On the weak convergence of kernel density estimators in Lp spaces

Since its introduction, the pointwise asymptotic properties of the kernel estimator of a probability density function in finite dimension, as well as the asymptotic behaviour of its integrated errors, have been studied in great detail. Its weak convergence in functional spaces, however, is a more difficult problem. In this paper, we show that any Borel measurable weak limit of the (properly centred and rescaled) kernel density estimator must be 0. We also provide simple conditions for proving or disproving the existence of this Borel measurable weak limit

Estimation of the conditional tail index using a smoothed local Hill estimator

International audienceFor heavy-tailed distributions, the so-called tail index is an important parameter that controls the behavior of the tail distribution and is thus of primary interest to estimate extreme quantiles. In this paper, the estimation of the tail index is considered in the presence of a finite-dimensional random covariate. Uniform weak consistency and asymptotic normality of the proposed estimator are established and some illustrations on simulations are provided

On a class of norms generated by nonnegative integrable distributions

We show that any distribution function on $\mathbb{R}^d$ with nonnegative, nonzero and integrable marginal distributions can be characterized by a norm on $\mathbb{R}^{d+1}$, called $F$-norm. We characterize the set of $F$-norms and prove that pointwise convergence of a sequence of $F$-norms to an $F$-norm is equivalent to convergence of the pertaining distribution functions in the Wasserstein metric. On the statistical side, an $F$-norm can easily be estimated by an empirical $F$-norm, whose consistency and weak convergence we establish. The concept of $F$-norms can be extended to arbitrary random vectors under suitable integrability conditions fulfilled by, for instance, normal distributions. The set of $F$-norms is endowed with a semigroup operation which, in this context, corresponds to ordinary convolution of the underlying distributions. Limiting results such as the central limit theorem can then be formulated in terms of pointwise convergence of products of $F$-norms. We conclude by showing how, using the geometry of $F$-norms, we may characterize nonnegative integrable distributions in $\mathbb{R}^d$ by simple compact sets in $\mathbb{R}^{d+1}$. We then relate convergence of those distributions in the Wasserstein metric to convergence of these characteristic sets with respect to Hausdorff distances

Asymptotic behaviour of extreme geometric quantiles and their estimation under moment conditions

A popular way to study the tail of a distribution is to consider its extreme quantiles. While this is a standard procedure for univariate distributions, it is harder for multivariate ones, primarily because there is no universally accepted definition of what a multivariate quantile should be. In this paper, we focus on extreme geometric quantiles. Their asymptotics are established, both in direction and magnitude, under suitable moment conditions, when the norm of the associated index vector tends to one. In particular, it appears that if a random vector has a finite covariance matrix, then the magnitude of its extreme geometric quantiles grows at a fixed rate. We take advantage of these results to define an estimator of extreme geometric quantiles of such a random vector. The consistency and asymptotic normality of the estimator are established and our results are illustrated on some numerical examples

Value-at-Risk estimation: A novel GARCH-EVT approach dealing with bias and heteroscedasticity

Open House, ISM in National Center of Sciences Building, 2019.6.05統計数理研究所オープンハウス（学術総合センター）、R1.6.5ポスター発

Estimating the conditional tail index with an integrated conditional log-quantile estimator in the random covariate case

It is well known that the tail behavior of a heavy-tailed distribution is controlled by a parameter called the tail index. Such a parameter is therefore of primary interest in extreme value analysis, particularly to estimate extreme quantiles. In various applications, the random variable of interest can be linked to a finite-dimensional random covariate. In such a situation, the tail index is function of the covariate and is referred to as the conditional tail index. The goal of this paper is to provide a class of estimators of this quantity. The pointwise weak consistency and asymptotic normality of these estimators are established. We illustrate the finite sample performance of our technique on a simulation study and on a real hurricane data set

Extreme geometric quantiles in a multivariate regular variation framework

International audienceConsidering extreme quantiles is a popular way to understand the tail of a distribution. While they have been extensively studied for univariate distributions, much less has been done for multivariate ones, primarily because there is no universally accepted definition of what a multivariate quantile or a multivariate distribution tail should be. In this paper, we focus on extreme geometric quantiles. In Girard and Stupfler (2014) "Intriguing properties of extreme geometric quantiles", their asymptotics are established, both in direction and magnitude, under suitable integrability conditions, when the norm of the associated index vector tends to one. In this paper, we study extreme geometric quantiles when the integrability conditions are not fulfilled, in a framework of regular variation

On a relationship between randomly and non-randomly thresholded empirical average excesses for heavy tails

Motivated by theoretical similarities between the classical Hill estimator of the tail index of a heavy-tailed distribution and one of its pseudo-estimator versions featuring a non-random threshold, we show a novel asymptotic representation of a class of empirical average excesses above a high random threshold, expressed in terms of order statistics, using their counterparts based on a suitable non-random threshold, which are sums of independent and identically distributed random variables. As a consequence, the analysis of the joint convergence of such empirical average excesses essentially boils down to a combination of Lyapunov's central limit theorem and the Cramér-Wold device. We illustrate how this allows to improve upon, as well as produce conceptually simpler proofs of, very recent results about the joint convergence of marginal Hill esti-mators for a random vector with heavy-tailed marginal distributions. These results are then applied to the proof of a convergence result for a tail index estimator when the heavy-tailed variable of interest is randomly right-truncated. New results on the joint convergence of conditional tail moment estimators of a random vector with heavy-tailed marginal distributions are also obtained

ExpectHill estimation, extreme risk and heavy tails

Risk measures of a financial position are traditionally based on quantiles. Replacing quantiles with their least squares analogues, called expectiles, has recently received increasing attention. The novel expectile-based risk measures satisfy all coherence requirements. We revisit their extreme value estimation for heavy-tailed distributions. First, we estimate the underlying tail index via weighted combinations of top order statistics and asymmetric least squares estimates. The resulting expectHill estimators are then used as the basis for estimating tail expectiles and Expected Shortfall. The asymptotic theory of the proposed estimators is provided, along with numerical simulations and applications to actuarial and financial data

Tail expectile process and risk assessment

Expectiles define a least squares analogue of quantiles. They are determined by tail expectations rather than tail probabilities. For this reason and many other theoretical and practical merits, expectiles have recently received a lot of attention, especially in actuarial and financial risk management. Their estimation, however, typically requires to consider non-explicit asymmetric least squares estimates rather than the traditional order statistics used for quantile estimation. This makes the study of the tail expectile process a lot harder than that of the standard tail quantile process. Under the challenging model of heavy-tailed distributions, we derive joint weighted Gaussian approximations of the tail empirical expectile and quantile processes. We then use this powerful result to introduce and study new estimators of extreme expectiles and the standard quantile-based expected shortfall, as well as a novel expectile-based form of expected shortfall. Our estimators are built on general weighted combinations of both top order statistics and asymmetric least squares estimates. Some numerical simulations and applications to actuarial and financial data are provided