive Dependence

We provide stochastic bounds for conditional distributions of individual risks in a portfolio, given that the aggregate risk exceeds its value at risk. Expectations of these conditional distributions can be interpreted as marginal risk contributions to the aggregate risk as measured by the tail conditional expectation. We first provide general lower and upper stochastic bounds and then we obtain further improvements of the bounds in the case of a portfolio consisting of dependent risks. We also derive new characterizations of comonotonic random vectors.Miguel A. Sordo and Alfonso Suarez-Llorens acknowledge the support of Ministerio de Ciencia e Innovación (grant MTM2009-08326) and Consejería de Economía Innovación y Ciencia (grant P09-SEJ-4739)

Comparison of conditional distributions in portfolios of dependent risks

Given a portfolio of risks, we study the marginal behavior of the i-th risk under an adverse event, such as an unusually large loss in the portfolio or, in the case of a portfolio with a positive dependence structure, to an unusually large loss for another risk. By considering some particular conditional risk distributions, we formalize, in several ways, the intuition that the i-th component of the portfolio is riskier when it is part of a positive dependent random vector than when it is considered alone. We also study, given two random vectors with a xed dependence structure, the circumstances under which the existence of some stochastic orderings among their marginals implies an ordering among the corresponding conditional risk distributions

Predicting failure times of coherent systems

The article is focused on studying how to predict the failure times of coherent systems from the early failure times of their components. Both the cases of independent and dependent components are considered by assuming that they are identically distributed (homogeneous components). The heterogeneous components' case can be addressed similarly but more complexly. The present study is for non-repairable systems, but the information obtained could be used to decide if a maintenance action should be carried out at time (Formula presented.). Different cases are considered regarding the information available at time (Formula presented.). We use quantile regression techniques to predict the system failure times and to provide prediction intervals. The theoretical results are applied to specific system structures in some illustrative examples.18 página

A Bayesian Model of COVID-19 Cases Based on the Gompertz Curve

The COVID-19 pandemic has highlighted the need for finding mathematical models to forecast the evolution of the contagious disease and evaluate the success of particular policies in reducing infections. In this work, we perform Bayesian inference for a non-homogeneous Poisson process with an intensity function based on the Gompertz curve. We discuss the prior distribution of the parameter and we generate samples from the posterior distribution by using Markov Chain Monte Carlo (MCMC) methods. Finally, we illustrate our method analyzing real data associated with COVID-19 in a specific region located at the south of Spain.Ministerio de Economía y Competitividad. Gobierno de España; 2014-2020 ERDF Operational Programme; Consejería de Economía, Conocimiento, Empresas y Universidad.Junta de Andalucí

Minimal repair of failed components in coherent systems

The minimal repair replacement is a reasonable assumption in many practical systems. Under this assumption a failed component is replaced by another one whose reliability is the same as that of the component just before the failure, i.e., a used component with the same age. In this paper we study the minimal repair in coherent systems. We consider both the cases of independent and dependent components. Three replacement policies are studied. In the first one, the first failed component in the system is minimally repaired while, in the second one, we repair the component which causes the system failure. A new technique based on the relevation transform is used to compute the reliability of the systems obtained under these replacement policies. In the third case, we consider the replacement policy which assigns the minimal repair to a fixed component in the system. We compare these three options under different stochastic criteria and for different system structures. In particular, we provide the optimal strategy for all the coherent systems with 1-4 independent and identically distributed components