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, $0<l<k<n,$ $P(p,h)$ is the set of all ordered parititions of $\ p>0$ 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.Comment: 22 page

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.Comment: 23 page

A simple proof of the theorem of Sklar and its extension to distribution functions

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.Comment:

How to use the functional empirical process for deriving asymptotic laws for functions of the sample

The functional empirical process is a very powerful tool for deriving asymptotic laws for almost any kind of statistics whenever we know how to express them into functions of the sample. Since this method seems to be applied more and more in the very recent future, this paper is intended to provide a complete but short description and justification of the method and to illustrate it with a non-trivial example using bivariate data. It may also serve for citation whithout repeating the arguments.Comment: 11 page

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.Comment: 17 page

A remark on the asymptotic tightness in ℓ∞([a,b])\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.Comment: 4 page

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. Cs\"{o}rg\H{o} and D.M. Mason (1985).Comment: 1

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} \beta_{n}(q_{t},t)=\beta_{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 $$ 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.Comment: 11 pag

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=\delta (\frac{A(Q_{n},n,Z)}{nB(Q,n)}\overset{Q_{n}}{\underset{j=1}{\sum}%}w(\mu_{1}n+\mu_{2}Q_{n}-\mu_{3}j+\mu_{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.Comment: 17 page

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 fileComment: 42 pages, 3 figure