We address the estimation of "extreme" conditional quantiles i.e. when their
order converges to one as the sample size increases. Conditions on the rate of
convergence of their order to one are provided to obtain asymptotically
Gaussian distributed kernel estimators. A Weissman-type estimator and kernel
estimators of the conditional tail-index are derived, permitting to estimate
extreme conditional quantiles of arbitrary order.Comment: arXiv admin note: text overlap with arXiv:1107.226

A Berlinet

BM Hill

F Caeiro

F Ferraty

I Gijbels

I Weissman

J. Pickands

J. Segers

JHJ Einmahl

L Gardes

L Haan de

MI Gomes

T Samanta

W Stute

English

arXiv

Laurent Gardes

Stèphane Girard

Crossref

Functional Kernel Estimators of Conditional Extreme Quantiles

National audienceWe address the estimation of conditional quantiles when the covariate is functional and when the order of the quantiles converges to one as the sample size increases. In a first time, we investigate to what extent these large conditional quantiles can still be estimated through a functional kernel estimator of the conditional survival function. Sufficient conditions on the rate of convergence of their order to one are provided to obtain asymptotically Gaussian distributed estimators. In a second time, basing on these result, a functional Weissman estimator is derived, permitting to estimate large conditional quantiles of arbitrary large order. These results are illustrated on finite sample situations

Girard, Stéphane

Gardes, Laurent

Hal - Université Grenoble Alpes

Functional kernel estimators of conditional extreme quantiles

We address the estimation of “extreme” conditional quantiles i.e. when their order converges to one as the sample size increases. Conditions on the rate of convergence of their order to one are provided to obtain asymptotically Gaussian distributed kernel estimators. A Weissman-type estimator and kernel estimators of the conditional tail-index are derived, permitting to estimate extreme conditional quantiles of arbitrary order

Stéphane Girard

CiteSeerX

International audienceWe address the estimation of ''extreme'' conditional quantiles i.e. when their order converges to one as the sample size increases. Conditions on the rate of convergence of their order to one are provided to obtain asymptotically Gaussian distributed kernel estimators. A Weissman-type estimator and kernel estimators of the conditional tail-index are derived, permitting to estimate extreme conditional quantiles of arbitrary order

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HAL Id: hal-00764317https://hal.archives-ouvertes.fr/hal-00764317v2Submitted on 5 Dec 2012HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestinée au dépôt et à la diffusion de documentsscientifiques de niveau recherche, publiés ou non,émanant des établissements d’enseignement et derecherche français ou étrangers, des laboratoirespublics ou privés.Functional kernel estimators of conditional extremequantilesLaurent Gardes, Stephane GirardTo cite this version:Laurent Gardes, Stephane Girard. Functional kernel estimators of conditional extreme quantiles.Frédéric Ferraty. Recent advances in functional data analysis and related topics, Springer, Physica-Verlag, pp.135-140, 2011, Contributions to Statistics, 978-3-7908-2735-4. ￿10.1007/978-3-7908-2736-1_21￿. ￿hal-00764317v2￿Second International Workshop onFunctional and Operatorial Statistics.Santander, June 16-18, 2011Functional kernel estimatorsof conditional extreme quantilesLaurent GARDES∗∗ and Stéphane GIRARD∗Team Mistis, INRIA Rhône-Alpes and LJK, Inovallée, 655, av. de l’Europe,Montbonnot, 38334 Saint-Ismier cedex, France.∗ Stephane.Girard@inrialpes.frAbstractWe address the estimation of “extreme” conditional quantiles i.e. when their orderconverges to one as the sample size increases. Conditions on the rate of convergenceof their order to one are provided to obtain asymptotically Gaussian distributed kernelestimators. A Weissman-type estimator and kernel estimators of the conditional tail-index are derived, permitting to estimate extreme conditional quantiles of arbitrary order.1. IntroductionLet (Xi, Yi), i = 1, . . . , n be independent copies of a random pair (X,Y ) in E × Rwhere E is a metric space associated to a distance d. We address the problem of esti-mating q(αn|x) ∈ R verifying P(Y > q(αn|x)|X = x) = αn where αn → 0 as n → ∞ andx ∈ E. In such a case, q(αn|x) is referred to as an extreme conditional quantile in con-trast to classical conditional quantiles (known as regression quantiles) for which αn = αis fixed in (0, 1). While the nonparametric estimation of ordinary regression quantileshas been extensively studied, see for instance the seminal papers (Roussas, 1969), (Stone,1977) or (Ferraty and Vieu, 2006, Chapter 5) less attention has been paid to extreme con-ditional quantiles despite their potential interest. Here, we focus on the setting where theconditional distribution of Y given X = x has an infinite endpoint and is heavy-tailed,an analytical characterization of this property being given in the next section. We show,under mild conditions, that extreme conditional quantiles q(αn|x) can still be estimatedthrough a functional kernel estimator of P(Y > .|x). We provide sufficient conditionson the rate of convergence of αn to 0 so that our estimator is asymptotically Gaussiandistributed. Making use of this, some functional kernel estimators of the conditionaltail-index are introduced and a Weissman type estimator (Weissman, 1978) is derived,permitting to estimate extreme conditional quantiles q(βn|x) where βn → 0 arbitrarilyfast.2. Notations and assumptionsThe conditional survival function (csf) of Y given X = x is denoted by F̄ (y|x) =P(Y > y|X = x). The kernel estimator of F̄ (y|x) is defined for all (x, y) ∈ E × R byˆ̄Fn(y|x) =n∑i=1K(d(x,Xi)/h)Q((Yi − y)/λ)/n∑i=1K(d(x,Xi)/h), (1)with Q(t) =∫ t−∞q(s)ds where K : R+ → R+ and q : R → R+ are two kernel functions,and h = hn and λ = λn are two nonrandom sequences such that h → 0 as n → ∞. In thiscontext, h and λ are called window-width. This estimator was considered for instancein (Ferraty and Vieu, 2006, page 56). In Theorem 1, the asymptotic distribution of (1)is established when estimating small tail probabilities, i.e when y = yn goes to infinitywith the sample size n. Similarly, the kernel estimators of conditional quantiles q(α|x)are defined via the generalized inverse of ˆ̄Fn(.|x):q̂n(α|x) = inf{t,ˆ̄Fn(t|x) ≤ α}, (2)for all α ∈ (0, 1). Many authors are interested in this type of estimator for fixed α ∈ (0, 1):weak and strong consistency are proved respectively in (Stone, 1977) and (Gannoun,1990), asymptotic normality being established when E is finite dimensional by (Stute,1986), (Samanta, 1989), (Berlinet et al., 2001) and by (Ferraty et al., 2005) when E isa general metric space. In Theorem 2, the asymptotic distribution of (2) is investigatedwhen estimating extreme quantiles, i.e when α = αn goes to 0 as the sample size n goesto infinity. The asymptotic behavior of such estimators depends on the nature of theconditional distribution tail. In this paper, we focus on heavy tails. More specifically,we assume that the csf satisfies(A1): F̄ (y|x) = c(x) exp{−∫ y1(1γ(x)− ε(u|x))duu},where γ(.) is a positive function of the covariate x, c(.) is a positive function and |ε(.|x)| iscontinuous and ultimately decreasing to 0. (A1) implies that the conditional distributionof Y given X = x is in the Fréchet maximum domain of attraction. In this context,γ(x) is referred to as the conditional tail-index since it tunes the tail heaviness of theconditional distribution of Y given X = x. Assumption (A1) also yields that F̄ (.|x) isregularly varying at infinity with index −1/γ(x). i.e for all ζ > 0,limy→∞F̄ (ζy|x)F̄ (y|x)= ζ−1/γ(x). (3)The function ε(.|x) plays an important role in extreme-value theory since it drives thespeed of convergence in (3) and more generally the bias of extreme-value estimators.Therefore, it may be of interest to specify how it converges to 0. In (Gomes et al.,2000), the auxiliary function is supposed to be regularly varying and the estimation ofthe corresponding regular variation index is addressed. Some Lipschitz conditions arealso required:(A2): There exist κε, κc, κγ > 0 and u0 > 1 such that for all (x, x′) ∈ E2 and u > u0,|log c(x)− log c(x′)| ≤ κcd(x, x′),|ε(u|x)− ε(u|x′)| ≤ κεd(x, x′),|1/γ(x)− 1/γ(x′)| ≤ κγd(x, x′).The last assumptions are standard in the kernel estimation framework.(A3): K is a function with support [0, 1] and there exist 0 < C1 < C2 < ∞ such thatC1 ≤ K(t) ≤ C2 for all t ∈ [0, 1].(A4): q is a probability density function (pdf) with support [−1, 1].One may also assume without loss of generality that K integrates to one. In this case,K is called a type I kernel, see (Ferraty and Vieu, 2006, Definition 4.1). Finally, letB(x, h) be the ball of center x and radius h. The small ball probability of X is definedby ϕx(h) = P(X ∈ B(x, h)). Under (A3), for all τ > 0, the τth moment is defined byµ(τ)x (h) = E{Kτ (d(x,X)/h)}.3. Main resultsLet us first focus on the estimation of small tail probabilities F̄ (yn|x) when yn → ∞as n → ∞. The following result provides sufficient conditions for the asymptotic normal-ity of ˆ̄Fn(yn|x).Theorem 1. Suppose (A1) – (A4) hold. Let x ∈ E such that ϕx(h) > 0 and introduceyn,j = ajyn for j = 1, . . . , J with 0 < a1 < a2 < . . . < aJ and where J is a positiveinteger. If yn → ∞ such that nϕx(h)F̄ (yn|x) → ∞, nϕx(h)F̄ (yn|x)(λ/yn)2 → 0 andnϕx(h)F̄ (yn|x)(h log yn)2 → 0 as n → ∞, then{√nµ(1)x (h)F̄ (yn|x)(ˆ̄Fn(yn,j |x)F̄ (yn,j |x)− 1)}j=1,...,Jis asymptotically Gaussian, centered, with covariance matrix C(x) where Cj,j′ (x) =a1/γ(x)j∧j′ for (j, j′) ∈ {1, . . . , J}2.Note that nϕx(h)F̄ (yn|x) → ∞ is a necessary and sufficient condition for the almost surepresence of at least one sample point in the region B(x, h) × (yn,∞) of E × R. Thus,this natural condition states that one cannot estimate small tail probabilities out of thesample using ˆ̄Fn. This result may be compared to (Einmahl, 1990) which establishesthe asymptotic behavior of the empirical survival function in the unconditional case butwithout assumption on the distribution. Letting σn(x) = (nµ(1)x (h)αn)−1/2, the asymp-totic normality of q̂n(αn|x) when αn → 0 as n → ∞ can be established under similarconditions.Theorem 2. Suppose (A1) – (A4) hold. Let x ∈ E such that ϕx(h) > 0 and introduceαn,j = τjαn for j = 1, . . . , J with τ1 > τ2 > . . . > τJ > 0 and where J is a positiveinteger. If αn → 0 such that σn(x) → 0, σ−1n (x)λ/q(αn|x) → 0 and σ−1n (x)h logαn → 0as n → ∞, then{σ−1n (x)(q̂n(αn,j |x)q(αn,j |x)− 1)}j=1,...,Jis asymptotically Gaussian, centered, with covariance matrix γ2(x)Σ where Σj,j′ = 1/τj∧j′for (j, j′) ∈ {1, . . . , J}2.The functional kernel estimator of extreme quantiles q̂n(αn|x) requires a stringent condi-tion on the order αn of the quantile, since by construction it cannot extrapolate beyondthe maximum observation in the ball B(x, h). To overcome this limitation, a Weissmantype estimator (Weissman, 1978) can be derived:q̂Wn (βn|x) = q̂n(αn|x)(αn/βn)γ̂n(x).Here, q̂n(αn|x) is the functional kernel estimator of the extreme quantile considered sofar and γ̂n(x) is a functional estimator of the conditional tail-index γ(x). As illustratedin the next theorem, the extrapolation factor (αn/βn)γ̂n(x) allows to estimate extremequantiles of order βn arbitrary small.Theorem 3. Suppose (A1)–(A4) hold. Let us introduce• αn → 0 such that σn(x) → 0, σ−1n (x)λ/yn → 0 and σ−1n (x)h logαn → 0 as n → ∞,• (βn) such that βn/αn → 0 as n → ∞,• γ̂n(x) such that σ−1n (x)(γ̂n(x)− γ(x))d−→ N (0, v2(x)) where v2(x) > 0.Then, for all x ∈ E,σ−1n (x)log(αn/βn)(q̂Wn (βn|x)q(βn|x)− 1)d−→ N (0, v2(x)).Note that, when K is the pdf of the uniform distribution, this result is consistent with(Gardes et al., 2010, Theorem 3), obtained in a fixed-design setting.Let us now give some examples of functional estimators of the conditional tail-index. Letαn → 0 and τ1 > τ2 > . . . > τJ > 0 where J is a positive integer. Two additionarynotations are introduced for the sake of simplicity: u = (1, . . . , 1)t ∈ RJ and v =(log(1/τ1), . . . , log(1/τJ))t ∈ RJ . The following family of estimators is proposedγ̂n(x) =ϕ(log q̂n(τ1αn|x), . . . , log q̂n(τJαn|x))ϕ(log(1/τ1), . . . , log(1/τJ)),where ϕ : RJ → R denotes a twice differentiable function verifying the shift and locationinvariance conditions ϕ(θv) = θϕ(v) for all θ > 0 and ϕ(ηu+x) = ϕ(x) for all η ∈ R andx ∈ RJ . For instance, introducing the auxiliary function mp(x1, . . . , xJ ) =∑Jj=1(xj −x1)p for all p > 0 and considering ϕH(x) = m1(x) gives rise to a kernel version of the Hillestimator (Hill, 1975):γ̂Hn(x) =J∑j=1[log q̂n(τjαn|x)− log q̂n(αn|x)]/J∑j=1log(1/τj) .Generalizations of the kernel Hill estimator can be obtained with ϕ(x) = mp(x)/mp−11 (x),see (Gomes and Martins, 2001, equation (2.2)), ϕ(x) = m1/pp (x), see e.g. (Segers, 2001,example (a)) or ϕ(x) = m1/θpθ (x)/mp−1(x), p ≥ 1, θ > 0, see (Caeiro and Gomes, 2002).In the case where J = 3, τ1 = 4, τ2 = 2 and τ3 = 1, the functionϕP(x1, x2, x3) = log(expx2 − expx1expx3 − expx2)leads us to a kernel version of Pickands estimator (Pickands, 1975)γ̂Pn(x) =1log 2log(q̂n(αn|x)− q̂n(2αn|x)q̂n(2αn|x)− q̂n(4αn|x)).We refer to (Gijbels and Peng, 2000) for a different variant of Pickands estimator in thecontext where the distribution of Y given X = x has a finite endpoint. The asymptoticnormality of γ̂n(x) is a consequence of Theorem 2.Theorem 4. Under assumptions of Theorem 2 and if σ−1n (x)ε(q(τ1αn|x)|x) → 0 asn → ∞, then, σ−1n (x)(γ̂n(x) − γ(x)) converges to a centered Gaussian random variablewith varianceV (x) =γ2(x)ϕ2(v)(∇ϕ(γ(x)v))tΣ(∇ϕ(γ(x)v)).As an illustration, in the case of the kernel Hill and Pickands estimators, we obtainVH(x) = γ2(x)J∑j=12(J − j) + 1τj− J2/J∑j=1log(1/τj)2.VP(x) =γ2(x)(22γ(x)+1 + 1)4(log 2)2(2γ(x) − 1)2.Clearly, VP(x) is the variance of the classical Pickands estimator, see for instance (deHaan and Ferreira, 2006, Theorem 3.3.5). Focusing on the kernel Hill estimator andchoosing τj = 1/j for each j = 1, . . . , J yields VH(x) = γ2(x)J(J−1)(2J−1)/(6 log2(J !)).In this case, VH(x) is a convex function of J and is minimum for J = 9 leading toVH(x) ≃ 1.25γ2(x).ReferencesA. Berlinet, A. Gannoun and E. Matzner-Løber. Asymptotic normality of convergentestimates of conditional quantiles. Statistics, 35:139–169, 2001.F. Caeiro and M.I. Gomes. Bias reduction in the estimation of parameters of rare events.Theory of Stochastic Processes, 8:67–76, 2002.J.H.J Einmahl. The empirical distribution function as a tail estimator, Statistica Neer-landica, 44:79–82, 1990.F. Ferraty and P. Vieu. Nonparametric functional data analysis. Springer, 2006.F. Ferraty, A. Rabhi and P. Vieu. Conditional quantiles for dependent functional datawith application to the climatic El Nino Phenomenon, Sankhya: The Indian Journal ofStatistics, 67(2):378–398, 2005.A. Gannoun. Estimation non paramétrique de la médiane conditionnelle, médianogrammeet méthode du noyau, Publications de l’Institut de Statistique de l’Université de Paris,XXXXVI:11–22, 1990.L. Gardes, S. Girard and A. Lekina. Functional nonparametric estimation of conditionalextreme quantiles. Journal of Multivariate Analysis, 101:419–433, 2010.I. Gijbels and L. Peng. Estimation of a support curve via order statistics. Extremes,3:251–277, 2000.M.I. Gomes and M.J. Martins and M. Neves. Semi-parametric estimation of the secondorder parameter, asymptotic and finite sample behaviour. Extremes, 3:207–229, 2000.M.I. Gomes and M.J. Martins. Generalizations of the Hill estimator - asymptotic versusfinite sample behaviour. Journal of Statistical Planning and Inference, 93:161–180, 2001.L. de Haan and A. Ferreira. Extreme Value Theory: An Introduction, Springer Series inOperations Research and Financial Engineering, Springer, 2006.B.M. Hill. A simple general approach to inference about the tail of a distribution. TheAnnals of Statistics, 3:1163–1174, 1975.J. Pickands. Statistical inference using extreme order statistics. The Annals of Statistics,3:119–131, 1975.G.G. Roussas. Nonparametric estimation of the transition distribution function of aMarkov process. Ann. Math. Statist., 40:1386–1400, 1969.T. Samanta. Non-parametric estimation of conditional quantiles. Statistics and Proba-bility Letters, 7:407–412, 1989.J. Segers. Residual estimators. Journal of Statistical Planning and Inference, 98:15–27,2001.C.J. Stone. Consistent nonparametric regression (with discussion). The Annals ofStatistics, 5:595–645, 1977.W. Stute. Conditional empirical processes. The Annals of Statistics, 14:638–647, 1986.I. Weissman. Estimation of parameters and large quantiles based on the k largest obser-vations, Journal of the American Statistical Association, 73:812–815, 1978.

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Functional kernel estimators of conditional extreme quantiles

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