Interval bounds for the optimal burn-in times for concave or convex reward functions

An interesting problem in reliability is to determine the optimal burn-in time. In a previous work, the authors studied the solution of such a problem under a particular cost structure. It has been shown there that a key role in the problem is played by a function $\rho$, representing the reward coming from the use of a component in the field. A relevant case in this investigation is the one when $\rho$ is linear. In this paper, we explore further the linear case and use its solutions as a benchmark for determining the locally optimal times when the function $\rho$ is not linear or under a different cost structure

Interactions between ageing and risk properties in the analysis of burn-in problems

Several relevant problems in reliability can be looked at as problems of risk management and of decisions in the face of uncertainty. However, in this frame, the so-called burn-in problem can be seen as a problem of risk taking par excellence. In this paper, we in particular point out some aspects concerning interactions between the probabilistic model for lifetimes and considerations of an economic kind. As one of the features of our work, we hinge on some unexplored connections between ageing properties of a one-dimensional survival function Formula and risk-aversion-type properties of the function u(t) = bG(t), b > 0, when the latter is seen as a utility function

Measuring Discrepancies Between Poisson and Exponential Hawkes Processes

Poisson processes are widely used to model the occurrence of similar and independent events. However they turn out to be an inadequate tool to describe a sequence of (possibly differently) interacting events. Many phenomena can be modelled instead by Hawkes processes. In this paper we aim at quantifying how much a Hawkes process departs from a Poisson one with respect to different aspects, namely, the behaviour of the stochastic intensity at jump times, the cumulative intensity and the interarrival times distribution. We show how the behaviour of Hawkes processes with respect to these three aspects may be very irregular. Therefore, we believe that developing a single measure describing them is not efficient, and that, instead, the departure from a Poisson process with respect to any different aspect should be separately quantified, by means of as many different measures. Key to defining these measures will be the stochastic intensity and the integrated intensity of a Hawkes process, whose properties are therefore analysed before introducing the measures. Such quantities can be also used to detect mistakes in parameters estimation

Optimal trial duration times for multiple change points products lifetime distributions

An interesting problem in reliability is to determine the optimal burn-in time. In Foschi and Spizzichino (Decis Anal 9:103-118, 2012), the solution of such a problem under a particular cost structure has been studied. It has been shown there that a key role in the problem is played by a function ρ, representing the reward coming from the use of a component in the field. A relevant case in this investigation is the one when ρ is linear. In view of more general applications to management of production processes, in this paper, we explore further the linear case and use its solutions as a benchmark for determining the locally optimal times when the function ρ is not linear or when the components lifetimes distribution is not bathtub (or upside down bathtub) shaped

Missing Links in Multiple Trade Networks

In this paper we develop a network model of international trade which is able to replicate the concentrated and sparse nature of trade data. Our model extends the preferential attachment (PA) growth model to the case of multiple networks. Countries trade a variety of goods of different complexity. Every country progressively evolves from trading less sophisticated to high-tech goods. The probability to capture more trade opportunities at a given level of complexity and to start trading more complex goods are both proportional to the number of existing trade links. We provide a set of theoretical predictions and simulative results. A calibration exercise shows that our model replicates the same concentration level of world trade as well as the sparsity pattern of the trade matrix. Moreover, we find a lower bound for the share of genuine missing trade links. We also discuss a set of numerical solutions to deal with large multiple networks

Preferential attachment in multiple trade networks

In this paper we develop a model for the evolution of multiple networks which is able to replicate the concentrated and sparse nature of world trade data. Our model is an extension of the preferential attachment growth model to the case of multiple networks. Countries trade a variety of goods of different complexity. Every country progressively evolves from trading less sophisticated to high-tech goods. The probabilities of capturing more trade opportunities at a given level of complexity and of starting to trade more complex goods are both proportional to the number of existing trade links. We provide a set of theoretical predictions and simulative results. A calibration exercise shows that our model replicates the same concentration level of world trade as well as the sparsity pattern of the trade matrix. We also discuss a set of numerical solutions to deal with large multiple networks

Aging functions and multivariate notions of NBU and IFR

For d≥2, let X=(X1, …, Xd) be a vector of exchangeable continuous lifetimes with joint survival function $\overline{F}$. For such models, we study some properties of multivariate aging of $\overline{F}$ that are described by means of the multivariate aging function $B_{\overline{F}}$, which is a useful tool for describing the level curves of $\overline{F}$. Specifically, the attention is devoted to notions that generalize the univariate concepts of New Better than Used and Increasing Failure Rate. These multivariate notions are satisfied by random vectors whose components are conditionally independent and identically distributed having univariate conditional survival function that is New Better than Used (respectively, Increasing Failure Rate). Furthermore, they also have an interpretation in terms of comparisons among conditional survival functions of residual lifetimes, given a same history of observed survivals

Diagonal sections of copulas, multivariate conditional hazard rates and distributions of order statistics for minimally stable lifetimes

As a motivating problem, we aim to study some special aspects of the marginal distributions of the order statistics for exchangeable and (more generally) for minimally stable non-negative random variables $T_{1},...,T_{r}$. In any case, we assume that $T_{1},...,T_{r}$ are identically distributed, with a common survival function $\overline{G}$ and their survival copula is denoted by $K$. The diagonal's and subdiagonals' sections of $K$, along with $\overline{G}$, are possible tools to describe the information needed to recover the laws of order statistics. When attention is restricted to the absolutely continuous case, such a joint distribution can be described in terms of the associated multivariate conditional hazard rate (m.c.h.r.) functions. We then study the distributions of the order statistics of $T_{1},...,T_{r}$ also in terms of the system of the m.c.h.r. functions. We compare and, in a sense, we combine the two different approaches in order to obtain different detailed formulas and to analyze some probabilistic aspects for the distributions of interest. This study also leads us to compare the two cases of exchangeable and minimally stable variables both in terms of copulas and of m.c.h.r. functions. The paper concludes with the analysis of two remarkable special cases of stochastic dependence, namely Archimedean copulas and load sharing models. This analysis will allow us to provide some illustrative examples, and some discussion about peculiar aspects of our results

Threshold copulas and positive dependence

Starting with a notion of positive dependence View the MathML source and with the family of the lower threshold copulas Ct associated with a bivariate distribution having copula C, we define different notions of positive dependence for C, reflecting the dependence properties of the copulas Ct for some t. Then, we analyze some structural aspects of lower threshold copulas and of the given definitions. Furthermore we consider several specific cases arising from relevant special choices of View the MathML source (e.g., PQD, LTD, TP2 and PLR). Our analysis, in particular, allows us to present a number of relevant examples and counter-examples, which can be useful in the study of the tail dependence for a bivariate distribution