Estimation of multivariate critical layers: Applications to rainfall data

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Estimation of multivariate critical layers: Applications to rainfall data

Elena Di Bernardino, Didier Rullière

To cite this version:

Elena Di Bernardino, Didier Rullière. Estimation of multivariate critical layers: Applications to rainfall data. Journal de la Societe Française de Statistique, Societe Française de Statistique et Societe Mathematique de France, 2015, 156 (1), pp. 11-50. �hal-00940089v3�

Estimation of multivariate critical layers:

Applications to rainfall data

Elena Di Bernardino

and Didier Rullière

1Conservatoire National des Arts et Métiers, Département IMATH, EA4629, Paris Cedex 03, France.

2Université de Lyon, Université Lyon 1, ISFA, Laboratoire SAF, EA2429, 69366 Lyon, France.

December 9, 2014

Abstract

Calculating return periods and critical layers (i.e. multivariate quantile curves) in a multivariate envi- ronment is a difficult problem. A possible consistent theoretical framework for the calculation of the return period, in a multi-dimensional environment, is essentially based on the notion of copula and level sets of the multivariate probability distribution. In this paper we propose a fast and parametric methodology to esti- mate the multivariate critical layers of a distribution and its associated return periods. The model is based on transformations of the marginal distributions and transformations of the dependence structure within the class of Archimedean copulas. The model has a tunable number of parameters, and we show that it is possible to get a competitive estimation without any global optimum research. We also get parametric ex- pressions for the critical layers and return periods. The methodology is illustrated on rainfall 5-dimensional real data. On this real data-set we obtain a good quality of estimation and we compare the obtained results with some classical parametric competitors. Finally we provide a simulation study.

keywords: Multivariate probability transformations level sets estimation copulas hyperbolic conversion func- tions risk assessment multivariate return periods.

1 Introduction

1.1 Return Periods

The notion ofReturn Period (RP) is frequently used in environmental sciences for the identification of danger- ous events, and provides a means for rational decision making and risk assessment. Roughly speaking, the RP can be considered as an analogue of the “Value-at-Risk” in Economics and Finance, since it is used to quantify and assess the risk (see, e.g., Nappo and Spizzichino (2009)). In engineering practice, finance, insurance and environmental science the choice of the RP depends on the impact/magnitude of the considered event and the consequences of its realisation.

Equally important is the related concept of design quantile, usually defined as “the value of the variable char- acterizing the event associated with a given RP”. In the univariate case the design quantile is usually identified without ambiguity. Conversely in the multivariate setting different definitions are possible (seeSerfling(2002)).

For this reason, the identification problem of design events in a multivariate context has recently attracted the attention of many researches. The interested reader is referred for example to Embrechts and Puccetti (2006),Belzunce et al.(2007),Nappo and Spizzichino(2009), in the economics and finance context; toChebana and Ouarda(2009),Chebana and Ouarda(2011) (and references therein) in the hydrological context.

During the last years, researchers in environmental fields joined efforts to properly answer the following cru- cial question: “How is it possible to calculate the critical design event(s) in the multivariate case?” (see for instanceSalvadori et al.(2007)). In this sense, a possible consistent theoretical framework for the calculation of the design event(s) and the associated return period(s) in a multi-dimensional environment, is proposed, e.g., by Salvadori et al. (2011), Salvadori et al. (2012), Gräler et al. (2013). In particular the authors define the multivariate return period using the notion of upper and lower level sets of multivariate probability distribution

∗elena.di_bernardino@cnam.fr

†didier.rulliere@univ-lyon1.fr

F and of the associated Kendall’s measure.

In the following, we will consider a sequence X = {X1,X2, . . .} of independent and identically distributed d−dimensional random vectors, with d > 1. Thus each Xk, k ∈ N, has the same multivariate distribution FX : R^d+→[0,1] as the nonnegative real-valued random vector X ∼ FX = C(FX₁, . . . , FX_d) describing the hydrological phenomenon under investigation. The functionCis thed-dimensionalcopula associated toF (see Nelsen (1999)). We write I = {1, . . . , d} the set of indexes of the considered random variables and of their associated cumulative distribution functions, i.e.,FX_i(xi) =P(Xi≤xi), fori∈I.

In the following we will consider multivariate distribution functionsF_X satisfying theseregularity conditions - for allu∈[0,1], the diagonal of the copulaC, i.e. C(u, . . . , u), is a strictly increasing function ofu;

- for anyi∈I, the marginal FX_i is continuous and strictly monotonic distribution function.

In this setting, we introduce the notion of critical layer (see, e.g., Salvadori et al. (2011), Salvadori et al.

(2012),Gräler et al.(2013)).

Definition 1.1 (Critical layer). The critical layer ∂L(α) associated to the multivariate distribution function FXof level α∈(0,1) is defined as

∂L(α) ={x∈R^d:FX(x) =α}.

Then∂L(α)is the iso-hyper-surface (with dimensiond−1) whereF equals the constant value α. Thus,∂L(α) is a (iso)line for bivariate distributions, a (iso)surface for trivariate ones, and so on. The critical layer ∂L(α) partitionsR^d into three non-overlapping and exhaustive regions:







L^<(α) = {x∈R^d:FX(x)< α},

∂L(α) = the critical layer itself, L^>(α) = {x∈R^d:FX(x)> α}.

Practically, at any occurrence of the hydrological considered phenomenon, only three mutually exclusive events may happen: either a realization of the considered hydrological event lies in one of these 3 Borel setsL^<(α),

∂L(α), or L^>(α).

In the applications, usually, the event of interest is of the type {X ∈ A}, whereA is a non-empty Borel set in R^d collecting all the values judged to be “dangerous” according to some suitable criterion. A natural choice forAis the setL^>(α)(seeSalvadori et al. (2011), Gräler et al.(2013)). The first random indexN where Xk

reaches the set L^>(α) isN = mink∈N{Xk ∈L^>(α)}. AssumingP[X∈L^>(α)]∈(0,1), one easily shows that N is a geometric random variable. The Return Period is defined as the average time required for reaching the setL^>(α), that is:

RP^>(α) = ∆t·E[N] = ∆t

P[X∈L^>(α)], (1) where∆t>0is the (deterministic) average time elapsing betweenXkandXk+1,k∈N. The probability that a realization of this vector belongs toL^<(α)is given by the Kendall’s function, which only depends on the copula C of this random vector, i.e.,

KC(α) =P

X ∈L^<(α)

=P[C(U1, . . . , Ud)≤α], forα∈(0,1). (2) Then, the considered Return Period can be expressed using Kendall’s function in (2),RP^>(α) = ∆t·_1−K¹

C(α). Obviously, Return Periods can naturally be associated to other sets thanL^>(α), the interested reader is referred for example toSalvadori et al.(2011).

This paper aims at:

• giving a parametric representation of the multivariate distribution F of a random vector X, here repre- senting rain measurements (for applications see Section6),

• giving direct estimation procedure for this representation,

• giving closed parametric expressions, both for critical layers in Definition 1.1and Return Periods in (1),

• adapting this methodology to some asymmetric dependencies (as, for instance, non-exchangeable random vectors; for a possible investigation in this sense see Section 7).

In the next section, we introduce the model used to answer the issues introduced above.

1.2 The model

We consider the following model, which is detailed inDi Bernardino and Rullière(2013a),

F(xe ₁, . . . , x_d) =T◦C₀(T₁⁻¹◦F₁(x₁), . . . , T_d⁻¹◦F_d(x_d)), (3) or equivalently







F(xe 1, . . . , xd) = C(e Fe1(x1), . . . ,Fed(xd)), with C(ue 1, . . . , ud) = T ◦C0(T⁻¹(u1), . . . , T⁻¹(u1)) Fei(x) = T ◦T_i⁻¹◦Fi(x), for i∈I,

(4)

where F1, . . . , Fd are given parametric initial marginal cumulative distribution functions, and where C0 is a given initial copula. Hence the distribution Fe(x1, . . . , xd) is built from transformed marginals Fei, i ∈ I and from a transformed copula C, under regularity conditions. Transformatione T permits to transform the initial dependence structure C0. For a given T, transformations Ti permit to transform marginals, i ∈I. All these transformations are described hereafter.

As we will see in the following, the initial copulaC0 in (3) is not estimated but it is chosen at the beginning of the estimation procedure. So it can represent some kind of a priori belief on dependence structure of the data or on the considered problem, that will be transformed in order to improve the fit.

Furthermore, as inDi Bernardino and Rullière(2013b), we will assume in the following thatC₀is an Archimedean copula. This means that in this paper, we mainly consider copulas that can be written as

Cφ(u1, . . . , ud) =φ(φ⁻¹(u1) +. . .+φ⁻¹(ud)),

where the functionφis called the generator of the Archimedean copulaCφ. Some conditions like d−monotony are given in McNeil and Nešlehová (2009). Here we choose strict generator, i.e., φ(t) > 0,∀t ≥ 0 and

t→+∞lim φ(t) = 0, with proper inverse φ⁻¹ such thatφ◦φ⁻¹(t) =t.

The function T : [0,1] → [0,1] is a continuous and increasing function on the interval [0,1], with T(0) = 0, T(1) = 1, with supplementary assumptions that will be chosen to guarantee that Ce is also a copula (detailed hereafter). Internal transformations T_i : [0,1] → [0,1] are continuous non-decreasing functions, such that T_i(0) = 0, T_i(1) = 1, fori∈I. Conditions on transformations such thatCeis a copula are discussed for example inDurante et al.(2010),Di Bernardino and Rullière(2013a), Di Bernardino and Rullière(2013b).

Remark that among problems generated by transformations of Archimedean copulas, one can point out in par- ticular the problem of uniqueness. Transformations of a given initial copula leading to a given target copula are not unique. This raises some problems for the analysis of the convergence of estimators of the transforma- tion. This also causes problems to compare transformations and to understand their impact on the dependence structure. A further analysis shows that also a generator of an Archimedean copula is not unique, causing the same kind of problems. Then in Di Bernardino and Rullière (2013b), the definition of equivalence classes for both transformations and generators is provided to select some standardized forms for practical use, for the comparison and the interpretation of obtained distribution functions. Firstly equivalent classes for transforma- tions can be characterized. Furthermore one can ensures the uniqueness of the transformation T by passing through the point (x₀, y₀), among the invariant class for transformations (see Lemma 2.2 and Corollary 2.1 of the aforementioned paper). Obviously, in an iterative procedure of estimation, the uniqueness of the transfor- mationTis essential in order to permit the convergence of the procedure and the identifiability of the considered transformation model.

In the following we detail the proposed semi-parametric estimation procedure in order to easily fit the model in (3). We show in particular how to estimate the transformationsT andTi,i∈I.

Organization of the paper

The paper is organized as follows. In Section2we developed the estimation procedure for the chosen model in (3).

In Section2.1we focus on a central tool in our estimation procedure: the estimation of the diagonal section of a copula. This estimated diagonal section is used in the non-parametric estimation of the external transformation T and the internal transformationsTi(see Section2.2). The parametric estimation using composite hyperbolic transformations is detailed in Section 3. Then in Section 4 we propose the parametric form for the desired

quantities: the multivariate distribution function F, its critical layers ∂L(α) and the associated Kendall’s function KC(α), for α∈ (0,1). Finally, Section 6 is devoted to a detailed study of a5−dimensional rainfall data-set. Furthermore, a nested model is proposed for this5−dimensional data-set, using the whole estimation procedure presented in the paper (see Section7). Finally we provide a simulation study in Section8. Directions for future research and the conclusion are in Section9.

2 Nonparametric estimation

2.1 Initial diagonal section estimation

In order to give a non-parametric estimation of the external transformationT, we will propose a non-parametric estimator of the copula within the Archimedean class of copulas. Several non-parametric estimators of the generator of an Archimedean copula are available. One can cite for example those of Dimitrova et al.(2008) orGenest et al. (2011) both based on empirical Kendall’s function. Here, proposed estimations will rely on an initial estimation of the diagonal section of the empirical copula. Firstly, we propose some (classical) estimators of the diagonal sectionδ1 of a copula,

δ1(u) =C(u, . . . , u), u∈[0,1].

Remark that if (U₁, U₂, . . . , U_d) has as distribution function the copulaC then

P[U1≤u, . . . , Ud≤u] =C(u, . . . , u) =P[max{U1, U2, . . . , Ud} ≤u] =δ1(u).

As a consequence, estimators based on the diagonal section of a copula are relying on the distribution of the maximum ofU1, . . . , Ud, as explained inSungur and Yang(1996). As pointed out inDi Bernardino and Rullière (2013b), the diagonal of an Archimedean copula, under some suitable conditions, is essential to describe the copula. So, in the following we recall important assumptions (which are fulfilled for many Archimedean copulas, including the independent copula) for the unique determination of an Archimedean copulas starting from its diagonal section (see, for instance,Erdely et al.(2014) and references therein). Some constructions of copulas starting from thediagonal section are given for example inNelsen et al.(2008) andWysocki(2012).

Proposition 2.1(Identity of Archimedean copulas, Theorem 3.5 byErdely et al.(2014)). LetCad−dimensional Archimedean copula whose diagonal sectionδ₁ satisfiesδ₁⁰(1⁻) =d. ThenC is uniquely determined by its diag- onal.

Condition in Proposition2.1is referred to asFrank’s condition in Erdely et al.(2014) (see their Theorems 1.2 and 3.5). Note that if|φ⁰(0)|<+∞then the condition on the diagonal in Proposition2.1is automatically sat- isfied. In this case, the functionφcan be reconstructed from the diagonalδ(see alsoSegers(2011)). As pointed out byEmbrechts and Hofert(2011) a possible limitation is that ifφhas finite right-hand derivative at zero, the Archimedean copula generated byφhas upper tail independent structure. To show that the situation of many Archimedean copulas having the same diagonal is far from exceptional, a recipe to construct further examples is given inSegers(2011). For further details the interested reader is referred to Section 3 inDi Bernardino and Rullière(2013b).

It has been shown however that if one aims at fitting both lower and upper tail dependence, then some specific dependence structures can be suited to this purpose, see Di Bernardino and Rullière(2014). Here we will not focus on very extreme tail behavior, but on the main part of the distribution. Using the necessary few data in the tails would require some specific estimators of tail dependence coefficients, as detailed in above mentioned article.

In the following we present different estimation of the diagonal section. Some empirical copula estimators for δ₁are given inDeheuvels(1979). A comparison between several more recent estimators is presented inOmelka et al.(2009).

Empirical diagonal

We present here an estimator that is detailed in Omelka et al.(2009), and directly inspired by the one of De- heuvels (1979). Consider observations {X^(k) = (X₁^(k), . . . , X_d^(k))}k∈{1,...,n} of the random vector X. Define pseudo-observationsU^(k)= (U₁^(k), . . . , U_d^(k))by setting every componentifor observation numberk to

U_i^(k)= 1 n+ 1

n

X

j=1

1ⁿ

X_i^(j)≤X_i^(k)o,

withi∈I,k∈ {1, . . . , n}. One can check that for any i∈I,k∈ {1, . . . , n},U_i^(k)∈(0,1).

The empirical estimatorCb of the copula of vectorX, is

C(ub 1, . . . , ud) = 1 n

n

X

k=1

1ⁿ

U₁^(k)≤u1,...,U_d^(k)≤ud

o.

We thus obtain for anyu∈[0,1], the empirical estimation for the diagonal ofC and its inverse, i.e., (

δb₁^emp(u) = C(u, . . . , u)b , δb₋₁^emp(u) = arginfn

x∈[0,1]; δb₁^emp(x)≥uo .

Smooth empirical diagonal

We do not present here all possible smooth estimators of a copula and the associated smoothed estimates of the diagonal section. We will restrict ourselves in estimators based on transformations, in the same spirit as the other parts of this paper. These estimators perform reasonably well (see Omelka et al. (2009)), especially considering Cramér-von Mises distance.

Using a smooth estimation of the empirical cumulative distribution of U could create some bias since the distribution support must be[0,1], this is the reason why we consider a transformation of pseudo-observations.

For given smoothing coefficientsh1, . . . , hd, we can defineL^(k)= (L^(k)₁ , . . . , L^(k)_d )where L^(k)_i =G⁻¹(U_i^(k)),

with i ∈ I, k ∈ {1, . . . , n}, and where G⁻¹ is the inverse of a cumulative distribution function, for example G⁻¹(x) = logit(x), and one can check that((G)⁰(x))²/G(x)is bounded, which is a required condition detailed inOmelka et al.(2009).

A smooth version ofCb can be defined by:

C(u1, . . . , ud) = 1 n

n

X

k=1

Y

i∈I

K G⁻¹(ui)−L^(k)_i h_i

! ,

whereK is a suited kernel function (we took here a multiplicative multivariate kernel).

We thus obtain for any u∈[0,1], a smooth estimator bδof the diagonal section of C, and a smooth estimator bδ₋₁ of the inverse function of this diagonal section:

(

δb₁(u) = C(u, . . . , u) δb₋₁(u) = arginfn

x∈[0,1];bδ1(x)≥uo

. (5)

Note that initial pseudo-data can also be smoothed, by defining U^(k)_i = 1

n+ 1 X

j∈{1,...,n}

Φ X_i^(k)−X_i^(j) h_i

! .

Discussions on this last choices can be found inChen and Huang(2007) andOmelka et al.(2009).

A discussion on the possible choices forh₁, . . . , h_d is given inChiu(1996) (ford= 1),Wand and Jones(1993), Wand and Jones(1994),Zhang et al.(2006) (ford≥2) and references therein. However, according toOmelka et al.(2009), “a good bandwidth selection rule is missing, for the moment, and is subject of further research.”

A summary of the needed input parameter and the estimation ofδ1 andδ₁⁻¹ is given in Algorithm1.

Algorithm 1Smooth initial diagonal estimation Input parameters

ChooseG⁻¹ inverse of a cdf,e.g., G⁻¹(x) = logit(x)

Choosehi,i∈I bandwidth sizes,e.g., the ones proposed by the Silverman’s Rule of Thumb Estimation

For anyx∈[0,1], getbδ₁(x)andδb₋₁(x)by Equation (5)

2.2 Nonparametric estimation of transformations T and T

, i ∈ I

A non parametric estimator of the external transformation T in (3) is given in Di Bernardino and Rullière (2013b), forx∈(0,1),by

Tb(x) =δb_r(x)(y₀), (6)

withr(x)such thatδ⁰_r(x)(x0) =x,

wherebδ_r refers to the estimator of the self-nested diagonalδ_r of the target copulaC, andδ_r(x)⁰ (x₀)refers to the self-nested diagonal of the initial copulaC0. These estimatorsδbrand self-nested diagonals are defined hereafter.

In the case where the initial copulaC0 is an Archimedean copula of generatorφ0, then δ_r(x)⁰ (x₀) =φ₀

d^r(x)φ⁻¹₀ (x₀)

and r(x) = 1 lndln

φ⁻¹₀ (x) φ⁻¹₀ (x0)

.

In particular, if C₀ is the independence copula, with generator φ₀(x) = exp(−x), and setting for example x₀=y₀= exp(−1), then

Tb(x) =δb_{ln(−ln(x))/lnd}(e⁻¹).

Let bδ₁ be an estimator of δ₁, and bδ₋₁ be an estimator of the inverse function δ₋₁ as in Equation (5). At a relative integer orderk∈Z, the self-nested diagonals estimators are defined as







bδ_k(u) = bδ₁◦. . .◦bδ₁(u), (ktimes), k∈N bδ−k(u) = bδ−1◦. . .◦δb−1(u), (ktimes), k∈N bδ0(u) = u.

(7)

At any real orderr∈R, an estimator bδ_r of the self-nested diagonal is bδ_r(x) =z

z⁻¹◦bδ_k(x)1−α

z⁻¹◦bδ_k+1(x)α

, x∈[0,1]

with α=r− brcand k=brc, where brcdenotes the integer part ofr, and where z is a function driving the interpolation, ideally (if known) the generator of the considered copulaC. Some consistency results about this estimatorTbare detailed inDi Bernardino and Rullière (2013b).

For invertible marginal transformationsT_i, one easily shows

T_i=F_i◦Fe_i⁻¹◦T, i∈I. (8)

Then, by replacing the transformed marginal distributionFe_i in (8) by an estimator of the targeti-th marginal distribution, denoted byFbi (e.g., empirical cdf), and given the external transformation or its consistent estima- tionTb, a non-parametric estimation ofTi, fori∈I, is provided by

Tbi(u) =Fi◦Fb_i⁻¹◦Tb(u), u∈(0,1), (9) whereF_i is the choseni−th initial marginal.

Following Definition2.1summaries the expression of non parametric estimators for bothT and T_i,i∈I.

Definition 2.1 (Non-parametric estimators of T and T_i). For a given arbitrary couple (x₀, y₀) ∈ (0,1)², a non-parametric estimator of T is given by

Tb(x) =bδ_r(x)(y₀), for allx∈(0,1) with r(x)such that δ_r(x)⁰ (x0) =x,

where δ_r(x)⁰ refers to the self nested diagonal of the initial copula C₀. In particular, if the initial copula C₀ is the independence copula, r(x) = _ln¹_dln

−lnx

−lnx0

. For anyi∈I, non-parametric estimatorsTi are

Tb_i(x) =F_i◦Fb_i⁻¹◦Tb(x).

A summary of the steps for the non-parametric estimation ofT andTi is given in Algorithm2.

Algorithm 2Non-parametric estimation ofTbandTbi

Input parameters

Choose(x0, y0)in (0,1)², initial copulaC0 and initial marginalsFi, fori∈I Choosekmaxpre-computation range

Eventual pre-calculations Getbδ1and bδ₋₁ by Algorithm1

Calculate and storebδk(y0),k∈ {−kmax, . . . ,0, . . . , kmax}by Equation (7) Non-Parametric estimation

GetTbby Equation (6), using previous precomputations GetTbi by Equation (9)

2.3 Subset of points of transformations

In practice, we will propose in Section 3 some parametric estimators for the transformations T and T_i, by requiring that these transformations are passing through a finite set of points. The interested reader is also referred toDi Bernardino and Rullière(2013a). We aim here at determining some good set of points to be chosen.

Firstly, we define some sets of points to be chosen, starting from given sets of quantile levels. The chosen expressions for these sets is motivated by several interesting properties (see Proposition2.2below).

Definition 2.2 (Set Ω). Let J ⊂ N be a finite set of indexes and Q = n q_j^(T)o

j∈J be a given set of targeted percentiles,q^(T_j ⁾∈(0,1),j∈J. One definesΩ(Q) ={(αj, βj}_j∈J, with

(

αj = q^(T_j ⁾ β_j = Tb(α_j),

whereTbis the estimator in Equation (6).

Definition 2.3 (Sets Ωi, i ∈ I). Let Ji ⊂ N be finite sets of indexes and Qi = n q⁽ⁱ⁾_j o

j∈Ji

be finite given sets of targeted percentiles, q⁽ⁱ⁾_j ∈ (0,1), j ∈ Ji, i ∈ I. For a given transformation τ, one defines the sets Ωi(Qi, τ) =

(α_jⁱ, β_jⁱ)

j∈Ji where

( αⁱ_j = τ⁻¹(q_j⁽ⁱ⁾),

βⁱ_j = Fi◦Fb_i⁻¹(q_j⁽ⁱ⁾), (10)

whereFb_i⁻¹ is an estimator of the target i-th marginal distribution, denoted byFb_i.

Given Ω and Ω_i, for i ∈I, some set of points as in Definitions 2.2 and 2.3, we firstly derive some properties of some parametric transformations TΩ and TΩ_i, i∈ I that are chosen to pass through these sets Ω and Ωi, for i∈I (see Proposition2.2 below). Theses properties justify the choice of sets Ωand Ωi in our estimation procedure.

Proposition 2.2 (Set of points forTi). Assume that estimators Tb andFbi are given continuous and invertible functions. LetJ_i⊂N be finite sets of indexes andQ_i =n

q⁽ⁱ⁾_j o

j∈Ji

be a finite given set of targeted percentiles, q⁽ⁱ⁾_j ∈(0,1),j∈Ji,i∈I. Then for transformationsTbi defined in Equation (9),i∈I

Tbi is passing through all points ofΩi(Qi,T),b

where Ωi(Qi,Tb) is defined as in Definition 2.3. Furthermore, for any invertible transformation TΩ_i passing through all points of Ω_i(Qi,Tb),i∈I

Fe_Ω⁻¹

i (q) =Fb_i⁻¹(q)for allq∈ Qi, (11) whereFeΩ_i =Tb◦T_Ω⁻¹

i ◦Fi is the transformed marginal distribution which usesTΩ_i.

Proof: Leti ∈ I, for any q ∈ Qi, one can define (α, β) ∈Ωi(Qi,T)b by Equation (10), with α=Tb⁻¹(q) and β=Fi◦Fb_i⁻¹(q). We assume here that all estimators are continuous and invertible functions, so that for example Tb◦Tb⁻¹=Id.

Firstly, one can check that Tbi(α) = Tbi ◦Tb⁻¹(q) and from Equation (9), Tbi(α) = Fi◦Fb_i⁻¹◦Tb◦Tb⁻¹(q) = F_i◦Fb_i⁻¹(q) =β, so thatTb_i(α) =β and the first result holds.

Secondly, assuming all estimators are invertible, one can write Fe_Ω⁻¹

i = F_i⁻¹◦TΩi ◦Tb⁻¹, so that Fe_Ω⁻¹

i(q) = F_i⁻¹◦T_Ωi(α). By assumptionT_Ωi(α) =β since T_Ωi is passing through all points ofΩ_i(Qi,Tb), so that finally Fe_Ω⁻¹

i (q) =F_i⁻¹(β) =F_i⁻¹◦Fi◦Fb_i⁻¹(q)andFe_Ω⁻¹

i (q) =Fb_i⁻¹(q)which gives the second result.

Once chosen the thresholdsq_j⁽ⁱ⁾, j ∈Ji, for which we want to identify the transformed margins with targeted margins, one thus get a finite set of passage points forT_i. Indeed, Proposition 2.2 shows that, usingΩ_i from Definition 2.3, one can select a further parametric estimator T_Ωi passing trough points of Ω_i and such that quantiles of the targetFb_iare identified with quantiles of the transformed margin usingT_Ωi (see Equation (11)), for each quantile levelq∈ Q_i and for any i∈I. Figure1illustrates this identification of quantile levels.

0.0 0.5 1.0 1.5

0.00.20.40.60.81.0

Univariate distribution F

x

F(x)

Figure 1: Graphical illustration of procedure described in Proposition2.2, ford= 1. Here an univariate data-set and the setQ={0.1,0.5,0.9}are chosen. The black line represents the empirical cdf of data, Fb, and the red one the cdfFe using the procedure in Proposition 2.2. As proved before, the quantiles ofFb are identified with quantiles of the obtained transformedFefor each quantile levelq∈ Q(black points and associated dotted lines).

Such a result would be difficult to establish for the external transformation T. A weaker form is given by following Proposition2.3.

Proposition 2.3 (Set of points forT). LetJ ⊂Nbe a finite set of indexes andQ=n q_j^(T)o

j∈J be a given set of targeted percentiles,q_j^(T)∈(0,1),j∈J. DefineΩ(Q)as in Definition2.2. Then obviously

Tb is passing through all points ofΩ(Q).

Furthermore, ifTΩ is another transformation passing through all points ofΩ(Q), then

max

(αj,βj)∈Ω

C(βb _j, . . . , β_j)−Ce_Ω(β_j, . . . , β_j) ≤ sup

u∈[0,1]

Tb(u)−T_Ω(u)

where CeΩ(u1, . . . , ud) = TΩ◦C0(T_Ω⁻¹(u1), . . . , T_Ω⁻¹(ud)) is the transformed copula using transformation TΩ, CbΩ(u1, . . . , ud) =Tb◦C0(Tb⁻¹(u1), . . . ,Tb⁻¹(ud)) is the transformed copula using the full non-parametric trans- formationTband C0 is the initial copula.

Proof: From Definition2.2, it is obvious thatTbis passing through all points ofΩ(Q). For any(αj, βj)∈Ω(Q), CeΩ(βj, . . . , βj) =TΩ◦C0(T_Ω⁻¹(βj), . . . , T_Ω⁻¹(βj))andCbΩ(βj, . . . , βj) =Tb◦C0(Tb⁻¹(βj), . . . ,Tb⁻¹(βj)). Since both

Tband TΩ are passing trough all points of Ω(Q), it follows that T_Ω⁻¹(βj) =Tb⁻¹(βj) =αj. As a consequence,

C(βb _j, . . . , β_j)−Ce_Ω(β_j, . . . , β_j) =

T(vb _j)−T_Ω(v_j)

, withv_j=C₀(α_j, . . . , α_j),v_j ∈[0,1]. Hence the result.

Proposition 2.3gives an interpretation of the set Qdefined in Definition 2.2. Indeed,Q can be interpreted as a set of percentiles for which we want to minimize the difference between diagonals of the transformed non- parametric copula and a transformed parametric copula using a transformationTΩinstead ofTb.

Once obtained these finite sets of reaching points forTbi andT, one can find parametric estimators without anyb optimization procedure. A summary of the estimation procedure for obtaining piecewise linear estimators ofT andTi,i∈I, is given in Algorithm 3.

Algorithm 3Subsets from non-parametric estimation Input parameters

ChooseQandQi,i∈I, initial set of quantile levels,e.g. Q=Q1=. . .=Qd={0.25,0.5,0.75}.

Subsets of points

GetTˆ andTˆi,i∈I, by Algorithm2 GetΩ(Q)by Definition2.2

GetΩi(Qi,T)b by Definition2.3

We have seen that, starting from given thresholds q⁽ⁱ⁾_j ∈ (0,1), it was possible to propose some set of points Ωi,i∈I and Ω, as in Proposition 2.2and2.3, such that eachTbi is passing trough points of Ωi, and each Tb is passing trough points ofΩ. Starting from these considerations, one can also introduce in the following definition for some piecewise linear estimators ofT andT_i,i∈I.

Definition 2.4 (Piecewise linear estimators of T and Ti). One can define two piecewise linear estimators of the external and internal transformationsT andTi,i∈I.

• Tb^PL is defined as a piecewise linear function passing trough the points(0,0), all points ofΩ(Q), and(1,1), where Ω(Q) is given in Definition2.2.

• for eachi∈I,Tb_i^PL is defined as a piecewise linear function passing through the points (0,0), all points of Ω_i(Q_i,Tb^PL), and(1,1), whereΩ_i(Q_i,Tb^PL),i∈I is given in Definition 2.3.

Piecewise linear estimatorsTb^PLandTb_i^PL,i∈I, are estimators relying on a chosen finite number of parameters.

However, these estimators are not differentiable everywhere on (0,1). In the following, we will try to find differentiable parametric estimators passing through the points of considered subsetsΩandΩi,i∈I.

3 Parametric estimation

3.1 Composite hyperbolic transformations

As initially defined in Bienvenüe and Rullière (2011) and Bienvenüe and Rullière (2012) we recall here the definition of hyperbolic composite transformations.

Definition 3.1 (Conversion and transformation functions). Letf any bijective increasing function,f :R→R. It is said to be a conversion function. Furthermore the associated transformation function T tof is defined by:

Tf : [0,1]→[0,1]such that

T_f(u) =







0 if u= 0,

logit⁻¹(f(logit(u))) if 0< u <1,

1 if u= 1.

Remark that T_f is a continuous non-decreasing function, such that T_f(0) = 0, T_f(1) = 1. Furthermore we remark that the transformation functions in Definition 3.1are chosen in a way to be easily invertible. In par- ticular in a way such that T_f ◦T_g =T_f◦g, T_f⁻¹ =T_f−1. When f is easily invertible, these readily invertible transformations help sampling transformed distributions (seeBienvenüe and Rullière(2011)).

In this section we consider the following particular class of hyperbolic conversion function (for further details seeBienvenüe and Rullière(2012)).

Definition 3.2 (A class of hyperbole). The considered hyperboleH is

Hm,h,ρ₁,ρ₂,η(x) =m−h+ (e^ρ1+e^ρ2)x−m−h

2 −(e^ρ1−e^ρ2) s

x−m−h 2

²

+e^{η−ρ1 +2ρ2},

withm, h, ρ1, ρ2∈R, and one smoothing parameterη∈R. After some calculations, one can check that

H_m,h,ρ⁻¹

1,ρ2,η(x) = H_{m,−h,−ρ1,−ρ2,η}(x).

FunctionsH in Definition3.2are thus readily invertible: a simple change of the sign of some parameters leads to the inverse function. As a consequence, transformationsTHbased on conversion functionsH will also be readily invertible. In the following, we consider the generic hyperbolic conversion function defined in Definition 3.2.

First remark that when the smoothing parameterη tends to−∞, the hyperboleH tends to the angle function:

Am,h,ρ₁,ρ₂(x) =m−h+ (x−m−h) e^ρ11_{x<m+h}+e^ρ21_{x>m+h}

. (12)

As remarked in Bienvenüe and Rullière (2011), it thus appears that hyperbolic transformations have the ad- vantage of being smooth versions of angle functions. They show in their paper that initial parameters for the estimation are easy to obtain with angle compositions. Another advantage of the consider hyperbolic transfor- mations is the flexibility in the tail parameters estimation. We discuss this point in Remark1.

Remark 1 (Tail behavior). In Di Bernardino and Rullière (2014), it is shown that for the hyperbolic trans- formations defined above (see Definition 3.2) and starting from some particular initial Archimedean copulas C₀, it is possible to produce Archimedean copulas having tunable regular variation properties, and thus to get specific targeted multivariate lower and upper tail coefficients. Indeed using the hyperbolic conversion functions H, the aforementioned paper proposes a generic way to construct families of Archimedean generators presenting a chosen couple of lower and upper tail coefficients. This construction is based on the fact that the conversion function H in Definition3.2 has an asymptote ax+b at−∞ with a= e^ρ1, and an asymptote αx+β at+∞

with α= e^ρ2. Illustrations proposed in Section 4 in the above mentioned article show that, when fitting some data, it is thus possible to propose a fit that respects some estimated tail dependence coefficients, by deducing parametersρ1 andρ2 from tail coefficients and by estimating other parametersm,handη.

For sake of clarity, we recall below some definitions inDi Bernardino and Rullière(2013a). These definitions of composite transformations and suited parameters will be useful in the following.

Definition 3.3 (Composite transformations). Let k ∈ N. Consider η ∈ R and a given parameter vector θ = (m, h, ρ₁, ρ₂, a₁, r₁, . . . , a_k, r_k) if k ≥ 1, or θ = (m, h, ρ₁, ρ₂) if k = 0. We define the angle composite transformationAθ as:

Aθ=T_fθ, withf_θ=

A_ak,0,0,rk◦ · · · ◦A_a1,0,0,r1◦A_m,h,ρ1,ρ2 if k≥1,

A_m,h,ρ1,ρ2 if k= 0,

and the hyperbolic composite transformationH_θ,η as:

Hθ,η=Tf_θ,η, withfθ,η=

Ha_k,0,0,r_k,η◦ · · · ◦Ha₁,0,0,r₁,η◦Hm,h,ρ₁,ρ₂,η if k≥1,

Hm,h,ρ1,ρ2,η if k= 0,

withAm,h,ρ1,ρ2 as in Equation (12)andHm,h,ρ1,ρ2,η as in Definition3.2.

Remark 2 (Inverse composite transformations). Let k ∈ N. Consider η ∈ Rand a given parameter vector θ= (m, h, ρ₁, ρ₂, a₁, r₁, . . . , a_k, r_k)ifk≥1, orθ= (m, h, ρ₁, ρ₂)ifk= 0. SinceT_f⁻¹=T_f−1, the angle composite transformationA⁻¹_θ is such that:

A⁻¹_θ =Tf_θ, withfθ=

A_{m,−h,−ρ1,−ρ2}◦Aa₁,0,0,−r1◦ · · · ◦Aa_k,0,0,−rk ifk≥1,

A_{m,−h,−ρ1,−ρ2} ifk= 0.

The hyperbolic inverse composite transformationHθ,η is such that:

H⁻¹_θ,η=T_fθ,η, withf_θ,η=

H_{m,−h,−ρ1,−ρ2,η}◦H_{a1,0,0,−r1,η}◦ · · · ◦H_{ak,0,0,−rk,η} ifk≥1,

H_{m,−h,−ρ1,−ρ2,η} ifk= 0.

Definition 3.4 (Suited parameters from Ω). Let k ∈ N. Consider one given setΩ = {ω1, . . . , ω3+k}, ωj ∈ (0,1)². Denote by uj and vj the two respective components of each ωj in the logit scale, such that ωj = (logit⁻¹uj,logit⁻¹vj),j∈ {1, . . . ,3 +k}. Assume thatuj andvj are increasing sequences ofj. We define:

Θ(Ω) =

(m, h, ρ₁, ρ₂, a₁, r₁, . . . , a_k, r_k) if k≥1, (m, h, ρ₁, ρ₂) if k= 0, where m= ^u2+v₂ ², h= ^u2−v₂ ²,ρ₁ = ln

v₂−v1

u₂−u1

, ρ₂ = ln

v₃−v2

u₃−u2

, r_k = ln_v

3+k−v_2+k u_3+k−u2+k

u_2+k−u_1+k v_2+k−v1+k

, a_k =v_2+k, k≥1.

Proposition 3.1(Suited composite transformations). Let k∈N. Consider one given set Ω ={ω₁, . . . , ω_3+k}, ω_j ∈(0,1)² and a smoothing parameterη∈R. Setθ= Θ(Ω), then

- the transformationAθ(x) is piecewise linear in the logit scale and will be called logit-piecewise linear. It links point(0,0), points ofΩ, and point(1,1).

- the transformationHθ,η converges pointwise toAθ asη tends to −∞. It results that the continuous and differentiable transformationHθ,η can fit as precisely as desired the set of points Ωwhenη tends to −∞.

Proof: The first result is proved in Bienvenüe and Rullière (2011). It simply comes from the fact that A_Θ(Ω)(uj) = vj for all j ∈ {1, . . . ,3 +k}, where ωj = (logit⁻¹uj,logit⁻¹vj). The convergence of the hyper- bole composite transformation toward the angle composite transformation is straightforward and also evoked inBienvenüe and Rullière (2011).

3.2 Smoothed parametric estimators

For the estimation of transformationsT andT_i, we assume that are given:

• Initial estimators ofδandδ⁻¹ (see Section2.1).

• The sets of quantile levels Q=n q_j^(T)o

j∈J,Qi=n q_j⁽ⁱ⁾o

j∈J_i,i∈I (see Section2.3).

• Some smoothing parametersη∈Randηi∈R,i∈I (see Section3.1).

In the following we introduce the smooth parametric estimators T and T_i, i ∈ I, for external and internal transformations respectively.

Estimation ofT

Using estimators of δandδ⁻¹ (see Section2.1), one easily gets the resulting set Ω(Q)by Proposition2.3 and suited parameters by Definition3.4:

θb= Θ(Ω(Q)).b (13)

Then one defines: T =H

bθ,η. Estimation ofTi, i∈I

OnceTestimated byT, one gets the passage setΩi, fori∈Iby Proposition2.2and suited associated parameters θi by Definition3.4, i.e.,

θb_i= Θ(bΩ_i(Qi, T))). (14)

Note that once given thresholds setsQ andQi and smooth parametersη andηi, all estimated parameters are directly and analytically defined, so that we do not need here any inversion or optimization procedure.

Then one defines: Ti=H

bθi,ηi, fori∈I.

A summary of the expressions of smooth estimatorsT andTi,i∈I, is given in following definition.

Definition 3.5 (Smooth estimation ofT andTi, i∈I). Let Q andQi, i∈I be given sets of quantile levels.

Let η∈Randηi∈R,i∈I be given smoothing parameters. One defines ( T = H

θ,ηb , Ti = H

θbi,ηi, i∈I, (15)

where parameters bθandθbi are given by (

θb = Θ(bΩ(Q))),

θb_i = Θ(bΩ_i(Qi, T)), i∈I.

The setsΩandΩi are defined in Definitions 2.2and2.3. The function Θis defined in Definition3.4.

Estimation procedure for obtaining smoothed parametric estimatorsT and Ti, i∈I is gathered in Algorithm 4.

Algorithm 4Parametric estimation Input parameters

ChooseQandQi,i∈I the sets of quantile levels,e.g. Q=Q1 =. . .=Qd={0.25,0.5,0.75}, Choose smoothing parametersη∈Randηi ∈R,i∈I,

Parametric estimation

GetΩ(Q))b andΩbi(Qi, T),i∈I by Algorithm3 Getθbby Equation (13)

Getθbi by Equation (14)

Get smooth estimatorsT andTi,i∈Iby Equation (15).

One can define the complete vector parameter presented in Algorithms3 and4:

Θ

ΘΘ = (bθ1, . . . ,θbd,θ, ηb 1, . . . , ηd, η).

What is noticeable here is that given thresholdsQ,Qi and smoothing parametersη,ηi, one have direct expres- sions for the parametric estimators θband θbi,i∈I. The estimation can also change, depending on some other estimation choices (generator among its equivalence class(x0, y0), bandwidthshi, i∈I or kernel functionK), but one aims here at finding good estimators whatever the choice of any reasonable value of these parameters.

In numerical Section6we will illustrate this point. Finding a global optimum in high dimension would lead to a curse a dimensionality and numerical problems. As a consequence, it is very important to get correct estimators without jointly optimizing a lot of parameters in high dimension.

4 Final parametric results

Once all parameters estimated as in Section3, the previous parametric model allows to get various analytical results for both the multivariate distribution function, its associated critical layers, Kendall’s function and mul- tivariate return periods.

Firstly we obtain the parametric expression for the transformed copulaC:e

C(ue ₁, . . . , u_d) =φ(ee φ⁻¹(u₁) +. . .+φe⁻¹(u_d)), (16) where φe is the final parametric transformed Archimedean generator (see (4)). This generator can be easily written in terms of the external transformationT, i.e.,

φ(t) =e T(φ₀(t)), (17)

whereφ0is the generator associated to the initial copulaC0in (4) andTis the smooth estimator of the external transformation presented in Section3.2.

Using model in (4), the corresponding estimated transformed multivariate distribution is given by:

Fe_ΘΘΘ(x₁, . . . , x_d) =H

θ,ηb ◦C₀(H⁻¹

θb1,η1

◦F₁(x₁), . . . ,H⁻¹

bθ_d,η_d◦F_d(x_d)), (18) whereΘΘΘis the complete estimated vector parameter presented in Algorithms3 and4.

Furthermore, using expression in (18) for the estimated transformed multivariate distributionFe, the associated parametricαcritical-layers are given by

∂LeΘΘΘ(α) ={(F₁⁻¹◦ H

θb1,η1(u1), . . . , F_d⁻¹◦ H

θb_d,η_d(ud)),(u1, . . . , ud)∈(0,1)^d, C0(u1, . . . , ud) =H⁻¹

θ,ηb (α)}, where a direct analytic expression H⁻¹

θ,ηb is given by Remark 2 (see also Proposition 2.4 in Di Bernardino and Rullière(2013a)).

As presented in the introduction, we aim at providing a parametric estimation of the multivariate Return Period

RP^>(α) = ∆t·_1−K¹

C(α).

Genest and Rivest (2001) obtained the following explicit expression for the Kendall distribution in the case of multivariate Archimedean copulas with a given generatorϕ, i.e.,

KC(α) =α+

d−1

X

i=1

1

i! −ϕ⁻¹(α)ⁱ

ϕ⁽ⁱ⁾ ϕ⁻¹(α)

, forα∈(0,1), where the notationf⁽ⁱ⁾corresponds to thei−th derivatives of a functionf.

Consider the case of transformed generatorφ(t) =e T(φ₀(t)), in (16)-(17). Then, the analytical formula for the estimated transformed Kendall distributionKe_C can be easily written as:

KeC(α) = α+

d−1

X

i=1

1 i!

−φ⁻¹₀ (T⁻¹(α))ⁱ

T◦φ0

⁽ⁱ⁾

φ⁻¹₀ (T⁻¹(α)

, forα∈(0,1). (19) Remark that the hyperbolic transformationT =H

θ,ηb ,in this paper are chosen in a way to be easily invertible.

Then estimated smooth inverse transformationT⁻¹ in (19) is straightforwardly obtained by changing the signs of some parameters of the hyperbolic composition (see Remark2).

5 Comprehensive scheme of the estimation procedure

The method presented in this paper involves different notations progressively introduced in different sections.

Moreover, the final scheme of the procedure relies in many tuning parameters. To improve clarity, a comprehen- sive scheme of the estimation procedure is presented below (see Table1). The relationship between the different algorithms presented above is pointed out using arrows and implications. Moreover, the nature of consid- ered parameters is specified, in particular, we distinguish between tuning parameters and estimated/calculated quantities.

As one can see in Table 1 (left column), the algorithm depends on many tuning parameters, we discuss here the sensitivity of the results on these choices. The results expressed below refer to variations of mean absolute errors and are based on our simulation study (see Section8, casen= 1000, using as a referencex₀=y₀= e⁻¹, Q=Q₁=Q₂={0.25,0.5,0.75},η =−1, η₁=η₂=−3, with very small bandwidths).

• We always used in our numerical illustrations (see Sections 6, 7 and 8), G⁻¹ = logit(x) in order to simplify inversion procedures and C0 the independent copula. We believe that these choices might have more important implications for the tails. A deep analysis in this sense is developed inDi Bernardino and Rullière(2014).

• The use of positive bandwidth sizes hi ensures continuity and possible inversion of fitted copula diagonal section and marginals distributions. Using small bandwidth sizes may increase the variance of some non-parametric estimators, but in practice this leads to estimate parametric copula and margins using empirical diagonal and empirical margins instead of smoothed ones. The impact is thus limited on the parametric estimation. As an example in further simulation study (see Section 8), using a bandwidth given by classical rule of thumbor dividing this bandwidth by100leads to an absolute variation of0.6%

on resulting mean absolute errors.

• Concerning the choice of the point(x0, y0), we observed absolute variations of order4%in mean absolute errors compared to the choice ofx0=y0= e⁻¹(which simplifies some calculations).

• Concerning the choice of quantile levels set Q, Qi, for i∈ I, using regular thresholds Q=Q1 =Q2 = {0.25,0.5,0.75}, without any optimization, leads to mean absolute error of M AE = 3.6% (instead of approx. 1%for optimized thresholds), which is still reasonable for non optimized parameters.

• At last, choices of η and ηi, i ∈ I, mostly depend on the desired smoothness of the final resulting distribution. Except for clearly excessive smoothing, the impact on resulting distribution is quite small.

As an example on simulated data of Section 8, using η = 0 and η1 = η2 = −2 or using η = −2 and η1=η2=−4 leads to absolute variations of0.5%on resulting mean absolute errors.

Scheme of the estimation procedure Chosen tuning param-

eters in Algorithm Calculated quantities

1) G⁻¹ (inverse of a cdf ) 2) hi,i∈I bandwidth sizes

3) Point(x0, y0)in(0,1)² 4) Initial copulaC0

5) Initial marginalsFi, fori∈I

6) QandQi,i∈I

(initial set of quantile levels)

7) η∈Randηi∈R,i∈I (smoothing parameters)

a) Using 1) and 2), estimate diag- onalbδ1(x)andbδ−1(x), for any x∈[0,1].

b) Using 3), 4) and a), getTb

c) Estimate of the targeti-th marginal distribution,Fbifor i∈I(e.g., empirical cdf )

d) Using 5), b) and c), getTbi, for i∈I

e) Using 6) and b), getΩ(Q)

f ) Using 5), 6), b) and c), get Ωi(Qi,T), forb i∈I

g) Using 7), e) and f ),

• Getbθ= Θ(bΩ(Q)).

• Getbθi = Θ(bΩi(Qi, T))for i∈I, whereT =H_θ,ηb .

h) Using 4), 5) and g), finally get FeΘΘΘ(x1, . . . , xd) =H_θ,ηb ◦C0(H⁻¹

θb1,η1◦F1(x1), . . . ,H⁻¹

bθd,ηd◦ Fd(xd)),

whereΘΘΘ = (bθ1, . . . ,bθd,θ, ηb 1, . . . , ηd, η).

Table 1: Comprehensive scheme of the global estimation procedure. We distinguish between tuning parameters (left column) and estimated/calculated quantities (right column).

This quite small sensitivity of these parameters is important. Indeed in high dimension, due to the curse of dimensionality, it is not possible in practice to find the global optimum of a function. It is thus essential that any chosen tuning parameter lead to small average errors, in order that calculated quantities are near a local optimizer of the considered error, and in order to reduce the difference between local optima and global one.

Resulting calculated parameters like (m, h, ρ1, ρ2)for each transformation can still be optimized once they are close to one optimizer.

In further numerical illustrations (see Sections 6, 7 and 8), we never optimize resulting calculated quantities, but we sometimes choose tuning parameters that permit to get reduced final errors.

6 Numerical results on the rainfall real data

In the following we illustrate our methodology presented in the previous sections using a5−dimensional rainfall data-set.

6.1 Presentation of the data-set

Data comes from the website CISL Research Data Archive (RDA), http://rda.ucar.edu, and is available for registered users. The user is granted the right to use the Site for non-commercial, non-profit research, or educational purposes only, without any fee or cost, as specified in the terms of use of the website (January 2014).

The name of the data in this website is: ds570.0, and the direct url to this data is http://rda.ucar.edu/

datasets/ds570.0/. The whole citation information for this data is: