Multivariate extremality measure

We propose a new multivariate order based on a concept that we will call extremality". Given a unit vector, the extremality allows to measure the "farness" of a point with respect to a data cloud or to a distribution in the vector direction. We establish the most relevant properties of this measure and provide the theoretical basis for its nonparametric estimation. We include two applications in Finance: a multivariate Value at Risk (VaR) with level sets constructed through extremality and a portfolio selection strategy based on the order induced by extremality.Extremality, Oriented cone, Value at risk, Portfolio selection

Clustering and classifying images with local and global variability

A procedure for clustering and classifying images determined by three classification variables is presented. A measure of global variability based on the singular value decomposition of the image matrices, and two average measures of local variability based on spatial correlation and spatial changes. The performance of the procedure is compared using three different databases.Images, Cluster, Classification

On identifiability of MAP processes

Two types of transitions can be found in the Markovian Arrival process or MAP: with and without arrivals. In transient transitions the chain jumps from one state to another with no arrival; in effective transitions, a single arrival occurs. We assume that in practice, only arrival times are observed in a MAP. This leads us to define and study the Effective Markovian Arrival process or E-MAP. In this work we define identifiability of MAPs in terms of equivalence between the corresponding E-MAPs and study conditions under which two sets of parameters induce identical laws for the observable process, in the case of 2 and 3-states MAP. We illustrate and discuss our results with examples.Batch Markovian Arrival process, Hidden Markov models, Identifiability problems

BAYESIAN ESTIMATION FOR THE M/G/1 QUEUE USING A PHASE TYPE APPROXIMATION

This article deals with Bayesian inference and prediction for M/G/1 queueing systems. The general service time density is approximated with a class of Erlang mixtures which are phase type distributions. Given this phase type approximation, an explicit evaluation of measures such as the stationary queue size, waiting time and busy period distributions can be obtained. Given arrival and service data, a Bayesian procedure based on reversible jump Markov Chain Monte Carlo methods is proposed to estimate system parameters and predictive distributions.

Non-identifiability of the two state Markovian Arrival process

In this paper we consider the problem of identifiability of the two-state Markovian Arrival process (MAP2). In particular, we show that the MAP2 is not identifiable and conditions are given under which two different sets of parameters, induce identical stationary laws for the observable process.Batch Markovian Arrival process, Markov Renewal process, Hidden Markov models, Identifiability problems

On the Conjecture of Kochar and Korwar

In this paper, we solve for some cases a conjecture by Kochar and Korwar (1996) in relation with the normalized spacings of the order statistics related to a sample of independent exponential random variables with different scale parameter. In the case of a sample of size n=3, they proved the ordering of the normalized spacings and conjectured that result holds for all n. We give the proof of this conjecture for n=4 and for both spacing and normalized spacings. We also generalize some results to n>4Heterogeneous exponential distribution, Hazard rate order, Normalized

On stochastic properties between some ordered random variables

A great number of articles have dealt with stochastic comparisons of ordered random variables in the last decades. In particular, distributional and stochastic properties of ordinary order statistics have been studied extensively in the literature. Sequential order statistics are proposed as an extension of ordinary order statistics. Since sequential order statistics models unify various models of ordered random variables, it is interesting to study their distributional and stochastic properties. In this work, we consider the problem of comparing sequential order statistics according to magnitude and location orders.Stochastic orderings, Reliability, Order statistics

Comparing quantile residual life functions by confidence bands

A quantile residual life function is the quantile of the remaining life of a surviving subject, as it varies with time. In this article we present a nonparametric method for constructing confidence bands for the difference of two quantile residual life functions. These bands provide evidence for two random variables ordering with respect to a quantile residual life order introduced in Franco-Pereira et al. (2010). A simulation study has been carried out in order to evaluate and illustrate the performance and the consistency of this new methodology. We also present applications to real data examples.Quantile residual life, Confidence bands