Mhamed Mesfioui

On the properties of some nonparametric concordance measures in the discrete case

It is shown here that Kendalls τ and Spearmans ρ are monotone with respect to the concordance ordering of pairs of discrete as well as continuous random variables. This extends and completes results of [Tchen, A.H., 1980, The Annals of Probability, 8, 814–827.] It is also shown that various relationships between Kendalls τ and Spearmans ρ mentioned in [Nelsen, R.B., 1999, An Introduction to Copulas. Lecture Notes in Statistics no. 139 (New York: Springer).] remain valid for discrete variables. In particular, a result of [Capéraà, P. and Genest, C., 1993, Journal of Nonparametric Statistics, 2, 183–194.] is extended to the case of discrete random pairs. Finally, an analytic expression is given for the most extreme values of Kendalls τ and Spearmans ρ associated with discrete uniform variates.

Compound Poisson approximations for individual models with dependent risks

Abstract This paper shows how compound Poisson distributions can be used to approximate the distribution of the total claim amount in the context of single- or multi-class individual risk models where dependence between the contracts arises through mixtures. Some of these models are generated by Archimedean copulas, and others are seen to fall under the purview of a general multi-class shock model whose structure is both intuitive and easily tractable. A numerical study is used to illustrate the quality of the approximation as a function of the heterogeneity and the dependence in the portfolio. A theoretical result is also provided which helps to explain the effect of dependence on the total claim amount when the contracts are linked through an Archimedean copula model.

Hydroclimatic variability and relation with flood events (Southern Québec, Canada)

In the context of global warming, some climatic models predict an increase in flooding in some regions of the world. It is therefore important to better define the high-risk areas and to limit the use of these areas by riverside communities as much as possible. The study deals with the historical and chronological reconstruction of flood events (from 1865 to 2005) in the southern Quebec basins, and compare with the hydroclimatic data (streamflow, temperature, precipitation) over the past century. Different statistic tests are used on hydroclimatic series and flood events to detect the trend observed. We note an important variability of hydrometric data series and the chronological flood events shows a significant trend in increased flooding in the last 100 years.

Upper stop-loss bounds for sums of possibly dependent risks with given means and variances

Consider non-negative random variables X1,...,Xn whose marginal means and variances are known. The purpose of this paper is to compare two different strategies for finding an upper bound on the stop-loss premium [pi](X1+...+Xn,d)=E{max (0,X1+...+Xn-d)} that are valid for all retention amounts d[greater-or-equal, slanted]0 in the absence of information concerning the type or degree of dependence between the risks Xi. One approach consists of maximizing the premium over all possible values [rho]ij=corr(Xi,Xj), 1[less-than-or-equals, slant]i

Concordance measures for multivariate non-continuous random vectors

A notion of multivariate concordance suitable for non-continuous random variables is defined and many of its properties are established. This allows the definition of multivariate, non-continuous versions of Kendalls tau, Spearmans rho and Spearmans footrule, which are concordance measures. Since the maximum values of these association measures are not +1 in general, a special attention is given to the computation of upper bounds. The latter turn out to be multivariate generalizations of earlier findings made by Neslehova (2007) [9] and Denuit and Lambert (2005) [2]. They are easy to compute and can be estimated from a data set of (possibly) discontinuous random vectors. Corrected versions are considered as well.

Gamma Kernel Estimators for Density and Hazard Rate of Right-Censored Data

The nonparametric estimation for the density and hazard rate functions for right-censored data using the kernel smoothing techniques is considered. The “classical” fixed symmetric kernel type estimator of these functions performs well in the interior region, but it suffers from the problem of bias in the boundary region. Here, we propose new estimators based on the gamma kernels for the density and the hazard rate functions. The estimators are free of bias and achieve the optimal rate of convergence in terms of integrated mean squared error. The mean integrated squared error, the asymptotic normality, and the law of iterated logarithm are studied. A comparison of gamma estimators with the local linear estimator for the density function and with hazard rate estimator proposed by Muller and Wang (1994), which are free from boundary bias, is investigated by simulations.

Organic carbon distribution in alluvial soils according to different flood risk zones

This study examines the spatial distribution of organic carbon in alluvial soils subjected to frequent flooding according to different flood risk zones, that is, interval recurrences of 0-20 years (FFz) and 20-100 years (MFz). Sites located outside of flood zones (NFz) were also selected to compare the soil organic carbon (SOC) in different zones. The selected sites are located in floodplains covered by forest dominated by silver maple (Acer saccharinum L.) and green ash (Fraxinus pennsylvanica Marsh.) in southern Quebec. These floodplains are affected by frequent flooding, especially in the last decades, which has a direct impact on pedogenic processes, particularly in terms of in situ soil biomass and organic matter. The soil samples (0-20 cm depth) collected in a frequent flood zone (FFz), generally show a lower content of soil organic carbon (SOC%) ranging from 1.74 to 2.59% (median values), and mean values between 1.79 and 2.83%, respectively. In areas not affected by the floods, levels of SOC (%) are generally higher, with values ranging between 2.86 and 3.73% (mean), and mean values between 3.18 and 5.17%. Loss of biomass (litter) during the flood recession causes a net loss of organic matter to the subsurface soils. Successive flooding leads to an impoverishment of alluvial soils and undermining of the pedogenic processes and soil development. This confirms the trends observed in our previous work on soil depletion in active floodplains in the study area. Key words: Alluvial soils, soil organic carbon (SOC), floods, spatial variability, climate change.

BIVARIATE EXTENSIONS OF SKELLAM'S DISTRIBUTION

Skellam’s name is traditionally attached to the distribution of the difference of two independent Poisson random variables. Many bivariate extensions of this distribution are possible, e.g., through copulas. In this paper, the authors focus on a probabilistic construction in which two Skellam random variables are affected by a common shock. Two different bivariate extensions of the Skellam distribution stem from this construction, depending on whether the shock follows a Poisson or a Skellam distribution. The models are nested, easy to interpret, and yield positive quadrant-dependent distributions, which share the convolution closure property of the univariate Skellam distribution. The models can also be adapted readily to account for negative dependence. Closed form expressions for Pearson’s correlation between the components make it simple to estimate the parameters via the method of moments. More complex formulas for Kendall’s tau and Spearman’s rho are also provided.

On Concordance Measures for Discrete Data and Dependence Properties of Poisson Model

We study Kendalls tau and Spearmans rho concordance measures for discrete variables. We mainly provide their best bounds using positive dependence properties. These bounds are difficult to write down explicitly in general. Here, we give the explicit formula of the best bounds in a particular Frechet space in order to understand the behavior of the ranges of these measures. Also, based on the empirical copula which is viewed as a discrete distribution, we propose a new estimator of the copula function. Finally, we give useful dependence properties of the bivariate Poisson distribution and show the relationship between parameters of the Poisson distribution and both tau and rho.

On a multivariate Lindley distribution

ABSTRACT This article proposes a multivariate extension of the generalized Lindley distribution introduced by Abouammoh et al. (2015). The proposed model is based on a probabilistic construction in which several Lindley random variables are connected by a common shock. Many statistical properties of this new distribution are explored. In particular, explicit forms of product moments, moment-generating function, and conditional moments are derived. Explicit expressions of moment-based estimators of the underlying parameters of the proposed model are established. Estimation using the maximum likelihood procedure is also investigated. Simulations attesting to the quality of the proposed estimators are presented. Application of the proposed model in reliability is also discussed.