arXiv subjects

Gane Samb Lo

Publications and source records attributed to Gane Samb Lo.

At least 73 records · Page 4Linked to original sources

arXiv2016

Uniform in bandwidth consistency for the transformation kernel estimator of copulas

In this paper we establish the uniform in bandwidth consistency for the transformation kernel estimator of copulas introduced in [Omelka et al.(2009)]. To this end, we first prove a uniform in bandwidth law of the iterated logarithm for the maximal deviation of this estimator from its expectation. We then show that, as n goes to infinity, the bias of the estimator converges to zero uniformly in the bandwidth h, varying over a suitable interval. A practical method of selecting the optimal bandwidth is also presented. Finally, we make conclusive simulation experiments showing the performance of the estimator in finite samples.

math.ST↗

arXiv2016

Sur la décomposabilité empirique des indicateurs de pauvreté

We study the empirical decomposition of poverty indicators. This property is very important and convenient in the context of the fight against poverty. Indeed, it makes it possible to put in place sectoral poverty reduction policies on the basis of a relevant stratification laid down at the outset. The simultaneous impacts of these policies, measured as reduction gains over the population as a whole, is then obtained by aggregating those obtained at each stratum by a relatively simple formula. It turns out that indicators as important as those of Sen and Shorrocks do not verify this property contrary to the elements of the class of Foster - Greer and Thorbecke. Given the data from the 1996 Senegalese Survey of Households (ESAM), we show that the lack of decomposability of these indicators on the income variable for several types of population stratification is practically zero , of the order of one to two per thousand. This makes it possible to use the decomposition of the Sen and Shorrocks indicators without any untoward consequences. An explanatory model of these results is presented for future research.

econ.GN↗

arXiv2016

Demmartingales and the functionnal Hill process for small parameters

Association of random variables and Demimartingales are recent fields for handling asymptotic behaviors of sums of dependent random variables. We apply their techniques to establish the asymptotic law of a demimartingale We next apply the results to find the asymptotic behavior the functional Hill process for small parameters within the Extreme Value Theory (EVT) field. Such a result would have been very hard to find whithout demimartingales techniques.

stat.OT↗

arXiv2016

A supermartingale argument for characterizing the Functional Hill process weak law for small parameters

The paper deals with the asymptotic laws of functional of standard random variables. These classes of statistics are closely related to estimators of the extreme value index when the underlying distribution function is in the Weibull domain of attraction. We use techniques based on martingales theory to describe the non Gaussian asymptotic distribution of the aforementioned statistics. We provide results of a simulation study as well as statistical tests that may be of interest with the proposed results.

stat.ME↗

arXiv2016

Convergence Vague (IA) - Suites de Vecteurs Aléatoires

This monograph aims at presenting the core weak convergence theory for sequences of random vectors with values in $\mathbb{R}^k$. In some places, a more general formulation in metric spaces is provided. It lays out the necessary foundation that paves the way to applications in particular subfields of the theory. In particular, the needs of Asymptotic Statistics are addressed. A whole chapter is devoted to weak convergence in $\mathbb{R}$ where specific tools, for example for handling weak convergence of sequences using independent and indentically distributed random variables such that the Renyi's representations by means of standard uniform or exponential random variables, are stated. The function empirical process is presented as a powerful tool for solving a considerable number of asymptotic problems in Statistics. The text is written in a self-contained approach whith the proofs of all used results at the exception of the general Skorohod-Wichura Theorem

math.PR↗

arXiv2016

General Central Limit Theorems for Associated Sequences

In this paper, we provide general central limit theorems (CLT's) for associated random variables (rv's) following the approaches used by Newman (1980) and Olivera et al.(2012). Given some assumptions, a Lyapounov-Feller-Levy type theorem is stated. We next specify different particular CLT versions of associated sequences based on moment conditions. A comparison study with available CTL's is performed. As a by-product, we complete an important available theorem where an assumption was missing.

math.PR↗

arXiv2016

Invariance principles for random sums of random variables

This note investigates invariance principles for sums of N(nt) iid radom variables, where n is an integer, t is a positive real number and N(u) is a stochastic process with nonnegative integer values. We show that the sequence of sums of these random variables denoted S(n,t), when appropriately centered and normalized, weakly converges to a Gaussian process. We give sufficient conditions depending on the expectation of N(nt) which allows to rescale S(n,t) into a stochastic S(n,a(t)) weakly converging to a Brownian motion.

math.PR↗

arXiv2016

A remark on the asymptotic tightness in $\ell^{\infty}([a,b])$

In this note, we extend a simple criteria for uniform tightness in $C(0,1)$, the class of real continuous functions defined on $(0,1)$, given in Theorem 8.3 of Billingsley to the asymptotic tightness in $\ell^{+\infty}([a,b])$, the class of real bounded functions defined on $[a,b] $ with $a<b$, in the lines of Theorems 1.5.6 and 1.5.7 in van der vaart and Wellner.

math.PR↗

arXiv2016

Asymptotic confidence bands for copulas based on the transformation kernel estimator

In this paper we establish asymptotic simultaneous confidence bands for the transformation kernel estimator of copulas introduced in Omelka et al.(2009). To this aim, we prove a uniform in bandwidth law of the iterated logarithm for the maximal deviation of this estimator from its expectation, under smoothness conditions on the copula function. We also study the bias, which tends asymptotically and uniformly to zero with the same precise rate. Some simulation experiments are finally provided to support our results

stat.ME↗

arXiv2016

A note on the asymptotic normality of sums of extreme values

Let $X_1$, $X_2$,... be a sequence of independent random variables with common distribution function $F$ in the domain of attraction of a Gumbel extreme value distribution and for each integer $n\geq 1$, let $X_{1,n} \leq ... X_{n,n}$ denote the order statistics based on the first $n$ of these random variables. Along with related results it is shown that for any sequence of positive integers $k_n \rightarrow +\infty$ and $k_{n}/n \rightarrow 0$ as $n \rightarrow 0$ the sum of the upper $k_n$ extreme values $X_{n-k_{n},n}+...+X_{n,n}$, when properly centered and normalized, converges in distribution to a standard normal random variable $N(0, 1)$. These results constitute an extension of results by S. Csörgő and D.M. Mason (1985).

stat.ME↗

arXiv2016

A double-indexed functional Hill process and applications

Let $X_{1,n} \leq .... \leq X_{n,n}$ be the order statistics associated with a sample $X_{1}, ...., X_{n}$ whose pertaining distribution function (% \textit{df}) is $F$. We are concerned with the functional asymptotic behaviour of the sequence of stochastic processes \begin{equation} T_{n}(f,s)=\sum_{j=1}^{j=k}f(j)\left(\log X_{n-j+1,n}-\log X_{n-j,n}\right)^{s}, \label{fme} \end{equation} indexed by some classes $\mathcal{F}$ of functions $f:\mathbb{N}% ^{\ast}\longmapsto \mathbb{R}_{+}$ and $s \in ]0,+\infty[$ and where $k=k(n)$ satisfies \begin{equation*} 1\leq k\leq n,k/n\rightarrow 0\text{as}n\rightarrow \infty . \end{equation*} \noindent We show that this is a stochastic process whose margins generate estimators of the extreme value index when $F$ is in the extreme domain of attraction. We focus in this paper on its finite-dimension asymptotic law and provide a class of new estimators of the extreme value index whose performances are compared to analogous ones. The results are next particularized for one explicit class $\mathcal{F}$.

stat.ME↗

arXiv2015

Consistency bands for the mean excess function and application to graphical goodness of fit test for financial data

In this paper, we use the modern setting of functional empirical processes and recent techniques on uniform estimation for non parametric objects to derive consistency bands for the mean excess function in the i.i.d. case. We apply our results for modelling financial data, in particular Dow Jones data basis to see how good the Generalized hyperbolic distribution models fit monthly data.

stat.ME↗

arXiv2015

Asymptotic confidence bands for copulas based on the local linear kernel estimator

In this paper we establish asymptotic simultaneous confidence bands for copulas based on the local linear kernel estimator proposed by Chen and Huang [1]. For this, we prove under smoothness conditions on the copula function, a uniform in bandwidth law of the iterated logarithm for the maximal deviation of this estimator from its expectation. We also show that the bias term converges uniformly to zero with a precise rate. The performance of these bands is illustrated in a simulation study. An application based on pseudo-panel data is also provided for modeling dependence.

stat.ME↗

arXiv2015

Probabilistic, statistical and algorithmic aspects of the similarity of texts and application to Gospels comparison

The fundamental problem of similarity studies, in the frame of data-mining, is to examine and detect similar items in articles, papers, books, with huge sizes. In this paper, we are interested in the probabilistic, and the statistical and the algorithmic aspects in studies of texts. We will be using the approach of $k$\textit{-shinglings}, a $k$\textit{-shingling} being defined as a sequence of $k$ consecutive characters that are extracted from a text ($k\geq 1$ ). The main stake in this field is to find accurate and quick algorithms to compute the similarity in short times. This will be achieved in using approximation methods. The first approximation method is statistical and, is based on the theorem of Glivenko-Cantelli. The second is the banding technique. And the third concerns a modification of the algorithm proposed by Rajaraman and al (% \cite{AnandJeffrey}), denoted here as (RUM). The Jaccard index is the one used in this paper. We finally illustrate these results of the paper on the four Gospels. The results are very conclusive.

stat.ME↗

arXiv2015

A general strong law of large numbers and applications to associated sequences and to extreme value theory

The purpose of this paper is to establish a general strong law of large numbers (SLLN) for arbitrary sequences of random variables (rv's) based on the squared indice method and to provide applications to SLLN of associated sequences. This SLLN is compared to those based on the Hájek-Rényi type inequality. Nontrivial examples are given. An interesting issue that is related to extreme value theory (EVT) is handled here.

math.PR↗

arXiv2014

High moments Jarque-Bera tests for arbitrary distribution functions

The Jarque-Bera's fitting test for normality is a celebrated and powerful one. In this paper, we consider general Jarque-Bera tests for any distribution function df having at least 4k finite moments for k greater than 2. The tests use as many moments as possible whereas the JB classical test is supposed to test only skewness and kurtosis for normal variates. But our results unveil the relations between the coeffients in the JB classical test and the moments, showing that it really depends on the first eight moments. This is a new explanation for the powerfulness of such tests. General Chi-square tests for an arbitraty model, not only normal, are also derived. We make use of the modern functional empirical processes approach that makes it easier to handle statistics based on the high moments and allows the generalization of the JB test both in the number of involved moments and in the underlying distribution. Simulation studies are provided and comparison cases with the Kolmogorov-Smirnov's tests and the classical JB test are given.

stat.ME↗

arXiv2014

Gaussian Approximations and Related Questions for the Spacings process

All the available results on the approximation of the k-spacings process to Gaussian processes have only used one approach, that is the Shorack and Pyke's one. Here, it is shown that this approach cannot yield a rate better than $% \left( N/\log \log N\right) ^{-\frac{1}{4}}\left( \log N\right) ^{\frac{1}{2}% }$. Strong and weak bounds for that rate are specified both where k is fixed and where $k\rightarrow +\infty $. A Glivenko-Cantelli Theorem is given while Stute's result for the increments of the empirical process based on independent and indentically distributed random variables is extended to the spacings process. One of the Mason-Wellner-Shorack cases is also obtained.

stat.ME↗

arXiv2014

Strong limits related to the oscillation modulus of the empirical process based on the k-spacing process

Recently, several strong limit theorems for the oscillation moduli of the empirical process have been given in the iid-case. We show that, with very slight differences, those strong results are also obtained for some representation of the reduced empirical process based on the (non-overlapping) k-spacings generated by a sequence of independent random variables (rv's) uniformly distributed on $(0,1)$. This yields weak limits for the mentioned process. Our study includes the case where the step k is unbounded. The results are mainly derived from several properties concerning the increments of gamma functions with parameters k and one.

stat.ME↗