On the Statistical Modeling and Analysis of Repairable Systems

We review basic modeling approaches for failure and maintenance data from repairable systems. In particular we consider imperfect repair models, defined in terms of virtual age processes, and the trend-renewal process which extends the nonhomogeneous Poisson process and the renewal process. In the case where several systems of the same kind are observed, we show how observed covariates and unobserved heterogeneity can be included in the models. We also consider various approaches to trend testing. Modern reliability data bases usually contain information on the type of failure, the type of maintenance and so forth in addition to the failure times themselves. Basing our work on recent literature we present a framework where the observed events are modeled as marked point processes, with marks labeling the types of events. Throughout the paper the emphasis is more on modeling than on statistical inference.Comment: Published at http://dx.doi.org/10.1214/088342306000000448 in the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org

On the Decreasing Failure Rate property for general counting process. Results based on conditional interarrival times

In the present paper we consider general counting processes stopped at a random time $T$, independent of the process. Provided that $T$ has the decreasing failure rate (DFR) property, we give sufficient conditions on the arrival times so that the number of events occurring before $T$ preserves the DFR property of $T$. These conditions involve the study of the conditional interarrival times. As a main application, we prove the DFR property in a context of maintenance models in reliability, by the consideration of Kijima type I virtual age models under quite general assumptions

Bayesian salamanders: analysing the demography of an underground population of the European plethodontid <i>Speleomantes strinatii</i> with state-space modelling

&lt;b&gt;Background&lt;/b&gt;: It has been suggested that Plethodontid salamanders are excellent candidates for indicating ecosystem health. However, detailed, long-term data sets of their populations are rare, limiting our understanding of the demographic processes underlying their population fluctuations. Here we present a demographic analysis based on a 1996 - 2008 data set on an underground population of Speleomantes strinatii (Aellen) in NW Italy. We utilised a Bayesian state-space approach allowing us to parameterise a stage-structured Lefkovitch model. We used all the available population data from annual temporary removal experiments to provide us with the baseline data on the numbers of juveniles, subadults and adult males and females present at any given time. &lt;b&gt;Results&lt;/b&gt;: Sampling the posterior chains of the converged state-space model gives us the likelihood distributions of the state-specific demographic rates and the associated uncertainty of these estimates. Analysing the resulting parameterised Lefkovitch matrices shows that the population growth is very close to 1, and that at population equilibrium we expect half of the individuals present to be adults of reproductive age which is what we also observe in the data. Elasticity analysis shows that adult survival is the key determinant for population growth. &lt;b&gt;Conclusion&lt;/b&gt;: This analysis demonstrates how an understanding of population demography can be gained from structured population data even in a case where following marked individuals over their whole lifespan is not practical

Mandelbrot's 1/f fractional renewal models of 1963-67: The non-ergodic missing link between change points and long range dependence

The problem of 1/f noise has been with us for about a century. Because it is so often framed in Fourier spectral language, the most famous solutions have tended to be the stationary long range dependent (LRD) models such as Mandelbrot's fractional Gaussian noise. In view of the increasing importance to physics of non-ergodic fractional renewal models, I present preliminary results of my research into the history of Mandelbrot's very little known work in that area from 1963-67. I speculate about how the lack of awareness of this work in the physics and statistics communities may have affected the development of complexity science, and I discuss the differences between the Hurst effect, 1/f noise and LRD, concepts which are often treated as equivalent.Comment: 11 pages. Corrected and improved version of a manuscript submitted to ITISE 2016 meeting in Granada, Spai

Temporal Locality in Today's Content Caching: Why it Matters and How to Model it

The dimensioning of caching systems represents a difficult task in the design of infrastructures for content distribution in the current Internet. This paper addresses the problem of defining a realistic arrival process for the content requests generated by users, due its critical importance for both analytical and simulative evaluations of the performance of caching systems. First, with the aid of YouTube traces collected inside operational residential networks, we identify the characteristics of real traffic that need to be considered or can be safely neglected in order to accurately predict the performance of a cache. Second, we propose a new parsimonious traffic model, named the Shot Noise Model (SNM), that enables users to natively capture the dynamics of content popularity, whilst still being sufficiently simple to be employed effectively for both analytical and scalable simulative studies of caching systems. Finally, our results show that the SNM presents a much better solution to account for the temporal locality observed in real traffic compared to existing approaches.Comment: 7 pages, 7 figures, Accepted for publication in ACM Computer Communication Revie

Building Loss Models

This paper is intended as a guide to building insurance risk (loss) models. A typical model for insurance risk, the so-called collective risk model, treats the aggregate loss as having a compound distribution with two main components: one characterizing the arrival of claims and another describing the severity (or size) of loss resulting from the occurrence of a claim. In this paper we first present efficient simulation algorithms for several classes of claim arrival processes. Then we review a collection of loss distributions and present methods that can be used to assess the goodness-of-fit of the claim size distribution. The collective risk model is often used in health insurance and in general insurance, whenever the main risk components are the number of insurance claims and the amount of the claims. It can also be used for modeling other non-insurance product risks, such as credit and operational risk.Insurance risk model; Loss distribution; Claim arrival process; Poisson process; Renewal process; Random variable generation; Goodness-of-fit testing