arXiv subjects

Gane Samb Lo

Publications and source records attributed to Gane Samb Lo.

88 records · Page 5Linked to original sources

arXiv2014

The weak limiting behavior of the de Haan-Resnick estimator of the exponent of a stable distribution

The problem of estimating the exponent of a stable law received a considerable attention in the recent literature. Here, we deal with an estimate of such a exponent introduced by De Haan and Resnick when the corresponding distribution function belongs to the Gumbel's domain of attraction. This study permits to construct new statistical tests. Examples and simulations are given. The limiting law are shown to be the Gumbel's law and particular cases are given with norming constants expressed with iterated logarithms and exponentials.

stat.ME↗

arXiv2014

A simple note on some empirical stochastic process as a tool in uniform L-statistics weak laws

In this paper, we are concerned with the stochastic process \begin{equation} β_{n}(q_{t},t)=β_{n}(t)=\frac{1}{\sqrt{n}}\sum_{j=1}^{n}\left\{G_{t,n}(Y(t))-G_{t}(Y_{j}(t))\right\} q_{t}(Y_{j}(t)), \tag{A} \end{equation} where for $n\geq1$ and $T>0$, the sequences $\{Y_{1}(t),Y_{2}(t),...,Y_{n}(t),t\in [0,T]\}$ are independant observations of some real stochastic process ${Y(t),t\in [0,T]}$, for each $t \in [0,T]$, $G_{t}$ is the distribution function of $% Y(t)$ and $G_{t,n}$ is the empirical distribution function based on $% Y_{1}(t),Y_{2}(t),...,Y_{n}(t)$, and finally $q_{t}$ is a bounded real fonction defined on $\mathbb{R}$. This process appears when investigating some time-dependent L-Statistics which are expressed as a function of some functional empirical process and the process (A). Since the functional empirical process is widely investigated in the literature, the process reveals itself as an important key for L-Statistics laws. In this paper, we state an extended study of this process, give complete calculations of the first moments, the covariance function and find conditions for asymptotic tightness.

stat.ME↗

arXiv2014

Asymptotic Representation Theorems for Poverty Indices

We set general conditions under which the general poverty index, which summarizes all the available indices, is asymptotically represented with some empirical processes. This representation theorem offers a general key, in most directions, for the asymptotics of the bulk of poverty indices and issues in poverty analysis. Our representation results uniformly hold on a large collection of poverty indices. They enable the continuous measure of poverty with longitudinal data.

stat.ME↗

arXiv2014

Another look at Second order condition in Extreme Value Theory

This note compares two approaches both alternatively used when establishing normality theorems in univariate Extreme Value Theory. When the underlying distribution function ($df$) is the extremal domain of attraction, it is possible to use representations for the quantile function and regularity conditions (RC), based on these representations, under which strong and weak convergence are valid. It is also possible to use the now fashion second order condition (SOC), whenever it holds, to do the same. Some authors usually favor the first approach (the SOC one) while others are fond of the second approach that we denote as the representational one. This note aims at comparing the two approaches and show how to get from one to the other. The auxiliary functions used in each approach are computed and compared. Statistical applications using simultaneously both approaches are provided. A final comparison is provided.

math.PR↗

arXiv2014

Functional weak laws for the weighted mean losses or gains and applications

We show in this paper that many risk measures arising in Actuarial Sciences, Finance, Medicine, Welfare analysis, etc. are garthered in classes of Weighted Mean Loss or Gain (WMLG) statistics. Some of them are Upper Threshold Based (UTH) or Lower Threshold Based (LTH). These statistics may be time-dependent when the scene is monitored in the time and depend on specific functions $w$ and $d$. This paper provides time-dependent and uniformly functional weak asymptotic laws that allow temporal and spatial studies of the risk as well as comparison between statistics in terms of dependence and mutual influence. The results are particularised for usual statistics of that kind such that the Kakwani and Shorrocks ones. Datadriven applications based on pseudo-panel data are provided.

stat.ME↗

arXiv2014

A note on a new exponential bound for M-acceptable random variables

We present a new exponential inequality as a generalization of that of Sung \textit{et al.} \cite{sun2011} for $M$-acceptable random variables, and hence for extended negative ones. Our result is based on the simple real inequality $e^{x} \leq 1+x+(|x|/2)e^{|x|}, x\in\mathbb{R}$, in place of the following one: $e^{x} \leq 1+x+(x^{2}/2)e^{|x|}, x\in\mathbb{R}$, used by many authors. We compare the given bound with former ones.

math.PR↗

arXiv2012

Besov regularity of the uniform empirical process

The paths of Brownian motion have been widely studied in the recent years relatively in Besov spaces $B_{p, \infty}^\a$. The results are the same as to the Brownian bridge. In fact these regularities properties are established in some sequence spaces $S_{p, \infty}^\a$ using an isomorphisim between them and $B_{p, \infty}^\a$. In this note, we are concerned with the regularity of the paths of the continuous version of the uniform empirical process in the space $S_{p, \infty}^\a$ and in one of his separable sub space $S_{p, \infty}^{\a, 0}$ for a suitable choice of $\a$ and $p$.

math.PR↗

arXiv2012

The General Poverty Index

We introduce the General Poverty Index (GPI), which summarizes most of the known and available poverty indices, in the form {equation*} GPI=δ(\frac{A(Q_{n},n,Z)}{nB(Q,n)}\overset{Q_{n}}{\underset{j=1}{\sum}%}w(μ_{1}n+μ_{2}Q_{n}-μ_{3}j+μ_{4})d(\frac{Z-Y_{j,n}}{Z}%)),{equation*} where {equation*} B(Q_{n},n)=\sum_{j=1}^{Q}w(j), {equation*} $A(\cdot),$ $w(\cdot),and$ $d(\cdot) $\ are given measurable functions, $Q_{n}$ is the number of the poor in the sample, Z is the poverty line and $Y_{1,n}\leq Y_{2,n}\leq ...\leq Y_{n,n}$\ are the ordered sampled incomes or expenditures of the individuals or households. We show here how the available indices based on the poverty gaps are derived from it. The asymptotic normality is then established and particularized for the usual poverty measures for immediate applications to poor countries data.

stat.ME↗

arXiv2012

On a discrete Hill's statistical process based on sum-product statistics and its finite-dimensional asymptotic theory

The following class of sum-product statistics T_n(p)=\frac{1}{k}\sum_{h=1}^p \sum_{(s_1...s_h)\in P(p,h)} \sum_{i_1=l+1}^{i_0} ... \sum_{i_h=l+1}^{i_{h-1}} i_h \prod_{i=i_1}^{i_h} \frac{(Y_{n-i+1,n}-Y_{n-i,n})^{s_i}}{s_i!} (where $l,$ $k=i_{0}$ and n are positive integers, $0 0$ into $\ h$ positive integers and $Y_{1,n}\leq ...\leq Y_{n,n}$ are the order statistics based on a sequence of independent random variables $Y_{1},$ $Y_{2},...$with underlying distribution $\mathbb{P}(Y\leq y)=G(Y)=F(e^{y})$), is introduced. For each p, $T_{n}(p)^{-1/p}$ is an estimator of the index of a distribution whose upper tail varies regularly at infinity. \ This family generalizes the so called Hill statistic and the Dekkers-Einmahl-De Haan one. We study the limiting laws of the process ${T_{n}(p),1\leq p<\infty}$ and completely describe the covariance function of the Gaussian limiting process with the help of combinatorial techniques. Many results available for Hill's statistic regarding asymptotic normality and laws of the iterated logarithm are extended to each margin $T_{n}(p,k)$, for $p$ fixed, and for any distribution function lying in the extremal domain. In the process, we obtain special classes of numbers related to those of paths joining the opposite coins within a parallelogram.

stat.ME↗

arXiv2012

On Hàjek - Rényi type inequality and application

In this paper, we establish a new Hájek - Rényi's type inequality and obtain strong law of large numbers (SLLN) for arbitrary random variables. We base our methods on demimartingales and convex functions techniques. We obtain wide extensions of avalaible results, particularly those of Amini et al. and Rao and related SLLN's.

math.PR↗

arXiv2012

How many statistics are needed to characterize the univariate extremes

Let $X_{1},X_{2},...$ be a sequence of independent random variables ($rv$) with common distribution function ($df$) $F$ such that $F(1)=0$. We consider the simple statistical problem : find a statistics family of size $m\geq 1$ whose convergence, in probability or almost surely, to a point of some domain $\mathcal{S} \in \mathbb{R}^{m}$ is equivalent that $F$ lies in the extremal domain of attraction $Γ$. Such a family, whenever it exists, is called an Empirical Characterizing Statistics Family for the EXTtremes (ECSFEXT). The departure point of this theory goes back to Mason, who proved that the Hill estimator converges a.s. to a positive real number for some particular sequences if and only $F$ lies in the attaction domain of a Fréchet's law. Considered for the whole attraction domain, the question becomes more complex. We provide here an ECSFEXT of nine (9) elements and also characterize the subdomains of $Γ$. The question of lowering m=9 to a minimum number is launched.

stat.ME↗

arXiv2012

Statistical estimation of gap of decomposability of the general poverty index

For the decomposability property is very a practical one in Welfare analysis, most researchers and users favor decomposable poverty indices such as the Foster-Greer-Thorbeck poverty index. This may lead to neglect the so important weighted indices like the Kakwani and Shorrocks ones which have interesting other properties in Welfare analysis. To face up to this problem, we give in this paper, statistical estimations of the gap of decomposability of a large class of such indices using the General Poverty Indice (GPI) and of a new asymptotic representation Theorem for it, in terms of functional empirical processes theory. The results then enable independent handling of targeted groups and next global reporting with significant confidence intervals. Data-driven examples are given with real data.

stat.ME↗

arXiv2011

On general strong laws of large numbers for fields of random variables

A general method to prove strong laws of large numbers for random fields is given. It is based on the Hájek - Rényi type method presented in Noszály and Tómács \cite{noszaly} and in Tómács and Líbor \cite{thomas06}. Noszály and Tómács \cite{noszaly} obtained a Hájek-Rényi type maximal inequality for random fields using moments inequalities. Recently, Tómács and Líbor \cite{thomas06} obtained a Hájek-Rényi type maximal inequality for random sequences based on probabilities, but not for random fields. In this paper we present a Hájek-Rényi type maximal inequality for random fields, using probabilities, which is an extension of the main results of Noszály and Tómács \cite{noszaly} by replacing moments by probabilities and a generalization of the main results of Tómács and Líbor \cite% {thomas06} for random sequences to random fields. We apply our results to establishing a logarithmically weighted sums without moment assumptions and under general dependence conditions for random fields.

math.PR↗

arXiv2011

On the Generalized Hill Process for Small Parameters and Applications

Let $X_{1},X_{2},...$ be a sequence of independent copies (s.i.c) of a real random variable (r.v.) $X\geq 1$, with distribution function $df$ $F(x)=\mathbb{P}% (X\leq x)$ and let $X_{1,n}\leq X_{2,n} \leq ... \leq X_{n,n}$ be the order statistics based on the $n\geq 1$ first of these observations. The following continuous generalized Hill process {equation*} T_{n}(τ)=k^{-τ}\sum_{j=1}^{j=k}j^τ(\log X_{n-j+1,n}-\log X_{n-j,n}), \label{dl02} {equation*} $τ>0$, $1\leq k \leq n$, has been introduced as a continuous family of estimators of the extreme value index, and largely studied for statistical purposes with asymptotic normality results restricted to $τ> 1/2$. We extend those results to $0 < τ\leq 1/2$ and show that asymptotic normality is still valid for $τ=1/2$. For $0 < τ<1/2$, we get non Gaussian asymptotic laws which are closely related to the Riemann function $% ζ(s)=\sum_{n=1}^{\infty} n^{-s},s>1$

stat.ME↗

arXiv2011

On the Pickands stochastic process

We consider the Pickands process {equation*} P_{n}(s)=\log (1/s)^{-1}\log \frac{X_{n-k+1,n}-X_{n-[k/s]+1,n}}{% X_{n-[k/s]+1,n}-X_{n-[k/s^{2}]+1,n}}, {equation*} {equation*} (\frac{k}{n}\leq s^2 \leq 1), {equation*} which is a generalization of the classical Pickands estimate $P_{n}(1/2)$ of the extremal index. We undertake here a purely stochastic process view for the asymptotic theory of that process by using the Csörgő-Csörgő-Horváth-Mason (1986) \cite{cchm} weighted approximation of the empirical and quantile processes to suitable Brownian bridges. This leads to the uniform convergence of the margins of this process to the extremal index and a complete theory of weak convergence of $P_n$ in $\ell^{\infty}([a,b])$ to some Gaussian process $$\{\mathbb{G},a\leq s \leq b\} $$ for all $[a,b] \subset]0,1[$. This frame greatly simplifies the former results and enable applications based on stochastic processes methods.

stat.ME↗

arXiv2011

A functional Generalized Hill process and applications

We are concerned in this paper with the functional asymptotic behaviour of the sequence of stochastic processes T_{n}(f)=\sum_{j=1}^{j=k}f(j)(\log X_{n-j+1,n}-\log X_{n-j,n}), indexed by some classes $\mathcal{F}$ of functions $f:\mathbb{N} \backslash {0} \longmapsto \mathbb{R}_{+}$ and where $k=k(n)$ satisfies 1\leq k\leq n,k/n\rightarrow 0\text{as}n\rightarrow \infty. This is a functional generalized Hill process including as many new estimators of the extremal index when $F$ is in the extremal domain. We focus in this paper on its functional and uniform asymptotic law in the new setting of weak convergence in the space of bounded real functions. The results are next particularized for explicit examples of classes $\mathcal{F} $.

stat.ME↗