Limiting behavior of the search cost distribution for the move-to-front rule in the stable case

Move-to-front rule is a heuristic updating a list of n items according to requests. Items are required with unknown probabilities (or popularities). The induced Markov chain is known to be ergodic. One main problem is the study of the distribution of the search cost defined as the position of the required item. Here we first establish the link between two recent papers of Barrera and Paroissin and Lijoi and Pruenster that both extend results proved by Kingman on the expected stationary search cost. Combining results contained in these papers, we obtain the limiting behavior for any moments of the stationary seach cost as n tends to infinity.Normalized random measure, Random discrete distribution, Stable subordinator, Problem of heaps

Limiting search cost distribution for the move-to-front rule with random request probabilities

Consider a list of $n$ files whose popularities are random. These files are updated according to the move-to-front rule and we consider the induced Markov chain at equilibrium. We give the exact limiting distribution of the search-cost per item as $n$ tends to infinity. Some examples are supplied.Comment: move-to-front, search cost, random discrete distribution, limiting distribution, size biased permutatio

Limiting behavior of the search cost distribution for the move-to-front rule in the stable case

Move-to-front rule is a heuristic updating a list of n items according to requests. Items are required with unknown probabilities (or ppopularities). The induced Markov chain is known to be ergodic [4]. One main problem is the study of the distribution of the search cost defined as the position of the required item. Here we first establish the link between two recent papers [3, 8] that both extend results proved by Kingman [7] on the expected stationary search cost. Combining results contained in these papers, we obtain the limiting behavior for any moments of the stationary seach cost as n tends to infinity.normalized random measure; random discrete distribution; stable subordinator; problem of heaps

A comparison of statistical models for short categorical or ordinal time series with applications in ecology

We study two statistical models for short-length categorical (or ordinal) time series. The first one is a regression model based on generalized linear model. The second one is a parametrized Markovian model, particularizing the discrete autoregressive model to the case of categorical data. These models are used to analyze two data-sets: annual larch cone production and weekly planktonic abundance.Comment: 18 page

Passage times of perturbed subordinators with application to reliability

We consider a wide class of increasing Lévy processes perturbed by an independent Brownian motion as a degradation model. Such family contains almost all classical degradation models considered in the literature. Classically failure time associated to such model is defined as the hitting time or the first-passage time of a fixed level. Since sample paths are not in general increasing, we consider also the last-passage time as the failure time following a recent work by Barker and Newby. We address here the problem of determining the distribution of the first-passage time and of the last-passage time. In the last section we consider a maintenance policy for such models

Joint signature of two or more systems with applications to multistate systems made up of two-state components

The structure signature of a system made up of $n$ components having continuous and i.i.d. lifetimes was defined in the eighties by Samaniego as the $n$-tuple whose $k$-th coordinate is the probability that the $k$-th component failure causes the system to fail. More recently, a bivariate version of this concept was considered as follows. The joint structure signature of a pair of systems built on a common set of components having continuous and i.i.d. lifetimes is a square matrix of order $n$ whose $(k,l)$-entry is the probability that the $k$-th failure causes the first system to fail and the $l$-th failure causes the second system to fail. This concept was successfully used to derive a signature-based decomposition of the joint reliability of the two systems. In the first part of this paper we provide an explicit formula to compute the joint structure signature of two or more systems and extend this formula to the general non-i.i.d. case, assuming only that the distribution of the component lifetimes has no ties. We also provide and discuss a necessary and sufficient condition on this distribution for the joint reliability of the systems to have a signature-based decomposition. In the second part of this paper we show how our results can be efficiently applied to the investigation of the reliability and signature of multistate systems made up of two-state components. The key observation is that the structure function of such a multistate system can always be additively decomposed into a sum of classical structure functions. Considering a multistate system then reduces to considering simultaneously several two-state systems

Étude probabiliste et statistique de quelques modèles principalement issus d'applications industrielles

La première partie de ce mémoire porte sur l'étude de modèles en fiabilité, essentiellement des modèles de dégradation pour un composant ou un système, mais également des modèles de durées de vie. On s'intéresse tant à l'étude des propriétés probabilistes de ces modèles qu'à des problèmes d'estimation et qu'à l'optimisation d'une politique de remplacement ou de maintenance. Les mesures de dégradation peuvent être soit qualitatives soit quantitatives. Dans le premier cas, on utilise alors des modèles multi-états. Différents modèles de ce type sont étudiés au chapitre 2 de ce mémoire. Dans le second cas, la dégradation est modélisée à l'aide de processus de Lévy (homogène ou non). Des questions de nature probabiliste ou statistique sont étudiées au chapitre 3 pour quelques modèles de ce type. Aux chapitres 4 et 5, on étudie des problèmes connexes. D'abord, on s'intéresse à la prise en compte des incertitudes pour un modèle de dégradation ou pour un modèle de durée de vie. Ensuite, des problèmes de comparaison de deux échantillons de durée de vie sont étudiés, l'un avec sous l'angle de la détection de rupture, l'autre sous l'angle des cartes de contrôles. Le chapitre 6 présente quelques perspectives liés aux travaux précédemment menés.La deuxième partie de ce mémoire est articulé autour des lois discrètes aléatoires (c'est-à-dire l'ensemble des lois de probabilités à valeurs dans un simplexe). Ce type de lois, en particulier la loi de Dirichlet, trouvent des applications dans divers domaines. Au chapitre 8, on les a utilisé pour étudier le coût de recherche asymptotique pour deux heuristiques auto-organisatrices pour lesquels les popularités des objets sont inconnues et aléatoires. Le chapitre 9 est consacré à des problèmes d'inférence statistique pour la loi de Dirichlet, avec des applications en écologie (estimation du nombre d'espèces à partir d'un échantillon).Enfin, la troisième et dernière partie de ce mémoire est consacrée à un aperçu de travaux plus appliquées, en collaboration avec des biologistes, des médecins, etc