Searcharxiv⌕ Search

arXiv subjects

Gane Samb Lo

Publications and source records attributed to Gane Samb Lo.

At least 55 records · Page 3Linked to original sources
arXiv2019

Divergence measures estimation and its asymptotic normality theory in the discrete case

In this paper we provide the asymptotic theory of the general of $ϕ$-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 estimators. Almost sure rates of convergence and asymptotic normality theorem are obtained in the general case, and next particularized for the Renyi and Tsallis families and for the Kullback-Leibler measure as well. Our theorical results are validated by simulations.

math.ST↗
arXiv2019

Entropies and their Asymptotic Theory in the discrete case

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.

math.ST↗
arXiv2018

On some properties of the new Sine-skewed Cardioid Distribution

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 the random generation of data from the \textit{ssc} distribution. Finally, we proceed to a simulation study to show the performance of the random generation method and the quality of the moments estimation of the parameters.

stat.OT↗
arXiv2018

Weak Convergence (IIA) - Functional and Random Aspects of the Univariate Extreme Value Theory

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 partial maximas of a sequence of independent and identically distributed random variables. In this monograph, this theory is comprehensively studied in the broad frame of weak convergence of random vectors as exposed in Lo et al.(2016). It has two main parts. The first is devoted to its nice mathematical foundation. Most of the materials of this part is taken from the most essential Loève(1936,177) and Haan (1970), based on the stunning theory of regular, pi or gamma variation. To prepare the statistical applications, a number contributions I made in my PhD and my Doctorate of Sciences are added in the last chapter of the last chapter of that part. Our real concern is to put these materials together with others, among them those of the authors from his PhD dissertations and Science doctorate thesis, in a way to have an almost full coverage of the theory on the real line that may serve as a master course of one semester in our universities. As well, it will help the second part of the monograph. This second part will deal with statistical estimations problems related to extreme values. It addresses various estimation questions and should be considered as the beginning of a survey study to be updated progressively. Research questions are tackled therein. Many results of the author, either unpublished or not sufficiently known, are stated and/or updated therein.

math.PR↗
arXiv2018

Weak Convergence (IA). Sequences of Random Vectors

(English) 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↗
arXiv2018

Mathematical Foundations of Probability Theory

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 fundamental aspects of Probability Theory, as described by the keywords and phrases below, are presented, not from experiences as in the book \textit{A Course on Elementary Probability Theory}, but from a pure mathematical view based on Measure Theory. Such an approach places Probability Theory in its natural frame of Functional Analysis and constitutes a firm preparation to the study of Random Analysis and Stochastic processes. At the same time, it offers a solid basis towards Mathematical Statistics Theory. The book will be continuously updated and improved on a yearly basis.

math.PR↗
arXiv2018

On the influence function for the Theil-like class of inequality measures

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 determined and analysed in various aspects for a large number of statistics. We proceed to a unifying study of the IF of all the members of the so-called Theil-like family and regroup those IF's in one formula. Comparative studies become easier.

stat.ME↗
arXiv2018

A Theil-like Class of Inequality Measures, its Asymptotic Normality Theory and Applications

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 in a unified approach rather than in a case by case one. The results are then applied to some UEMOA countries databases.

stat.ME↗
arXiv2018

Asymptotic Representations of Statistics in the Functional Empirical process : A portal and some applications

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. This treatise is conceived to be a kind of \textbf{spaceship} on which modules are hanged. The spaceship is a functional Gaussian process and each module is the asymptotic representation of one statistic in terms of that Gaussian process. In that way, it is possible to navigate from one module to another, that is, to find the joint distribution of any pair of statistics, to compare them with respect to the areas and the times. In order to be able to do so, we should have a broad conception at the beginning. Within the constructed frame, the asymptotic joint law of any finite number of other statistics is automatically given as well as the joint distribution of its spatial variation or temporal variation, in absolute or relative values. We also deal with the general problem of decomposability of statistics by comparing statistical decomposability, a new view we introduce, versus functional decomposability. A general result only based on the GRI is provided. \noindent This monograph is also the portal of a handbook of GRI that will cover the largest number possible of statistics. In prevision of that, we treat three important examples as show cases. It is expected that this portal and the handbook will attract the attention of researchers working in the asymptotic area and will furnish useful tools to scientists who are interested in application of asymptotic tests, completed by computer packages.

stat.ME↗
arXiv2017

Measuring inequality: application of semi-parametric methods to real life data

A number of methods have been introduced in order to measure the inequality in various situations such as income and expenditure. In order to curry out statistical inference, one often needs to estimate the available measures of inequality. Many estimators are available in the literature, the most used ones being the non parametric estimators. kpanzou(2011) has developed semi-parametric estimators for measures of inequality and showed that these are very appropriate especially for heavy tailed distributions. In this paper we apply such semi-parametric methods to a practical data set and show how they compare to the non parametric estimators. A guidance is also given on the choice of parametric distributions to fit in the tails of the data

stat.ME↗
arXiv2017

VB and R codes using Households databases available in the NSI's : A prelude to statistical applied studies

We describe the main features of the households databases we can find in most of our National Statistics Institute. We provide algorithms aimed at extracting a diversity of variables on which different statistical procedures may be applied. Here, we particularly focus on the scaled income, as a beginning. Associated codes (MS Visual Basic and R codes) have been successfully tested and delivered in the text and in a separate file

cs.CY↗
arXiv2017

Uniform weak convergence of poverty measures with relative poverty lines

This paper introduces a general continuous form of poverty index that encompasses most of the existing formulas in the literature. We then propose a consistent estimator for this index in case the poverty line is a functional of the distribution. We also establish a uniform functional Central Limit Theorem for the proposed estimator over a suitable product class of real-valued functions. As a consequence, testing procedures based either on single or simultaneously several poverty indices can be developed. A simulation study showing the asymptotic normality of the estimator is given as well as an application to real data for estimating the effect of relative poverty lines on the variance of the poverty estimates.

stat.AP↗
arXiv2017

On the joint distribution of variations of the Gini index and Welfare indices

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 Mergane and Lo(2013). The results are given as representation theorems using the Gaussian fields of the unidimensional or the bidimensional functional Brownian bridges. Such representations, when combined with those already available, lead to joint asymptotic distributions with other statistics of interest like growth, welfare and inequality indices and then, unveil interesting results related to the mutual influence between them. The results are also appropriate for studying whether a growth is fair or not, depending on the variation of the inequality measure. Datadriven applications are also available. Although the variances may seem complicated at a first sight, their computations which are needed to get confidence intervals of the indices, are possible with the help of R codes we provide. Beyond the current results, the provided representations are useful in connection with different ones of other statistics.

stat.ME↗
arXiv2017

Divergence Measures Estimation and Its Asymptotic Normality Theory Using Wavelets Empirical Processes

In this paper we provide the asymptotic theory of the general of $ϕ$-divergences measures, which includes the most common divergence measures : Renyi and Tsallis families and the Kullback-Leibler measure. Instead of using the Parzen nonparametric estimators of the probability density functions whose discrepancy is estimated, we use the wavelets approach and the geometry of Besov spaces. One-sided and two-sided statistical tests are derived as well as symmetrized estimators. Almost sure rates of convergence and asymptotic normality theorem are obtained in the general case, and next particularized for the Renyi and Tsallis families and for the Kullback-Leibler measure as well. The applicability of the results to usual distribution functions is addressed.

stat.ME↗
arXiv2017

Asymptotic Theory and Statistical Decomposability gap Estimation for Takayama's Index

In the spirit of recent asymptotic works on the General Poverty Index (GPI) in the field of Welfare Analysis, the asymptotic 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 allows also to extend to it, recent results of statistical decomposability gaps estimations. The theoretical results are applied to real databases. The conclusions of the undertaken applications recommend to use Takayama's index as a practically decomposable one, in virtue of the low decomposability gaps with respect to the large values of the index.

stat.ME↗