A Double-indexed Functional Hill Process and Applications

Abstract

<div>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</div><div> </div><div>\begin{equation}<br />T_{n}(f,s)=\sum_{j=1}^{j=k}f(j)\left( \log X_{n-j+1,n}-\log<br />X_{n-j,n}\right)^{s} , \label{fme}<br />\end{equation}</div><div> </div><div>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</div><div> </div><div>\begin{equation*}<br />1\leq k\leq n,k/n\rightarrow 0\text{ as }n\rightarrow \infty .<br />\end{equation*}</div><div> </div><div>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 $\mathcal{F}$.</div>

Extremes, Extremal Index Estimation, Records, Moment Problem for the Pseudo-Lindley Distribution and Applications

“…But that one could not hold if S n (f, s) contains a sub-sequence converging to a finite and positive number. That remark should be recalled in interpreting the results in [12].…”

“…is the most celebrated estimator of the extreme value index γ = 1/θ of Z = exp(X). Among a significant number of generalizations of the Hill's estimator, the Ngom-Lo generalization ( [12], 2016), called the functional Double-indexed Hill estimator, is one the sharpest one. It is defined as…”

“…In [12], we gave a direct proof of the asymptotic normality of S n (f, s) by using the two hypotheses B n (f, s) → 0 and s n (f, s) → +∞. Here, it seems that we only used the first one.…”

Return Levels Approach and Periods of Currency Crises

“…Many studies extended the Hill estimator to more general case ξ ∈ R such that adapted Hill estimator, moment estimator and the QQ-estimator ( (Embrechts et al 1999), (Gencay & Selcuk 2004) and (Marimoutou et al2009)). Some other popular and remarkable estimators are the Moments estimators of (Dekkers et al 1989), the Generalized Hill estimator of (Diop & Lo, 2006, (Lo & Dème, 2012) and recently the (Ngom & LO, 2016).…”

The Pseudo-Lindley Alpha Power transformed distribution, mathematical characterizations and asymptotic properties

“…The extremal domain of attraction of PL-APT is found and its quantile and extremal quantile functions studied. Finally, the extremal value index is estimated by the double-indexed Hill's estimator (Ngom and Lo, 2016) and related asymptotic statistical tests are provided and characterized.…”