Lo Gane Samb

• Phd mathematics and statistics, 1986, university Paris VI, France
• Managing Director at Gaston Berger University

The objective of this paper is to establish a general asymptotic representation ( GAR ) for a wide range of statistics, employing two fundamental processes: the functional empirical process ( fep ) and the residual functional empirical process introduced by Lo and Sall (2010a, 2010b), denoted as ( lrfep ). The functional empirical process ( fep ) i...

We briefly introduce this special issue of Afrika Statistika on the proceedings of the annual Imhotep seminar on functional empirical and residual processes and their applications.

In this paper we investigate the Burr distributions family which contains twelve members. Second order expansions of quantiles of the Burr’s distributions are provided on which may be based statistical methods, in particular in extreme value theory. Beyond the proper interest of these expansions, we apply them to characterize the asymptotic laws of...

In this paper, we study some aspects on random analysis on the L\'eevy stochastic processes with margins following generalized hyperbolic distributions generated by gamma laws. In particular we study the boundedness of its total variations and the quadratic variations. Next we give an empirical construction that enables the graphical representation...

La distribution pseudo-Lindley a été introduite comme une généralisation utile de la distribution Lindley qui a montré sa pertinence en Analyse de fiabilité et dans l’étude de survie. Dans cet article, nous étudions les estimateurs des moments de la paire de paramètres de ladite distribution et établissons leurs lois asymptotiques en utilisant le p...

According to the Chinese Health Statistics Yearbook, in 2005, the number of traffic accidents was 187781 with total direct property losses of 103691.7 (10000 Yuan). This research aims to fill the gap in the literature by investigating the extreme claim sizes not only for the entire portfolio. This empirical study investigates the behavior of the up...

We propose a probabilistic setting in which we study the probability law of the Rajaraman and Ullman \textit{RU} algorithm and a modified version of it denoted by \textit{RUM}. These algorithms aim at estimating the similarity index between huge texts in the context of the web. We give a foundation of this method by showing, in the ideal case of ca...

The Pseudo-Lindley distribution is generalized in this paper by using the Kumaraswamy-G distribution developed by Cordeiro et al. (2010a). By fusing the Pseudo-Lindley distribution in the Kumaraswamy generator of distribution, we present a new unique four parameters of continuous model of distribution titled the Kumaraswamy Pseudo-Lindley distribut...

Here, we present an automatic data generation method which is fully computer-based for a variate $X$ with an absolutely continuous probability density function ( pdf ) $f$ exactly computable. The method uses computer-based on calculations of integrals (trapezoidal and/or the Monte-Carlo method) for approximating the cumulative distribution function...

The investigation asymptotic limits on associated data mainly focused on limit theorems of summands of associated data and on the related invariance principles. In a series of papers, we are going to set the general frame of the theory by considering an arbitrary infinitely decomposable (divisible) limit law for summands and study the associated fu...

The simple L\'evy Poisson process and scaled forms are explicitly constructed from partial sums of independent and identically distributed random variables and from sums of non-stationary independent random variables. For the latter, the weak limits are scaled Poisson processes. The method proposed here prepares generalizations to dependent data, t...

In earlier stages in the introduction to asymptotic methods in probability theory, the weak convergence of sequences (Xn)n≥1 of binomial random variables (rv’s) to a Poisson law is classical and easy to prove. A version of such a result concerning sequences (Yn)n≥1 of negative binomial rv’s also exists. In both cases, Xn and Yn −n are by-row sums S...

The paper [Lo G.S. and Ahsanullah M. (2019). An introduction to general records theory both for dependent and high dimensions. Afrika Statistika. Vol. 14 (2), pp. 2019-2056. Doi : dx.doi.org/10.16929/as/2019.2019.147] served as the basic material in an international course held in September 2022. A few number of misprints and wrongly written formul...

In earlier stages in the introduction to asymptotic methods in probability theory, the weak convergence of sequences $(X_n)_{n\geq 1}$ of Binomial of random variables (\textit{rv}'s) to a Poisson law is classical and easy-to prove. A version of such a result concerning sequences $(Y_n)_{n\geq 1}$ of negative binomial \textit{rv}'s also exists. In b...

We introduce a new generalization of the Pseudo-Lindley distribution by applying alpha power transformation. The obtained distribution is referred as the Pseudo-Lindley alpha power transformed distribution (\textit{PL-APT}). Some tractable mathematical properties of the \textit{PL-APT} distribution as reliability, hazard rate, order statistics and...

The simple Lévy Poisson process and scaled forms are explicitly constructed from partial sums of independent and identically distributed random variables and from sums of non-stationary independent random variables. For the latter, the weak limits are scaled Poisson processes. The method proposed here prepares generalizations to dependent data, to...

The simple Lévy Poisson process and scaled forms are explicitly constructed from partial sums of independent and identically distributed random variables and from sums of non-stationary independent random variables. For the latter, the weak limits are scaled Poisson processes. The method proposed here prepares generalizations to dependent data, to...

For many probability laws, in parametric models, the estimation of the parameters can be done in the frame of the maximum likelihood method, or in the frame of moment estimation methods, or by using the plug-in method, etc. Usually, for estimating more than one parameter, the same frame is used. We focus on the moment estimation method in this pape...

For many probability laws, in parametric models, the estimation of the parameters can be done in the frame of the maximum likelihood method, or in the frame of moment estimation methods, or by using the plug-in method, etc. Usually, for estimating more than one parameter, the same frame is used. We focus on the moment estimation method in this pape...

A. A uniformly continuously integrable sequence of real-valued measurable functions , defined on some probability space, is relatively compact in the σ(L 1 , L ∞) topology. In this paper, we link such a result to weak convergence theory of bounded measures as exposed in Billingsley (1968) and in Lo(2021) to offer a detailed and new proof using the...

This paper investigates the probability density function (\textit{pdf}) of the $(2n-1)$-vector ($n\geq 1$) of both lower and upper record values for a sequence of independent random variables with common \textit{pdf} $f$ defined on the same probability space, provided that the lower and upper record times are finite up to $n$. A lot is known about...

A. This paper is part of series on self-contained papers in which a large part, if not the full extent, of the asymptotic limit theory of summands of independent random variables is exposed. Each paper of the series may be taken as review exposition but specially as a complete exposition expect a few exterior resources. For graduate students and fo...

This paper investigates the probability density function ($pdf$) of the $(2n-1)$-vector $(n\geq 1)$ of both lower and upper record values for a sequence of independent random variables with common $pdf f$ defined on the same probability space, provided that the lower and upper record times are finite up to $n$. A lot is known about the lower or the...

General Central limit theorem deals with weak limits (in type) of sums of row-elements of array random variables. In some situations as in the invariance principle problem, the sums may include only parts of the row-elements. For strictly stationary arrays (stationary for each row), there is no change to the asymptotic results. But for non-stationa...

This paper is part of series on self-contained papers in which a large part, if not the full extent, of the asymptotic limit theory of summands of independent random variables is exposed. Each paper of the series may be taken as review exposition but specially as a complete exposition expect a few exterior resources. For graduate students and for r...

A uniformly continuously integrable sequence of real-valued measurable functions, defined on some probability space, is relatively compact in the $\sigma(L^1,L^\infty)$ topology. In this paper, we link such a result to weak convergence theory of bounded measures as exposed in Billingsley (1968) and in Lo(2021) to offer a detailed and new proof usin...

In this monograph, our final objective is to provide second order expansions of quantile functions of as many probability laws as possible. Second order expansions of quantile functions are important tools for finding extreme value domain of attraction of probability laws and for discovering rates of convergence in extreme value theory. We hope tha...

According to the Chinese Health Statistics Yearbook, in 2005, the number of traffic accidents was 187781 with total direct property losses of 103691.7 (10000 Yuan). This research aims to fill the gap in the literature by investigating the extreme claim sizes not only for the entire portfolio. This empirical study investigates the behavior of the up...

A method of estimating the joint probability mass function of a triplet of discrete random variables is described. This estimator is used to construct the joint‐conditional entropies and mutual information estimates involving three random variables. From there almost sure rates of convergence and asymptotic normality are established. The theorical...

In this paper we investigate the Burr distributions Family which contains twelve members. Second order expansions of quantiles of the Burr's distributions are provided on which may be based statistical methods, in particular in extreme value theory. Beyond the proper interest of these expansions, we apply them to characterize the asymptotic laws of...

In this paper we investigate the Burr distributions Family which contains twelve members. Second order expansions of quantiles of the Burr's distributions are provided on which may be based statistical methods, in particular in extreme value theory. Beyond the proper interest of these expansions, we apply them to characterize the asymptotic laws of...

Asymptotic laws of record values have usually been investigated as limits in type. In this paper, we use functional representations of the tail of cumulative distribution functions in the extreme value domain of attraction to directly establish asymptotic laws of record values, not necessarily as limits in type and their rates of convergences. Resu...

In this paper we want to find a statistical rule that assigns a passing or failing grade to students who undertook at least three exams out of four in a national exam, instead of completely dismissing them students. While it is cruel to declare them as failing, especially if the reason for their absence it not intentional, they should have demonstr...

As the first paper of a series of exploratory analysis and statistical investigation works on the Gambian \textit{GABECE} data based on a variety of statistical tools, we wish to begin with a thorough unsupervised learning process through descriptive and exploratory methods. This will lead to a variety of discoveries and hypotheses that will direct...

This book introduces to the theory of probabilities from the beginning. Assuming that the reader possesses the normal mathematical level acquired at the end of the secondary school, we aim to equip him with a solid basis in probability theory. The theory is preceded by a general chapter on counting methods. Then, the theory of probabilities is pres...

In this paper we want to find a statistical rule that assigns a passing or failing grade to students who undertook at least three exams out of four in a national exam, instead of completely dismissing them students. While it is cruel to declare them as failing, especially if the reason for their absence it not intentional, they should have demonstr...

In this note, we combine the two approaches of Billingsley (1998) and Cs\H{o}rg\H{o} and R\'ev\'esz (1980), to provide a detailed sequential and descriptive for creating s standard Brownian motion, from a Brownian motion whose time space is the class of non-negative dyadic numbers. By adding the proof of Etemadi's inequality to text, it becomes sel...

Asymptotic theories on record values and times, including central limit theorems, make sense only if the sequence of records values (and of record times) is infinite. If not, such theories could not even be an option. In this paper, we give necessary and/or sufficient conditions for the finiteness of the number of records. We prove, for example for...

The moment problem is an important problem in Functional Analysis and in Probability measure. It goes back to Stieltjes, around 1890. There is still an important ongoing interest in the recent literature. But, up today, the main theoretical resource (Shohat and Tamarkin, 1934) does not have the modern exposure it deserves, especially in the current...

The representation Skorohod theorem of weak convergence of random variables on a metric space goes back to Skorohod (1956) in the case where the metric space is the class of real-valued functions defined on [0,1] which are right-continuous and have left-hand limits when endowed with the Skorohod metric. Among the extensions of that to metric spaces...

Integrating with respect to functions which are constant on intervals whose bounds are discontinuity points (of those functions) is frequent in many branches of Mathematics, specially in stochastic processes. For such functions and alike extension, a comparison between Riemann-Stieltjes and Lebesgue-Stieltjes integration and the integrals formulas...

In this work a new nonparametric estimator of joint probability mass function of a pair of discrete random variables is described. This estimator is used to construct conditional Shannon-R\'eyni-Stallis entropies estimates. Almost sure rates of convergence and asymptotic normality results are established. Our theorical results are validated by simu...

This paper proposes a new method for estimating the joint probability mass function of a pair of discrete random variables. This estimator is used to construct joint Shannon Rényi-Tsallis entropies, and the mutual information estimates of a pair of discrete random variables. Almost sure consistency and central limit Theorems are established. Our th...

The pseudo-Lindley distribution which was introduced in Zeghdoudi and Nedjar (2016) is studied with regards to its upper tail. In that regard, and when the underlying distribution function follows the Pseudo-Lindley law, we investigate the behavior of its values, the asymptotic normality of the Hill estimator and the double-indexed generalized Hill...

The pseudo-Lindley distribution which was introduced in Zeghdoudi and Nedjar (2016) is studied with regards to its upper tail. In that regard, and when the underlying distribution function follows the Pseudo-Lindley law, we investigate the behavior of its values, the asymptotic normality of the Hill estimator and the double-indexed generalized Hill...

The pseudo-Lindley distribution was introduced as a useful generalization of the Lindley distribution in Zeghdoudi and Nedjar (2016) who showed interesting properties of their new laws and efficiencies in modeling data in Reliability and Survival Analysis. In this paper we study the estimators of the pair of parameters and determine their asymptoti...

The probabilistic investigation on record values and record times of a sequence of random variables defined on the same probability space has received much attention from 1952 to now. A great deal of such theory focused on \textit{iid} or independent real-valued random variables. There exists a few results for real-valued dependent random variables...

The pseudo-Lindley distribution was introduced as a useful generalization of the Lindley distribution in Zeghdoudi and Nedjar (2016) who showed interesting properties of their new law, in particular its efficiency in modeling data in Reliability and Survival Analysis. In this paper we study the estimators of the pair of parameters and determine the...

Asymptotic theories on record values and times, including central limit theorems, make sense only if the sequence of records values (and of record times) is infinite. If not, such theories could not even be an option. In this paper, we give necessary and/or sufficient conditions for the finiteness of the number of records. We prove, for example for...

On one hand, a large class of inequality measures, which includes the generalized entropy, the Atkinson, the Gini, etc., for example, has been introduced in P.D. Mergane, G.S. Lo, Appl. Math. 4 (2013), 986–1000. On the other hand, the influence function (IF) of statistics is an important tool in the asymptotics of a nonparametric statistic. This fu...

In this paper we provide the asymptotic theory of the general of φ-divergences measures, which include the most common divergence measures : R´enyi and Tsallis families and the Kullback-Leibler measure. We are interested in divergence measures in the discrete case. One sided and two-sided statistical tests are derived as well as symmetrized estimat...

Using a general strong law of large number proved by Sangare and Lo in \cite% {sanglo} and the entropy numbers, we provide a functional Glivenko-Cantelli theorem for arbitrary random variables (rv's).

This paper proposes a new method for estimating the joint probability mass function of a pair of discrete random variables. This estimator is used to construct joint entropy and Shannon mutual information estimates of a pair of discrete random variables. Almost sure consistency and central limit Theorems are established. Theorical results are valid...

The like-Lebesgue integral of real-valued measurable functions (abbreviated as \textit{RVM-MI})is the most complete and appropriate integration Theory. Integrals are also defined in abstract spaces since Pettis (1938). In particular, Bochner integrals received much interest with very recent researches. It is very commode to use the \textit{RVM-MI}...

When applying the classical Stone-Weierstrass common version in Probability Theory for example, and in other fields as well, problems may arise if all points of the compact set are not separated. A solution may consist in going back to the proof and finding alternative versions. In this note, we did it and come back with two flexible versions which...

Asymptotic laws of records values have usually been investigated as limits in type. In this paper, we use functional representations of the tail of cumulative distribution functions in the extreme value domain of attraction to directly establish asymptotic laws of records value, not necessarily as limits in type. Results beyond the extreme value va...

We present some new nonparametric estimators of entropies and we establish almost sure consistency and central limit Theorems for some of the most important entropies in the discrete case. Our theorical results are validated by simulations.

In the two previous papers of this series, the main results on the asymptotic behaviors of empirical divergence measures based on wavelets theory have been established and particularized for important families of divergence measures like Rényi and Tsallis families and for the Kullback-Leibler measures. While the proofs of the results in the second...

In this paper we provide the asymptotic theory of the general of $\phi$-divergences measures, which include the most common divergence measures : Renyi and Tsallis families and the Kullback-Leibler measure. We are interested in divergence measures in the discrete case. One sided and two-sided statistical tests are derived as well as symmetrized est...

In this paper, we consider a coherent theory about the asymptotic representations for a family of inequality indices called Theil-Like Inequality Measures (TLIM), within a Gaussian field. The theory uses the functional empirical process approach. We provide the finite-distribution and uniform asymptotic normality of the element of the TLIM class in...

In the spirit of recent asymptotic works on the General Poverty Index (GPI) in the field of Welfare Analysis, the asymptotic statistical representation of the non-decomposable Takayama’s index, which has failed to be incorporated in the unified GPI approach, is addressed and established here. This representation also allows to extend to it, recent...

The aim of this paper is to establish the asymptotic behavior of the mutual influence of the Gini index and the poverty measures by using the Gaussian fields described in Lo and Mergane (2013). (See the full abstract in next page). Full Abstract. The aim of this paper is to establish the asymptotic behavior of the mutual influence of the Gini index...

Left and right-continuous functions play an important role in Real analysis, especially in Measure Theory and Integration on the real line and in Stochastic processes indexed by a continuous real time. Semi-continuous functions are also of major interest in the same way. This paper aims at presenting a useful handling of semi-continuous function in...

The new Sine Skewed Cardioid (ssc) distribution been just introduced and characterized by Ahsanullah (2018). Here, we study the asymptotic properties of its tails by determining its extreme value domain, the characteristic function, the moments and likelihood estimators of the two parameters, the asymptotic normality of the moments estimators and t...

Un certain nombre de mesures d'inégalité ont été développées afin d'apprécier l'inégalité dans le revenu et la pauvreté. De telles mesures incluent l'entropie généralisée, la mesure d'Atkinson, l'indice de Gini et le rapport des quintiles. Récemment, une nouvelle mesure, à savoir l'indice de Zenga, a été introduite et s'est montrée plus appropriée...

The univariate extreme value theory deals with the convergence in type of powers of elements of sequences of cumulative distribution functions on the real line when the power index gets infinite. In terms of convergence of random variables, this amounts to the the weak convergence, in the sense of probability measures weak convergence, of the parti...

In this paper, we consider a coherent theory about the asymptotic rep- resentations for a family of inequality indices called Theil-Like Inequality Measures (TLIM), within a Gaussian field. The theory uses the functional empirical process approach. We provide the finite-distribution and uniform asymptotic normality of the element of the TLIM class...

In the footsteps of the book \textit{Measure Theory and Integration By and For the Learner} of our series in Probability Theory and Statistics, we intended to devote a special volume of the very probabilistic aspects of the first cited theory. The book might have assigned the title : From Measure Theory and Integration to Probability Theory. The fu...

On one hand, a large class of inequality measures, which includes the generalized entropy, the Atkinson, the Gini, etc., for example, has been introduced in Mergane and Lo (2013). On the other hand, the influence function of statistics is an important tool in the asymptotics of a nonparametric statistic. This function has been and is being determin...

In this paper, we consider a coherent theory about the asymptotic representations for a family of inequality indices called Theil-Like Inequality Measures (TLIM), within a Gaussian field. The theory uses the functional empirical process approach. We provide the finite-distribution and uniform asymptotic normality of the elements of the TLIM class i...

We deal with the asymptotic normality theory of empirical divergences measures based on wavelets in a series of three papers. In this first paper, we provide the asymptotic theory of the general of phi-divergences measures, which includes the most common divergence measures : Renyi and Tsallis families and the Kullback-Leibler measures. Instead of...

In this research monograph, we deal with a very general asymptotic representation for statistics named GRI expressed in the functional empirical process, both one-dimensional and multidimensional, and another call residual empirical process. Most of statistics in form of combination of L-statistics are covered by the asymptotic theory dealt here. T...

In this paper, we consider the Zenga index, one of the most recent inequality index. We keep the finite-valued original form and address the asymptotic theory. The asymptotic normality is established through a multinomial representation. The Influence function is also given. Th results are simulated and applied to Senegalese data.

In this note we provide a quick proof of the Sklar's Theorem on the existence of copulas by using the generalized inverse functions as in the one dimensional case, but a little more sophisticated.

In this research monograph, we deal with a very general asymptotic representation for statistics named GRI expressed in the functional empirical process, both one-dimensional and multidimensional, and another call residual empirical process. Most of statistics in form of combination of L-statistics are covered by the asymptotic theory dealt here. T...

Abstract. We briefly introduce to this special issue of Afrika Statistika devoted to selectedPapers presented at the First East African Conference of Statistical Mathematics with Ap-plications (EACSMA-2017). Résumé (French). Ce numéro spécial de Afrika Statistika est consacré à la publication d’articles présentés à la Conférence Est Africaine de S...