Elena Di Bernardino
email: elena.di-bernardino@univ-lyon1.fr

Thomas Laloë
email: laloe@unice.fr

Véronique Maume-Deschamps
email: veronique.maume@univ-lyon1.fr

Clémentine Prieur
email: clementine.prieur@imag.fr

PLUG-IN ESTIMATION OF LEVEL SETS IN A NON-COMPACT SETTING WITH APPLICATIONS IN MULTIVARIATE RISK THEORY

Keywords: 1991 Mathematics Subject Classification. 62G05, 62G20, 60E05, 91B30 Level sets, distribution function, plug-in estimation, Hausdorff distance, Conditional Tail Expectation

This paper deals with the problem of estimating the level sets L(c) = {F (x) ≥ c}, with c ∈ (0, 1), of an unknown distribution function F on R 2 + . A plug-in approach is followed. That is, given a consistent estimator Fn of F , we estimate L(c) by Ln(c) = {Fn(x) ≥ c}. In our setting, noncompactness property is a priori required for the level sets to estimate. We state consistency results with respect to the Hausdorff distance and the volume of the symmetric difference. Our results are motivated by applications in multivariate risk theory. In particular we propose a new bivariate version of the Conditional Tail Expectation by conditioning the two-dimensional random vector to be in the level set L(c). We also present simulated and real examples which illustrate our theoretical results.

Introduction

The Conditional Tail Expectation (CTE) is a commonly used tool in the univariate risk theory. Given a random variable X with continuous and strictly monotonic distribution function F X , the CTE at level α is defined by

CTE α (X) = E[ X | X ≥ Q X (α) ], (1)
with α ∈ (0, 1) and Q X (α) the univariate quantile function of X at level α. In other words the CTE α (X) is defined by the expectation value of X conditionally to the fact that X belongs to a certain one-dimensional level set of F X . For further details about this univariate risk measure we refer the reader e.g. to Denuit et al. [START_REF] Denuit | Actuarial Theory for Dependent Risks[END_REF].

In this paper we are interested to introduce a new bivariate generalization of CTE in [START_REF] Artzner | Coherent measures of risk[END_REF] and to provide a consistent procedure to estimate it. Consequently we essentially need a generalization of the notion of quantile in dimension higher than one. Therefore we remark that the quantile function is a relevant tool in statistics and probability, but its practical use is mostly confined to the univariate case where it is easily interpretable and calculable. The bivariate (or multivariate) characterization has received some attention however, it is much less common, due to a number of theoretical and practical reasons. In particular the main difficulty is that there does not exist a natural ordering in n-dimensions, for n > 1. For this reason, in the last decade several attempts have been made to provide a multidimensional generalization of the univariate quantile function. We refer to Serfling [START_REF] Serfling | Quantile functions for multivariate analysis: approaches and applications[END_REF] for a complete review on the topic. For example Massé and Theodorescu [START_REF] Massé | Halfplane trimming for bivariate distributions[END_REF] defined multivariate quantiles as halfplanes and Koltchinskii [START_REF] Koltchinskii | M -estimation, convexity and quantiles[END_REF] provided a general treatment of multivariate quantiles as inversions of mappings.

As a possible generalization of the univariate quantile function, Tibiletti [START_REF] Tibiletti | On a new notion of multidimensional quantile[END_REF], Fernández-Ponce and Suárez-Lloréns [START_REF] Fernández-Ponce | Central regions for bivariate distributions[END_REF] and Belzunce et al. [START_REF] Belzunce | Quantile curves and dependence structure for bivariate distributions[END_REF] defined a multivariate quantile as a set of points which accumulate the same probability for a fixed orthant. They called it level curves or quantile curves. In this sense de Haan and Huang [START_REF] De Haan | Large quantile estimation in a multivariate setting[END_REF] model a risk-problem of flood in the bivariate setting using an estimator of level curves of bivariate distribution function. In this paper we will follow this approach in order to propose a new risk measure which generalizes the CTE in [START_REF] Artzner | Coherent measures of risk[END_REF]. To this end we need to estimate the level sets of a bivariate distribution function F . This leads us into the general field of level sets estimation.

The problem of estimating level sets of an unknown function (for instance of a density function and more recently a regression function) has received attention recently. In particular the estimation of density level sets has been studied in Polonik [START_REF] Polonik | Measuring mass concentrations and estimating density contour clusters-an excess mass approach[END_REF], Tsybakov [START_REF] Tsybakov | On nonparametric estimation of density level sets[END_REF], Baíllo et al. [START_REF] Baíllo | Convergence rates in nonparametric estimation of level sets[END_REF], Baíllo [START_REF] Baíllo | Total error in a plug-in estimator of level sets[END_REF], Cadre [START_REF] Cadre | Kernel estimation of density level sets[END_REF], Rigollet and Vert [START_REF] Rigollet | Optimal rates for plug-in estimators of density level sets[END_REF], Biau et al. [START_REF] Biau | A graph-based estimator of the number of clusters[END_REF]. The estimation of regression level sets in a compact setting has been analyzed in Cavalier [START_REF] Cavalier | Nonparametric estimation of regression level sets[END_REF], Laloë [START_REF] Laloë | Sur Quelques Problèmes d'Apprentissage Supervisé et Non Supervisé[END_REF]. An alternative approach, based on the geometric properties of the compact support sets, has been presented by Hartigan [START_REF] Hartigan | Estimation of a convex density contour in two dimensions[END_REF], Cuevas and Fraiman [START_REF] Cuevas | A plug-in approach to support estimation[END_REF], Cuevas and Rodríguez-Casal [START_REF] Cuevas | On boundary estimation[END_REF]. The problem of estimating general level sets under compactness assumptions has been discussed by Cuevas et al. [START_REF] Cuevas | Plug-in estimation of general level sets[END_REF]. The asymptotic behavior of minimum volume sets and of a generalized quantile process is analyzed by Polonik [START_REF] Polonik | Minimum volume sets and generalized quantile processes[END_REF].

Considering the level sets of a bivariate distribution function, the commonly assumed property of compactness for these sets is no more reasonable. Then, differing from the literature cited above, we need to work in a non-compact setting and this requires special attention in the statement of our problem.

Our general approach will be the following: first, we provide a consistent estimator of the level set L(c) = {F (x) ≥ c}, for c ∈ (0, 1).

Therefore we consider a plug-in approach (e.g. see Baíllo et al. [START_REF] Baíllo | Convergence rates in nonparametric estimation of level sets[END_REF]; Rigollet and Vert [START_REF] Rigollet | Optimal rates for plug-in estimators of density level sets[END_REF]; Cuevas et al. [START_REF] Cuevas | Plug-in estimation of general level sets[END_REF]), that is, L(c) is estimated by

L n (c) = {F n (x) ≥ c}, for c ∈ (0, 1),
where F n is a consistent estimator of F . The regularity properties of F and F n as well as the consistency properties of F n will be specified in the statements of our theorems.

In order to provide a consistency results we define two proximity criteria between sets. A standard choice is the volume of the symmetric difference. Another natural criterion is given by the Hausdorff distance that corresponds to an intuitive notion of "physical proximity" between sets. Our consistency properties are stated with respect to these two criteria under reasonable assumptions on F and F n (Theorems 2.1 -3.1). These results are based on a slight modification of Proposition 3.1 in the PhD Thesis of Rodríguez-Casal [START_REF] Rodríguez-Casal | Estimacíon de conjuntos y sus fronteras. Un enfoque geometrico[END_REF] (Proposition 1.1).

We then introduce our new bivariate version of the Conditional Tail Expectation by conditioning the twodimensional random vector to be in the level set L(c). This new risk measure is based on the bivariate Valueat-Risk proposed in Embrechts and Puccetti [START_REF] Embrechts | Bounds for functions of multivariate risks[END_REF] and Nappo and Spizzichino [START_REF] Nappo | Kendall distributions and level sets in bivariate exchangeable survival models[END_REF] and it is different from the existing bivariate generalizations of the univariate Conditional Tail Expectation (e.g. see Cai and Li [START_REF] Cai | Conditional tail expectations for multivariate phase-type distributions[END_REF]). Indeed we do not use any aggregate risk in the portfolio in order to analyze the bivariate risk's issue. Our bivariate version of the CTE deals with the simultaneous joint damages considering the dependence structure of data in a bivariate specific risk's area (L(c)). We propose an estimator for this new risk measure using plug-in approach for level sets and provide a consistency result (Theorem 4.1).

The paper is organized as follows. We introduce some notation, tools and technical assumptions in Section 1.

Consistency and asymptotic properties of our estimator of L(c) are given in Sections 2 and 3. In Section 4 we introduce a new bivariate generalization of CTE in (1) and we provide a consistent estimator for this risk measure. Illustrations with simulated and real data are presented in Section 5. Section 6 summarizes and briefly mentions directions for future research. Finally, some auxiliary results and more technical proofs are postponed to Section 7.

Notation and preliminaries

In this section we introduce some notation and tools which will be useful later.

Let

N * = N \ {0}, R 2 + * = R 2 + \ (0, 0), F the set of continuous distribution functions R 2 + → [0, 1] and F ∈ F. Given an i.i.d sample {X i } n i=1 in R 2 + with distribution function F , we denote by F n (•) = F n (X 1 , X 2 , . . . , X n ,
•) an estimator of F based on the finite sample (X 1 , X 2 , . . . , X n ). We restrict ourselves to R 2 + for convenience but the following results are completely adaptable in R 2 . This choice is motivated essentially by our application in risk theory proposed in Section 4, where random variables will be losses then defined in a positive support.

Define, for c ∈ (0, 1), the upper c-level set of F ∈ F and its plug-in estimator

L(c) = {x ∈ R 2 + : F (x) ≥ c}, L n (c) = {x ∈ R 2 + : F n (x) ≥ c},
and

{F = c} = {x ∈ R 2 + : F (x) = c}. In addition, given T > 0, we set L(c) T = {x ∈ [0, T ] 2 : F (x) ≥ c}, L n (c) T = {x ∈ [0, T ] 2 : F n (x) ≥ c}, {F = c} T = {x ∈ [0, T ] 2 : F (x) = c}.
Furthermore, for any A ⊂ R 2 + we denote by ∂A its boundary.

Note that {F = c} can be a portion of quadrant R 2 + instead of a set of Lebesgue measure null in R 2 + (that is the presence of plateau at level c). In the following we will introduce suitable conditions in order to avoid the presence of plateau in the graph of F .

In the metric space (R 2 + , d), where d stands for the Euclidean distance, we denote by B(x, ρ) the closed ball centered on x and with positive radius ρ. Let B(S, ρ) = x∈S B(x, ρ), with S a closed set of R 2 + . For r > 0 and ζ > 0, define

E = B({x ∈ R 2 + : | F (x) -c |≤ r}, ζ)
, and, for a twice differentiable function F ,

m ▽ = inf x∈E (∇F ) x , M H = sup x∈E (HF ) x ,
where (∇F ) x is the gradient vector of F evaluated at x and (∇F ) x its Euclidean norm, (HF ) x the Hessian matrix evaluated in x and (HF ) x its matrix norm induced by the Euclidean norm.

For sake of completeness, we recall that if A 1 and A 2 are compact sets in (R 2 + , d), the Hausdorff distance between A 1 and A 2 is defined by

d H (A 1 , A 2 ) = inf{ρ > 0 : A 1 ⊂ B(A 2 , ρ), A 2 ⊂ B(A 1 , ρ)},
or equivalently by

d H (A 1 , A 2 ) = max sup x∈A1 d(x, A 2 ), sup x∈A2 d(x, A 1 ) , where d(x, A 2 ) = inf y∈A2 x -y .
The above expression is well-defined even when A 1 and A 2 are just closed (not necessarily compact) sets but in this case the value d H (A 1 , A 2 ) could be infinity. Then in our setting, in order to avoid these situations we need a truncated version of Assumption T in Cuevas et al. [START_REF] Cuevas | Plug-in estimation of general level sets[END_REF] or Tsybakov [START_REF] Tsybakov | On nonparametric estimation of density level sets[END_REF]. More precisely we introduce the following assumption.

H: There exist γ > 0 and

A > 0 such that, if | t -c | ≤ γ then ∀ T > 0 such that {F = c} T = ∅ and {F = t} T = ∅, d H ({F = c} T , {F = t} T ) ≤ A | t -c | .
Assumption H is satisfied under mild conditions. Proposition 1.1 below is a slight modification of Proposition 3.1 in the PhD Thesis of Rodríguez-Casal [START_REF] Rodríguez-Casal | Estimacíon de conjuntos y sus fronteras. Un enfoque geometrico[END_REF] in order to deal with non-compact sets.

Proposition 1.1. Let c ∈ (0, 1). Let F ∈ F be twice differentiable on R 2 * + . Assume there exist r > 0, ζ > 0 such that m ▽ > 0 and M H < ∞. Then F satisfies Assumption H, with A = 2 m ▽ . The proof is postponed to Section 7.

Remark 1. Under assumptions of Proposition 1.1, F is continuous and m ▽ > 0, there is no plateau in the graph of F for each level t such that | tc | ≤ r. Furthermore from Theorem 1 in Rossi [START_REF] Rossi | Proprietà geometriche delle superficie di ripartizione[END_REF] we know that each half-line in R 2 + , parallel to one of the axis, meets {F = t} in a connected set of points. As a consequence we obtain that {F = t} is a curve in the quadrant R 2 + . In particular, from m ▽ > 0, for a fixed x we have to consider all corresponding values y in a specific interval (we refer the interested reader to Remark 2.1 in Rossi [START_REF] Rossi | Proprietà geometriche delle superficie di ripartizione[END_REF] or Remark 3.1 in Rossi [START_REF] Rossi | Sulle curve di livello di una superficie di ripartizione in due variabili; on level curves of two dimensional distribution function[END_REF]). In this case the plane curve {F = t} is not the graph of a function but it has the following monotonic property. We consider (x, y), (x ′ , y ′ ) ∈ {F = t}, if x < x ′ then y ≥ y ′ , if y < y ′ then x ≥ x ′ . In particular if we suppose that each component of (∇F ) x is greater than zero in E then {F = t} is a monotone decreasing curve in R 2 + . Finally under assumptions of Proposition 1.1 we obtain ∂L(c) T = {F = c} T = {F = c} ∩ [0, T ] 2 (we refer for details to Theorem 3.2 in Rodríguez-Casal [START_REF] Rodríguez-Casal | Estimacíon de conjuntos y sus fronteras. Un enfoque geometrico[END_REF]).

Consistency in terms of the Hausdorff distance

In this section we study the consistency properties of L n (c) T with respect to the Hausdorff distance between ∂L n (c) T and ∂L(c) T . The Hausdorff distance corresponds to an intuitive notion of "physical proximity" between sets (see Figure 1(left) for a graphical illustration). However the metric d H is not always completely successful in capturing the shape properties: two sets can be very close in d H and still have quite different shapes. For instance in Figure 1(right) we have X, the outside ball, and Y the set of balls inside X. The distance d H (X, Y ) is small even if X and Y are quite different in many important aspects. In order to avoid these situations, following Cuevas and Rodríguez-Casal [START_REF] Cuevas | On boundary estimation[END_REF] and Cuevas et al. [START_REF] Cuevas | Plug-in estimation of general level sets[END_REF], a way to reinforce the notion of visual proximity between two sets provided by d H is to impose the proximity of the respective boundaries. We will follow this approach in Theorem 2.1 below.

From now on we note, for n ∈ N * ,

F -F n ∞ = sup x ∈ R 2 + | F (x) -F n (x) |,
and for T > 0 F -F n T ∞ = sup x ∈ [0,T ] 2 | F (x) -F n (x) | .
The following result can be considered an adapted version of Theorem 2 in Cuevas et al. [START_REF] Cuevas | Plug-in estimation of general level sets[END_REF].

Theorem 2.1. Let c ∈ (0, 1). Let F ∈ F be twice differentiable on R 2 * + . Assume that there exist r > 0, ζ > 0 such that m ▽ > 0 and M H < ∞. Let T 1 > 0 such that for all t : | tc | ≤ r, ∂L(t) T1 = ∅. Let (T n ) n∈N * be an increasing sequence of positive values. Assume that, for each n and for almost all samples of size n, F n is a continuous function and that

F -F n ∞ → 0, a.s. (2)
Then d H (∂L(c) Tn , ∂L n (c) Tn ) = O( F -F n ∞ ), a.s.
The proof is postponed to Section 7. Under assumptions of Theorem 2.1, d H (∂L(c) Tn , ∂L n (c) Tn ) converges to zero and the quality of our plug-in estimator is obviously related to the quality of the estimator F n .

Remark 2. Theorem 2.1 provides an asymptotic result for a fixed level c. In particular following the proof of Theorem 2.1 (postponed to Section 7) we remark that, for n large enough,

d H (∂L(c) Tn , ∂L n (c) Tn ) ≤ 6 A F -F n Tn ∞ , a.s.,
where A = 2 m ▽ . Note that in the case c is close to one the constant A could be large. In this case, we will need a large number of data to get a reasonable estimation.

Note that the empirical distribution estimator does not satisfy continuity assumption imposed in Theorem 2.1. However, in order to overcome this problem it can be considered a smooth version of this estimator (e.g. see Chaubey and Sen [START_REF] Chaubey | Smooth estimation of multivariate survival and density functions[END_REF]), which satisfies convergence in [START_REF] Baíllo | Total error in a plug-in estimator of level sets[END_REF]. However we remark that the continuity assumption, imposed in Theorem 2.1, will not be necessary in the following results.

L 1 consistency

The previous section was devoted to the consistency of L n in terms of the Hausdorff distance. We consider now another consistency criterion: the consistency of the volume (in the Lebesgue measure sense) of the symmetric difference between L(c) Tn and L n (c) Tn . This means that we define the distance between two subsets A 1 and

A 2 of R + 2 by d λ (A 1 , A 2 ) = λ(A 1 △ A 2 ),
where λ stands for the Lebesgue measure on R 2 and △ for the symmetric difference.

Let us introduce the following assumption:

A1 There exist positive increasing sequences

(v n ) n∈N * and (T n ) n∈N * such that v n [0,Tn] 2 | F -F n | p λ(dx) P → n→∞ 0,
for some 1 ≤ p < ∞.

We now establish our consistency result in terms of the volume of the symmetric difference. We can interpret the following theorem as a generalization of Theorem 3 in Cuevas et al. [START_REF] Cuevas | Plug-in estimation of general level sets[END_REF] (Section 3) in the case of non-compact level sets. In addition, we provide also a convergence rate for the symmetric difference between L(c) Tn and L n (c) Tn .

Theorem 3.1. Let c ∈ (0, 1). Let F ∈ F be twice differentiable on R 2 * + . Assume that there exist r > 0, ζ > 0 such that m ▽ > 0 and M H < ∞. Assume that for each n, with probability one, F n is measurable. Let (v n ) n∈N * and (T n ) n∈N * positive increasing sequences such that Assumption A1 is satisfied and that for all t : | t-c | ≤ r, ∂L(t) T1 = ∅. Then it holds that

p n d λ (L(c) Tn , L n (c) Tn ) P → n→∞ 0, with p n an increasing positive sequence such that p n = o v 1 p+1 n /T p p+1 n .
The proof is postponed to Section 7. Theorem 3.1 provides a convergence rate, which is closely related to the choice of the sequence T n . A sequence T n whose divergence rate is large implies a convergence rate p n quite slow. Note that, as in Theorem 3 in Cuevas et al. [START_REF] Cuevas | Plug-in estimation of general level sets[END_REF], Theorem 3.1 above does not require any continuity assumption on F n . Remark 3. Assumptions on gradient vector and Hessian matrix of F , in Theorems 2.1 and 3.1, are satisfied for a quite large class of classical bivariate distribution functions; for instance independent copula and exponential marginals, Farlie-Gumbel-Morgenstern (FGM), Clayton or Survival Clayton copulas and Burr marginals.

We now suppose that there exists a positive increasing sequences

(v n ) n∈N * such that v n F -F n ∞ P → n→∞ 0. Then ∀ p ≥ 1, w n [0,Tn] 2 | F -F n | p λ(dx) P → n→∞ 0, with w n = v p n T 2 n .
In this case, from Theorem 3.1 we get

p n = o(w 1 p+1 n /T p p+1 n ) = o(v p p+1 n /T p+2 p+1 n ). (3)
From these considerations we can introduce the following remark. n ). For instance, for F (x, y) = (1e -x ) (1e -2 y ) and using the bivariate empirical distribution function F n we obtain p n = o(n

1 3 / ln(n) 4
3 ), with p = 2 and T n = ln(n).

Application in bivariate risk theory

From the usual definition in the univariate setting we know that the quantile function Q X provides a point which accumulates a probability α to the left tail and 1α to the right tail. More precisely, given an univariate continuous and strictly monotonic loss distribution function F X ,

Q X (α) = F -1 X (α), ∀ α ∈ (0, 1). ( 4
)
The notion of univariate quantile function Q X is used in risk theory to define an univariate measure of risk: the Value-at-Risk (VaR). This measure is defined as

VaR α (X) = Q X (α), ∀ α ∈ (0, 1). ( 5
)
As presented in the introduction of this paper, following the general ideas of Embrechts and Puccetti [START_REF] Embrechts | Bounds for functions of multivariate risks[END_REF] and Nappo and Spizzichino [START_REF] Nappo | Kendall distributions and level sets in bivariate exchangeable survival models[END_REF] an intuitive generalization of the VaR measure in the case of a bidimensional loss distribution function F is given by its α-quantile curves. More precisely: Definition 4.1. For α ∈ (0, 1) and F ∈ F, the bidimensional Value-at-Risk at probability level α is the boundary of its α-level set, i.e. VaR α (F ) = ∂L(α).

As pointed out in Tibiletti [START_REF] Tibiletti | On a new notion of multidimensional quantile[END_REF], imposing α = 1 2 in Definition 4.1, we get a natural extension of the bidimensional median. For details about a parametric formulation of the quantile curve ∂L(α) see Belzunce et al. [START_REF] Belzunce | Quantile curves and dependence structure for bivariate distributions[END_REF]. For details about its properties see for instance Fernández-Ponce and Suárez-Lloréns [START_REF] Fernández-Ponce | Central regions for bivariate distributions[END_REF] (and references therein) and Nappo and Spizzichino [START_REF] Nappo | Kendall distributions and level sets in bivariate exchangeable survival models[END_REF].

Then, using a bivariate estimator F n as in Sections 2 -3, we can define our plug-in estimator of the bivariate Value-at-Risk by VaR α (F n ) = ∂L n (α), for α ∈ (0, 1).

Moreover, under assumptions of Theorem 2.1, we obtain a consistency result, with respect to the Hausdorff distance, for the VaR α (F n ) on the quadrant R 2 + i.e.

d H (VaR α (F ) Tn , VaR α (F n ) Tn ) = O( F -F n ∞ ), a.s.
As in the univariate case, the two-dimensional VaR at a predetermined level α does not give any information about thickness of the upper tail of the distribution function. This is a considerable shortcoming of VaR measure because in practice we are not only concerned with frequency of the default but also with the severity of loss in case of default. In other words we are interested to analyze the behavior of X and Y not only on the boundary but also in the whole α-level set.

In dimension one, in order to overcome this problem, another risk measure has recently received growing attention in insurance and finance literature: Conditional Tail Expectation (CTE). Following Artzner et al. [START_REF] Artzner | Coherent measures of risk[END_REF] and Dedu and Ciumara [START_REF] Dedu | Restricted optimal retention in stop-loss reinsurance under VaR and CTE risk measures[END_REF], for a continuous loss distribution function F X the CTE at level α is defined by

CTE α (X) = E[ X | X ≥ VaR α (X) ],
where VaR α (X) is the univariate VaR in [START_REF] Biau | A graph-based estimator of the number of clusters[END_REF]. For a comprehensive treatment and for references to the extensive literature on VaR α (X) and CTE α (X) one may refer to Denuit et al. [START_REF] Denuit | Actuarial Theory for Dependent Risks[END_REF].

Several bivariate generalizations of the classical univariate CTE have been proposed; mainly using as conditioning events the total risk or some univariate aggregate extreme risk in the portfolio (for instance the sum, the max or the min between two risks):

E[ (X, Y ) | X + Y > Q X+Y (α) ], E[ (X, Y ) | min{X, Y } > Q min{X,Y } (α) ] and E[ (X, Y ) | max{X, Y } > Q max{X,Y } (α) ].
The interested reader is referred to Cai and Li [START_REF] Cai | Conditional tail expectations for multivariate phase-type distributions[END_REF] for further details. Using the same approach of Definition 4.1, we are interested to preserve the complete information about dependence structure between X and Y and the bi-dimensional aspect of our problem. To this end, starting from these general considerations, we propose to study a new bivariate version of the Conditional Tail Expectation (Definition 4.2 below).

Let us first introduce the following assumption:

A2: X and Y are positive random variables fulfilling E(X 2 ) < ∞ and E(Y 2 ) < ∞. The random vector (X, Y ) has a λ-density function f such that f 1+r dλ < ∞, for some r > 0.

Definition 4.2. Consider a random vector (X, Y ) satisfying Assumption A2, with associate distribution function F ∈ F. For α ∈ (0, 1), we define

(1) the bivariate α-Conditional Tail Expectation

CTE α (X, Y ) = E[(X, Y )|(X, Y ) ∈ L(α)] = E[ X | (X, Y ) ∈ L(α) ] E[ Y | (X, Y ) ∈ L(α) ] .
(2) the estimated bivariate α-Conditional Tail Expectation

CTE α (X, Y ) =      n i=1 X i 1 {(Xi,Yi)∈Ln(α)} n i=1 1 {(Xi,Yi)∈Ln(α)} n i=1 Y i 1 {(Xi,Yi)∈Ln(α)} n i=1 1 {(Xi,Yi)∈Ln(α)}      . (6)
Note that this bivariate Conditional Tail Expectation is a natural extension of the univariate one. Moreover, if X and Y are identically distributed with a symmetric copula then E

[ X | (X, Y ) ∈ L(α) ] = E[ Y | (X, Y ) ∈ L(α) ].
Contrarily to the generalizations of the univariate CTE above, our CTE α (X, Y ) does not use an aggregate variable (sum, min, max . . .) in order to analyze the bivariate risk's issue. By contrast, using a geometric approach, CTE α (X, Y ) rather deals with the simultaneous joint damages considering the bivariate dependence structure of data in a specific risk's area (L(α)).

Let α ∈ (0, 1) and F ∈ F. We introduce truncated versions of the theoretical and estimated CTE α :

CTE T α (X, Y ) = E[(X, Y )|(X, Y ) ∈ L(α) T ], CTE T α (X, Y ) =      n i=1 X i 1 {(Xi,Yi)∈Ln(α) T } n i=1 1 {(Xi,Yi)∈Ln(α) T } n i=1 Y i 1 {(Xi,Yi)∈Ln(α) T } n i=1 1 {(Xi,Yi)∈Ln(α) T }      ,
where L(α) T and L n (α) T are the truncated versions of theoretical and estimated upper α-level set defined in Section 1.

Theorem 4.1. Under Assumption A2, Assumptions of Theorem 3.1 and with the notation of Theorem 3.1, it holds that

β n CTE Tn α (X, Y ) -CTE Tn α (X, Y ) P → n→∞ 0, ( 7
)
where

β n = min{ p r 2(1+r) n , a n }, with r > 0 satisfying Assumption A2 and a n = o ( √ n).
The convergence in [START_REF] Cadre | Kernel estimation of density level sets[END_REF] must be interpreted as a componentwise convergence. In the case of a bounded density function f (X, Y ) we obtain β n = min{ √ p n , a n }.

Using Theorem 4.1 and Remark 4, we obtain the following result in the case of the bivariate empirical distribution function. ), with a choice of sequence T n = ln(n). We remark that (P[X ≥ T n or Y ≥ T n ]) -1 is increasing in T n , whereas the speed of convergence is decreasing in T n . This kind of compromise provides an illustration on how to choose T n , apart from satisfying the assumptions of consistency results in Section 3 (see Section 5.2.1 for several numerical illustrations of this aspect).

Illustrations

Estimation of the level sets

In this section we confront our estimator of level sets of the distribution function with various simulated samples. The plug-in estimation of level sets is constructed using the empirical estimator F n . We consider two distribution functions which satisfy assumptions of Theorem 3.1: independent copula with exponential marginals and Survival Clayton copula with Burr marginals.

Following Remark 4 we consider T n = ln(n) and we obtain p n = o(n

1 3 / ln(n) 4
3 ). We also take T n = n 0.45 . This second sequence provides an illustration for the case of T n too large (see Tables 1 We take a random grid of 10000 points in [0, T n ] 2 and a distribution with independent and exponentially distributed marginals with parameter 1 and 2 respectively. We provide a Monte Carlo approximation for λ(L(α) Tn △ L n (α) Tn ) (averaged on 100 iterations), for different values of α and n. The results are gathered in Table 1. 1. Distribution with independent and exponentially distributed marginals with parameter 1 and 2 respectively. Approximated λ(L(α) Tn △ L n (α) Tn ).

In order to illustrate the influence of the choice of T n , in Table 2 we show p n λ(L(α) Tn △ L n (α) Tn ), with p n = o(n

1 3 / ln(n) 4
3 ) in the case T n = ln(n) (Table 2(a)) and

T n = n 0.45 (Table 2(b)). Results in Table 2(a) set out how p n = o(n 1 3 / ln(n) 4
3 ) is at least the convergence rate of λ(L(α) Tn △ L n (α) Tn ) in this particular case. This illustrates not only the consistency property but also the speed of convergence of our plug-in estimator. As expected in Table 2(b) we see that taking T n too large, we can not expect to get a good approximation of λ(L(α) Tn △ L n (α) Tn ).

3 ).

In Tables 3(a) and 3(b) we calculate the theoretical λ(L(α) Tn ), for T n = ln(n) and T n = n 0.45 respectively, and for different values of n and α. These quantities provide an illustration of the truncation of level sets L(α), with respect to T n and α. Table 4 provides a Monte Carlo approximation for λ(L(α) Tn △ L n (α) Tn ) (averaged on 100 iterations), in the case of a Survival Clayton copula with parameter 1 and Burr(2, 1) marginals. In Table 5 we show p n λ(L(α) Tn △ L n (α) Tn ), with p n = o(n

1 3 / ln(n) 4
3 ) in the case T n = ln(n) (Table 5(a)) and T n = n 0.45 (Table 5(b)). We obtain results that are analogous to those Table 2.

3 ).

As expected, the greater n is, the better the estimations are (see Tables 1(a) and 4(a)). Moreover we note that for big values of α we need more data to get a good estimation of the level sets. This may come from the fact that when α grows the gradient of the distribution function decreases to zero and the constant A grows significantly (see proof of Theorem 3.1 in Section 7).

Remark 6. As remarked in Section 3 the choice of the sequence T n is essentially a compromise: if T n is large we have a quite slow convergence rate p n for λ(L(α) Tn △ L n (α) Tn ) and if T n is quite small we obtain a faster convergence rate. However choosing T n too small we substantially truncate our level sets. This aspect can be problematic in the case c is close to one. Finally the theoretical optimal criterion for the choice of sequence T n in Theorem 3.

1 is an open interesting problem that we leave for a future work.

5.2. Estimation of CTE α (X, Y )

Simulated data

In order to evaluate the performance of our estimator we present here some simulated cases of estimation of CTE α (X, Y ), for different values of level α. To compare the estimated results with the theoretical ones we consider cases for which we can calculate (with Maple) the explicit value of the theoretical CTE α (X, Y ). However our estimator can be applied to much more general cases. By being able to get a good estimator for the level sets, we can expect a good estimator for CTE α (X, Y ).

In the following we compare CTE Tn α (X, Y ) with the theoretical CTE α (X, Y ), for different distributions: Independent copula with exponentially distributed marginals; Clayton copula with parameter 1, with exponential and Burr(4, 1) univariate marginals. The plug-in estimation of level sets is constructed using the empirical estimator F n of the bivariate distribution function, with n = 1000.

In the following we denote CTE α Tn (X, Y ) = CTE α Tn,1 (X, Y ), CTE α Tn,2 (X, Y ) the mean (coordinate by coordinate) of CTE Tn α (X, Y ) on 100 simulations. We denote σ = ( σ 1 , σ 2 ) the empirical standard deviation (coordinate by coordinate) with

σ 1 = 1 99 100 j=1 CTE α Tn,1 (X, Y ) j -CTE α Tn,1 (X, Y ) 2
relatives to the first coordinate (resp. σ 2 relatives to the second one).

We denote RMSE = (RMSE 1 , RMSE 2 ) the relative mean square error (coordinate by coordinate) with

RMSE 1 = 1 100 100 j=1 CTE Tn ,1 α (X,Y )j -CTE Tn ,1 α (X,Y ) CTE Tn ,1 α (X,Y ) 2
relatives to the first coordinate of CTE Tn α (X, Y ) (resp. RMSE 2 relatives to the second one).

For the Independent copula with exponentially distributed marginals, from Corollary 4.1 and Remark 5, we take T n = ln(n) (see Tables 6 and8). This choice provides a convergence rate β n = o(n

1 6 / ln(n) 2
3 ). Furthermore, as in Section 5.1, we consider the case with T n = n 0.45 . This second sequence provides an illustration for the case of T n too large (see Tables 7 and9). Following the same approach of Remark 5, in the case of Clayton copula with parameter 1, with exponential and Burr(4, 1) univariate marginals we have

β n = o(n 1 6 /(n 0.2 ) 2 3
), with a sequence T n = n 0.2 (see Table 10). Also in this case, we show the behavior of our estimator for T n = n 0.45 (see Table 11. (X, Y ) with Clayton copula with parameter 1, F X exponential distribution with parameter 1, F Y Burr(4, 1) distribution; T n = n 0.45 .

Following we show that for high levels (here α = 0.9), one needs to use large samples to get reasonable estimates of CTE α . We consider for the purpose (X, Y ) independent and exponentially distributed with respective parameters 1 and 2. The theoretical value is CTE 0.9 (X, Y ) = (3.78, 1.89). We take T n = ln(n) (see Table 12).

In this case we need between 2000 and 2500 data to get the same performances as for lower level (see Table 8 above). Furthermore we consider the case T n = n 0.45 (see Table 13). In this case we need between 2500 and 5000 data to get the same performances as for lower level (see 13. Evolution of σ and RMSE in terms of sample size n for α = 0.9; (X, Y ) independent and exponentially distributed components with parameter 1 and 2 respectively. T n = n 0.45 .

The bad properties of the estimate for sample sizes less than 2000 (in the case of T n = ln(n)) can be explained by the fact that for high levels, the constant A is large (see proof of Theorem 3.1 in Section 7), but also by the fact that for large values of α the empirical estimate F n of F may not be the most appropriate.

Real data

We consider here the estimation of CTE α in a real case: Loss-ALAE data in the log scale (for details see Frees and Valdez [START_REF] Frees | Understanding relationships using copulas[END_REF]). Each claim consists of an indemnity payment (the loss, X) and an allocated loss adjustment expense (ALAE, Y ). Examples of ALAE are the fees paid to outside attorneys, experts, and investigators used to defend claims.

The data size is n = 1500. Let T n = n 0.4 . Again our estimator is constructed using the empirical estimator F n . Results about the estimation of CTE Tn α are gathered in Table 14. In Figure 2 In this real setting the estimation of CTE α can be used in order to quantify the mean of the Loss (resp. ALAE) in the log scale conditionally to the fact that Loss and ALAE data belong jointly to the specific risk's area L(α). (star) for different values of α.

Conclusion

In this paper we have provided convergence results for the plug-in estimator of the level sets of an unknown distribution function on R 2 + in terms of Hausdorff distance and volume of the symmetric difference. In this setting we have proposed and estimated a new bivariate risk measure: CTE α (X, Y ). A future work comparing our bivariate Conditional Tail Expectation with existing bivariate generalizations of the CTE in terms of classical properties (monotonicity, translation invariance, homogeneity, . . .), dependence structure, behavior with respect to different risk scenarios, is in preparation.

Proofs

Proof of Proposition 1.1 Take T > 0 such that for all t : | tc | ≤ r, {F = t} T = ∅, (from assumptions of Proposition 1.1 we know that such an T there exists).

Let x ∈ {z ∈ [0, T ] 2 : | F (z) -c |≤ r}. Define for λ ∈ R y λ ≡ y λ,x = x + λ (∇F ) x (∇F ) x , so y λ -x =| λ |.
From the differentiability properties of F and using Taylor's formula we have, for

| λ |< ζ F (y λ ) = F (x) + (∇F ) T x (y λ -x) + 1 2 (y λ -x) T (HF ) x (y λ -x),
with x a point on the segment between x and y λ . So

F (y λ ) = F (x) + λ (∇F ) x + λ 2 2 (∇F ) x 2 (∇F ) T x (HF ) x (∇F ) x .
By the Cauchy Schwarz Inequality we obtain

F (y λ ) ≥ F (x) + λ (∇F ) x - λ 2 2 (∇F ) x (HF ) x (∇F ) x and
F (y λ ) ≤ F (x) + λ (∇F ) x + λ 2 2 (∇F ) x (HF ) x (∇F ) x .
Since (HF ) x (∇F ) x ≤ (HF ) x (∇F ) x , we have

F (x) + λ (∇F ) x - λ 2 2 (HF ) x ≤ F (y λ ) ≤ F (x) + λ (∇F ) x + λ 2 2 (HF ) x . Since x ∈ E and M H < ∞ we obtain F (x) + λ (∇F ) x - λ 2 2 M H ≤ F (y λ ) ≤ F (x) + λ (∇F ) x + λ 2 2 M H . (8)
For 0 < λ < ζ, we have from the left side of ( 8)

F (y λ ) ≥ F (x) + λ (∇F ) x - λ 2 2 M H ≥ F (x) + λm ▽ - λ 2 2 M H .
We assume now that M H > 0 (the case M H = 0 is trivial).

For x ∈ {z ∈ [0, T ] 2 : | F (z) -c |≤ r} and 0 < λ < ζ ∧ m ▽ M H we have then F (y λ ) ≥ F (x) + λ 2 m ▽ , (9)
Similarly using the right side of ( 8) we obtain for 0

< λ < ζ ∧ m ▽ M H and x ∈ {z ∈ [0, T ] 2 : | F (z) -c |≤ r}, F (y -λ ) ≤ F (x) - λ 2 m ▽ . ( 10
) Define γ = m ▽ 4 ζ ∧ m ▽ M H ∧ r > 0. Suppose that t = c + ε, 0 < ε ≤ γ. Let x ∈ [0, T ] 2 such that F (x) = t = c + ε then x ∈ {z ∈ [0, T ] 2 : | F (z) -c |≤ r}. Take now 0 < λ = 2ε m ▽ < ζ ∧ m ▽ M H .
We obtain from ( 10)

F (y -λ ) ≤ F (x) - λ 2 m ▽ = c + ε -ε = c.
From the continuity property of F , we deduce that there exists y between x and y -λ such that F (y) = c and we have

x -y ≤ x -y -λ =| λ |= 2ε m ▽ = 2 m ▽ | t -c | . So we have proved that sup x∈{F =t} T d(x, {F = c} T ) ≤ 2 m ▽ | t -c | .
Similarly, take x ∈ [0, T ] 2 such that F (x) = c and use [START_REF] Cavalier | Nonparametric estimation of regression level sets[END_REF] to obtain sup

x∈{F =c} T d(x, {F = t} T ) ≤ 2 m ▽ | t -c | .
The proof in case t < c is completely analogous. So F satisfies Assumption H (see Section 1) with A = 2 m ▽ .

Proof of Theorem 2.1

Under assumptions of Theorem 2.1, we can always take T 1 > 0 such that for all t : | tc | ≤ r, ∂L(t) T1 = ∅. Then for each n, for all t : | tc | ≤ r, ∂L(t) Tn is a non-empty (and compact) set on R 2 + . In each [0, T n ] 2 , from Proposition 1.1, Assumption H (Section 1) is satisfied with

γ = m ▽ 4 ζ ∧ m ▽ M H ∧ r > 0 and A = 2 m ▽ .
First we have to find a bound for sup x∈∂L(c) Tn d(x, ∂L n (c) Tn ).

Take x ∈ ∂L(c) Tn and define ε n = 2 F -F n Tn ∞ . Using F -F n ∞ → 0, a.s., for n → ∞, then ε n → 0, a.s., for n → ∞. So with probability one there exists n 0 such that ∀ n ≥ n 0 , ε n ≤ γ.

Since for all t : | tc | ≤ r ∂L(t) Tn = ∅, from Assumption H, there exist u n ≡ u εn x and

l n ≡ l εn x in [0, T n ] 2 such that F (u n ) = c + ε n ; d(x, u n ) ≤ A ε n , F (l n ) = c -ε n ; d(x, l n ) ≤ A ε n .
Suppose now F -F n Tn ∞ > 0 (the other case is a trivial one). In this case

F n (u n ) = c + ε n + F n (u n ) -F (u n ) ≥ c + ε n -F -F n Tn ∞ = c + 2 F -F n Tn ∞ -F -F n Tn ∞ > c,
and in a similar way we can prove that F n (l n ) < c.

As F n (l n ) < c and F n (u n ) > c, with u n and l n in [0, T n ] 2 , there exists

z n ∈ ∂L n (c) Tn ∩ B(u n , d(u n , l n )) with d(z n , x) ≤ d(z n , u n )+d(u n , x) ≤ d(u n , l n )+d(u n , x) ≤ d(u n , x)+d(x, l n )+d(u n , x) ≤ 3 A ε n = 6 A F -F n Tn ∞ .
Hence, for n ≥ n 0 sup

x∈∂L(c) Tn d(x, ∂L n (c) Tn ) ≤ 6 A F -F n Tn ∞ .
Let us now bound sup x∈∂Ln(c) Tn d(x, ∂L(c) Tn ).

Take x ∈ ∂L n (c) Tn . From the a.s. continuity of F n we obtain F n (x) = c, a.s., so

| F (x) -c | ≤ | F (x) -F n (x) | ≤ F -F n Tn ∞ ≤ ε n , a.s. Remember that ∀ n ≥ n 0 , ε n ≤ γ, a.s. Then from Assumption H d(x, ∂L(c) Tn ) ≤ A | F (x)-c |≤ A F -F n
Take ε n such that

ε n = p n v n 1 p . ( 14
)
Then, from Assumption A1, in Section 3, we obtain p n λ( Q εn ) We only prove the result for the first coordinate of CTE α (X, Y ) (the proof is similar for the second one). First we remark that, in the following, probabilities involving events which depend on L n (α) Tn , are conditional probabilities to the sample (X i , Y i ), for i = 1, . . . n. We will denote P[(X, Y ) ∈ L n (α) Tn ] purely for reasons of notational convenience.

We introduce these two preliminary results (Lemma 7.1 and 7. Using ( 15)-( 16) we obtain that (17) converges to zero in probability, for n → ∞. Hence the result. Hence the result. Then to prove Theorem 4.1 we can write (7) as

Figure 1 .

 1 Figure 1. (left) Hausdorff distance between sets; (right) particular case where d H (X, Y ) does not capture the shape properties of X and Y .

Remark 4 .

 4 Let F n the bivariate empirical distribution function. Then it holds that v n F -F n ∞ P → n→∞ 0, with v n = o( √ n). From Theorem 3.1 and using the convergence rate in (3) with p = 2, we obtain p n = o(n

Corollary 4 . 1 .,Remark 5 . 1 6

 4151 Let F n the bivariate empirical distribution function. Under Assumptions Theorem 4.1 and with the notation of Theorem 4.1, it holds that β n CTE Tn α (X, Y ) -CTE Tn α (X, Y ) with r > 0 satisfying Assumption A2.In the case of a bounded density function f (X,Y ) , β n = o n Starting from Corollary 4.1, it could be interesting to consider the convergenceCTE α (X, Y ) -CTE Tn α (X, Y ) .We remark that in this case the speed of convergence will also depend on the convergence rate to zero of CTEα (X, Y ) -CTE Tn α (X, Y ) , then, in particular of P[(X, Y ) ∈ L(α) \ L(α) Tn ] for n → ∞. More precisely CTE α (X, Y ) -CTE Tnα (X, Y ) decays to zero at least with a convergence rate (P[X ≥ T n or Y ≥ T n ]) -1 . We now provide an illustration (in a simple case) on how to choose T n in order toconsider the convergence CTE α (X, Y ) -CTE Tn α (X, Y ) . Let F (x, y) = (1e -x )(1e -y ), i.e. independent and exponentially distributed marginals with parameter 1, with bounded density function in R 2 + . Let F n the bivariate empirical distribution function. In this case all assumptions of Corollary 4.1 are satisfied. Then we obtain that CTE α (X, Y ) -CTE Tn α (X, Y ) decays to zero at least with a convergence rate β n = o(n / ln(n) 2 3

  (b)-2(b) and 4(b)-5(b)).

  (a) Tn = ln(n) α n= 500 n= 1000 n= 2000 0.10 0

( a )

 a Tn = ln(n) α n= 500 n= 1000 n= 2000 0.10 0

Figure 2 .

 2 Figure 2. Loss-ALAE data in log scale: boundary of estimated level sets (line) and CTE Tn α

2 |

 2 can conclude that with probability one, for n ≥ n 0 sup x∈∂Ln(c) Tn d(x, ∂L(c) Tn ) ≤ A F -F n Tn ∞ . We obtain for n ≥ n 0 , d H (∂L(c) Tn , ∂L n (c) Tn ) ≤ 6 A F -F n Tn ∞ , then d H (∂L(c) Tn , ∂L n (c) Tn ) = O( F -F n Tn ∞ ), a.s. Hence the result. Proof of Theorem 3.1Under assumptions of Theorem 3.1, we can always take T 1 > 0 such that for all t : | tc | ≤ r, ∂L(t) T1 = ∅. Then for each n, for all t : | tc | ≤ r, ∂L(t) Tn is a non-empty (and compact) set on R 2 + . We consider a positive sequence ε n such that ε n → n→∞ 0. For each n ≥ 1 the random sets L(c) Tn △ L n (c) Tn ,Q εn = {x ∈ [0, T n ] 2 : | F -F n |≤ ε n } and Q εn = {x ∈ [0, T n ] 2 : | F -F n |> ε n } are measurable and λ(L(c) Tn △ L n (c) Tn ) = λ(L(c) Tn △ L n (c) Tn ∩ Q εn ) + λ(L(c) Tn △ L n (c) Tn ∩ Q εn ). (11) Since L(c) Tn △ L n (c) Tn ∩ Q εn ⊂ {x ∈ [0, T n ] 2 : cε n ≤ F < c + ε n } we obtain λ(L(c) Tn △ L n (c) Tn ) ≤ λ({x ∈ [0, T n ] 2 : cε n ≤ F < c + ε n }) + λ( Q εn ).From Assumption H (Section 1) and Proposition 1.1, if 2 ε n ≤ γ then d H (∂L(c + ε n ) Tn , ∂L(cε n ) Tn ) ≤ 2 ε n A. So we can write λ({x ∈ [0, T n ] 2 : cε n ≤ F < c + ε n }) ≤ 2 ε n A 2 T n . If we now choose ε n = o 1 p n T n(12)we obtain that for n large enough 2 ε n ≤ γ andp n λ({x ∈ [0, T n ] 2 : cε n ≤ F < c + ε n }) → n→∞ 0.Let us now prove that p n λ( Q εn ) write p n λ( Q εn ) = p n 1 {x∈[0,Tn] 2 : |F -Fn|>εn} λ(dx) F -F n | p λ(dx).

  ε n that satisfies (12) and (14). Hence the result.Proof of Theorem 4.1

2 ): Lemma 7 . 1 . 1 From 1 2

 27111 Under Assumption A2, Assumptions of Theorem 3.1 and with the notation of Theorem 3.1, it holds thatp r 2(1+r) n E[ X | (X, Y ) ∈ L(α) Tn ] -E[ X | (X, Y ) ∈ L n (α) Tn ] r > 0 satisfying Assumption A2.Proof of Lemma 7.Assumption A2 and Theorem 3.1 we obtainp r 2(1+r) n P[(X, Y ) ∈ L(α) Tn △ L n (α) Tn ] ≤ p r 2(1+r) n d λ L(α) Tn , L n (α) Tn r X, Y ) ∈ L(α) Tn ] -P[(X, Y ) ∈ L n (α) Tn ] Tn x f X,Y (x, y)λ(dx dy) -Ln(α) Tnx f X,Y (x, y)λ(dx dy) d λ L(α) Tn , L n (α)Tn

1 -

 1 (X, Y ) ∈ L(α) Tn ] -E[X|(X, Y ) ∈ L n (α) Tn ] Tn x f X,Y (x, y)λ(dx dy) P[(X, Y ) ∈ L(α) Tn ] -Ln(α) Tn x f X,Y (x, y)λ(dx dy) P[(X, Y ) ∈ L n (α) Tn ] X, Y ) ∈ L(α) Tn ] P[(X, Y ) ∈ L n (α) Tn ] P[(X, Y ) ∈ L(α) Tn ] L(α) Tnx f X,Y (x, y)λ(dx dy) -Ln(α) Tn x f X,Y (x, y)λ(dx dy) + L(α) Tn x f X,Y (x, y)λ(dx dy) P[(X, Y ) ∈ L(α) Tn ] -P[(X, Y ) ∈ L n (α) Tn ] .(17)

Lemma 7 . 2 . 2 X i 1 { 1 {

 72211 Under Assumption A2, Assumptions of Theorem 3.1 and with the notation of Theorem 3.1, it holds thata n E[ X | (X, Y ) ∈ L n (α) Tn ] -n i=1 X i 1 {(Xi,Yi)∈Ln(α) Tn } n i=1 1 {(Xi,Yi)∈Ln(α) Tn }We can writea n E[X|(X, Y ) ∈ L n (α) Tn ] -n i=1 (Xi,Yi)∈Ln(α) Tn } n i=1 1 {(Xi,Yi)∈Ln(α) Tn } = a n Ln(α) Tn x f X,Y (x, y)λ(dx dy) P[(X, Y ) ∈ L n (α) Tn ] -n i=1 X i 1 {(Xi,Yi)∈Ln(α) Tn } n i=1 (Xi,Yi)∈Ln(α) Tn } .Under assumptions of Lemma 7.2, from the central limit theorem for triangular arrays (e.g. Theorem 27.2 in Billingsley[START_REF] Billingsley | Probability and measure[END_REF]) we obtaina n P[(X, Y ) ∈ L n (α) Tn ] -Tn x f X,Y (x, y)λ(dx dy) -1 n n i=1X i 1 {(Xi,Yi)∈Ln(α) Tn } P → n→∞ 0.

Table 2 .

 2 Distribution

				(b) Tn = n 0.45
				α n= 500 n= 1000 n= 2000
	.069	0.068	0.065	0.10 0.229	0.248	0.188
	0.24 0.156	0.134	0.063	0.24 0.361	0.298	0.209
	0.38 0.172	0.139	0.121	0.38 0.411	0.357	0.339
	0.52 0.225	0.169	0.183	0.52 0.734	0.689	0.743
	0.66 0.298	0.199	0.195	0.66 0.849	0.752	0.762
	0.80 0.426	0.282	0.279	0.80 1.071	1.039	1.124

with independent and exponentially distributed marginals with parameter 1 and 2 respectively. Approximated p n λ(L(α) Tn △ L n (α) Tn ), with p n = o(n 1

3

/ ln(n)

Table 3 .

 3 Theoretical λ(L(α) Tn ), for different values of n and α. Distribution with independent and exponentially distributed marginals with parameter 1 and 2 respectively.

				(b) Tn = n 0.45
				α n= 500 n= 1000 n= 2000
	10 37.472	46.459	56.405	0.10 142.205 248.516 433.872
	0.24 35.777	44.588	54.358	0.24 139.039 244.378 428.449
	0.38 33.837	42.436	51.995	0.38 135.328 239.495 422.021
	0.52 31.505	39.837	49.129	0.52 130.771 233.464 414.045
	0.66 28.513	36.487	45.421	0.66 124.779 225.488 403.451
	0.80 24.201	31.621	40.003	0.80 115.851 213.505 387.436

Table 4 .

 4 Distribution with Survival Clayton copula with parameter 1 and Burr(2, 1) marginals. Approximated λ(L(α) Tn △ L n (α) Tn ).

				(b) Tn = n 0.45
				α n= 500 n= 1000 n= 2000
	.208	0.171	0.126	0.10 0.697	0.633	0.536
	0.24 0.252	0.241	0.149	0.24 0.893	0.872	0.809
	0.38 0.339	0.288	0.132	0.38 0.971	0.911	0.879
	0.52 0.349	0.335	0.245	0.52 1.001	0.982	1.229
	0.66 0.499	0.364	0.353	0.66 1.569	1.522	1.413
	0.80 0.771	0.644	0.605	0.80 2.377	2.269	2.175
	(a) Tn = ln(n)		(b) Tn = n 0.45
	α n= 500 n= 1000 n= 2000	α n= 500 n= 1000 n= 2000
	0.10 0.144	0.129	0.106	0.10 0.484	0.481	0.452
	0.24 0.175	0.183	0.126	0.24 0.620	0.663	0.682
	0.38 0.235	0.219	0.111	0.38 0.674	0.692	0.741
	0.52 0.242	0.254	0.206	0.52 0.695	0.746	1.036
	0.66 0.347	0.277	0.297	0.66 1.089	1.157	1.191
	0.80 0.535	0.489	0.501	0.80 1.651	1.725	1.834

Table 5 .

 5 Distribution

with Survival Clayton copula with parameter 1 and Burr(2, 1) marginals. Approximated p n λ(L(α) Tn △ L n (α) Tn ), with p n = o(n 1 3 / ln(n)

Table 11 )

 11 .

	α	CTE α (X, Y )	CTE α	Tn	(X, Y )	σ	RMSE
	0.10 (0.627, 0.627)	(0.585, 0.637)	(0.022, 0.021) (0.035, 0.039)
	0.24 (0.761, 0.761)	(0.766, 0.759)	(0.031, 0.031) (0.041, 0.046)
	0.38 (0.896, 0.896)	(0.864, 0.917)	(0.036, 0.037) (0.053, 0.049)
	0.52 (1.051, 1.051)	(1.027, 1.071)	(0.051, 0.053) (0.055, 0.052)
	0.66 (1.246, 1.246)	(1.252, 1.238)	(0.075, 0.067) (0.063, 0.065)
	0.80 (1.531, 1.531)	(1.497, 1.549)	(0.123, 0.117) (0.083, 0.077)

Table 6 .

 6 (X, Y ) with independent and exponentially distributed components with parameter 2; T n = ln(n).

	α	CTE α (X, Y )	CTE α	Tn	(X, Y )	σ	RMSE
	0.10 (0.627, 0.627)	(0.603, 0.656)	(0.031, 0.031) (0.062, 0.068)
	0.24 (0.761, 0.761)	(0.774, 0.731)	(0.061, 0.071) (0.082, 0.130)
	0.38 (0.896, 0.896)	(0.927, 0.871)	(0.072, 0.076) (0.087, 0.119)
	0.52 (1.051, 1.051)	(1.086, 1.128)	(0.094, 0.082) (0.095, 0.107)
	0.66 (1.246, 1.246)	(1.281, 1.322)	(0.127, 0.101) (0.102, 0.101)
	0.80 (1.531, 1.531)	(1.545, 1.611)	(0.157, 0.161) (0.105, 0.117)

Table 7 .

 7 (X, Y ) with independent and exponentially distributed components with parameter 2; T n = n 0.45 .

	α	CTE α (X, Y )	CTE α	Tn	(X, Y )	σ	RMSE
	0.10 (1.255, 0.627)	(1.222, 0.638)	(0.044, 0.022) (0.043, 0.039)
	0.24 (1.521, 0.761)	(1.488, 0.811)	(0.069, 0.023) (0.051, 0.042)
	0.38 (1.792, 0.896)	(1.797, 0.911)	(0.075, 0.038) (0.044, 0.046)
	0.52 (2.102, 1.051)	(2.082, 1.047)	(0.104, 0.052) (0.052, 0.052)
	0.66 (2.492, 1.246)	(2.461, 1.255)	(0.139, 0.071) (0.057, 0.056)
	0.80 (3.061, 1.531)	(3.011, 1.544)	(0.251, 0.125) (0.084, 0.082)

Table 8 .

 8 (X, Y ) with independent and exponentially distributed components with parameter 1 and 2 respectively; T n = ln(n).

	α	CTE α (X, Y )	CTE α	Tn	(X, Y )	σ	RMSE
	0.10 (1.255, 0.627)	(1.253, 0.624)	(0.061, 0.043) (0.051, 0.054)
	0.24 (1.521, 0.761)	(1.514, 0.803)	(0.074, 0.049) (0.048, 0.075)
	0.38 (1.792, 0.896)	(1.793, 0.948)	(0.096, 0.055) (0.053, 0.084)
	0.52 (2.102, 1.051)	(2.087, 1.111)	(0.118, 0.076) (0.056, 0.092)
	0.66 (2.492, 1.246)	(2.497, 1.311)	(0.169, 0.108) (0.068, 0.101)
	0.80 (3.061, 1.531)	(3.056, 1.602)	(0.313, 0.153) (0.102, 0.111)

Table 9 .

 9 (X, Y ) with independent and exponentially distributed components with parameter 1 and 2 respectively; T n = n 0.45 .

	α	CTE α (X, Y )	CTE α	Tn	(X, Y )	σ	RMSE
	0.10 (1.188, 1.229)	(1.049, 1.192)	(0.032, 0.021) (0.019, 0.033)
	0.24 (1.448, 1.366)	(1.283, 1.379)	(0.053, 0.224) (0.019, 0.063)
	0.38 (1.727, 1.505)	(1.525, 1.471)	(0.046, 0.031) (0.019, 0.031)
	0.52 (2.049, 1.666)	(1.803, 1.625)	(0.058, 0.041) (0.023, 0.034)
	0.66 (2.454, 1.875)	(2.129, 1.823)	(0.071, 0.054) (0.035, 0.039)
	0.80 (3.039, 2.202)	(2.591, 2.144)	(0.111, 0.105) (0.055, 0.054)

Table 10 .

 10 (X, Y ) with Clayton copula with parameter 1, F X exponential distribution with parameter 1, F Y Burr(4, 1) distribution; T n = n 0.2 .

	α	CTE α (X, Y )	CTE α	Tn	(X, Y )	σ	RMSE
	0.10 (1.188, 1.229)	(1.189, 1.238)	(0.061, 0.035) (0.039, 0.029)
	0.24 (1.448, 1.366)	(1.462, 1.365)	(0.065, 0.037) (0.046, 0.031)
	0.38 (1.727, 1.505)	(1.751, 1.536)	(0.082, 0.046) (0.049, 0.037)
	0.52 (2.049, 1.666)	(2.063, 1.713)	(0.091, 0.061) (0.051, 0.045)
	0.66 (2.454, 1.875)	(2.457, 1.951)	(0.117, 0.104) (0.057, 0.068)
	0.80 (3.039, 2.202)	(3.037, 2.322)	(0.192, 0.165) (0.063, 0.108)

Table 12 .

 12 Table 9 above). Evolution of σ and RMSE in terms of sample size n for α = 0.9; (X, Y ) independent and exponentially distributed components with parameter 1 and 2 respectively. T n = ln(n).

	n	500	1000	1500	2000	2500	5000
	σ	(0.614, 0.359) (0.444, 0.308) (0.431, 0.295) (0.377, 0.168) (0.241, 0.123) (0.216, 0.121)
	RMSE (0.168, 0.189) (0.123, 0.163) (0.115, 0.161) (0.099, 0.089) (0.077, 0.079) (0.063, 0.057)
	n	500	1000	1500	2000	2500	5000
	σ	(0.919, 0.419) (0.568, 0.319) (0.511, 0.294) (0.382, 0.239) (0.348, 0.223) (0.307, 0.151)
	RMSE (0.242, 0.221) (0.151, 0.172) (0.133, 0.165) (0.101, 0.144) (0.093, 0.129) (0.092, 0.108)
		Table

Table 14 .

 14 we represent data, estimated level sets and estimated bivariate Conditional Tail Expectation for several values of α. CTE Tn α for Loss-ALAE data in log scale, with different values of level α.

	α		0.10	0.24	0.38
	CTE	Tn α	(9.937, 9.252)	(10.361, 9.566)	(10.731, 9.728)
	α		0.52	0.66	0.80
	CTE	Tn α	(11.096, 10.011) (11.518, 10.315) (12.057, 10.758)

Acknowledgements:

The authors thank two anonymous referees for relevant remarks and constructive comments on a previous version of the paper. The authors thank Alberto Rodríguez-Casal for his fruitful comments. They thank also Stéphane Loisel and Ragnar Norberg for helpful discussions. This work has been partially supported by the French research national agency (ANR) under the reference ANR-08BLAN-0314-01. The fourth author thanks the BNP Paribas Cardif Chair "Management de la modélisation" for financial support. The views expressed in this document are the author's own and do not necessarily reflect those endorsed by BNP Paribas Cardif.

The result is a straightforward application of Lemma 7.1 and Lemma 7.2.


On rotationally invariant codes


Werner To

Ballmann

VOLUME ENTROPY RIGIDITY OF NON-POSITIVELY CURVED SYMMETRIC SPACES FRANC ¸OIS LEDRAPPIER

Keywords: 2000 Mathematics Subject Classification. 53C24, 53C20, 58J65 volume entropy, rank one manifolds

We characterize symmetric spaces of non-positive curvature by the equality case of general inequalities between geometric quantities.

Introduction

Let (M, g) be a closed connected Riemannian manifold, and π : ( M , g) → (M, g) its universal cover endowed with the lifted Riemannian metric. We denote p(t, x, y), t ∈ R + , x, y ∈ M the heat kernel on M , the fundamental solution of the heat equation ∂u ∂t = Div ∇u on M . Since we have a compact quotient, all the following limits exist as t → ∞ and are independent of x ∈ M : where B M (x, t) is the ball of radius t centered at x in M and Vol is the Riemannian volume on M . All these numbers are nonnegative. Recall λ 0 is the Rayleigh quotient of M , ℓ the linear drift, h the stochastic entropy and v the volume entropy. There is the following relation:

λ 0 = inf f ∈C 2 c ( M ) |∇f | 2 |f | 2 =
(1) 4λ 0 (a) ≤ h (b) ≤ ℓv (c) ≤ v 2 .
See [L1] for (a), [Gu] for (b). Inequality (c) is shown in [L3] as a corollary of (b) and ( 2):

(2)

ℓ 2 ≤ h
If ( M , g) is a locally symmetric space of nonpositive curvature, all five numbers 4λ 0 , ℓ 2 , h, ℓv and v 2 coincide and are positive unless ( M , g) is (R n , Eucl.). Our result is a partial converse:

Theorem 1.1. Assume (M, g) has nonpositive curvature. With the above notation, any of the equalities ℓ = v, h = v 2 and 4λ 0 = v 2 hold if, and only if, ( M , g) is a symmetric space.

As recalled in [L3], Theorem 1.1 is known in negative curvature and follows from [K], [BFL], [FL], [BCG] and [L1]. The other possible converses are delicate: even for negatively curved manifolds, in dimension greater than two, it is not known that h = ℓv holds only for locally symmetric spaces. This is equivalent to a conjecture of Sullivan (see [L2] for a discussion). Sullivan conjecture holds for surfaces of negative curvature ([L1], [Ka]). It is not known either whether 4λ 0 = h holds only for locally symmetric spaces. This would follow from the hypothetical 4λ 0 (d) ≤ ℓ 2 by the arguments of this note.

We assume henceforth that (M, g) has nonpositive sectional curvature. Given a geodesic γ in M , Jacobi fields along γ are vector fields t → J(t) ∈ T γ(t) M which describe infinitesimal variation of geodesics around γ. By nonpositive curvature, the function t → J(t) is convex. Jacobi fields along γ form a vector space of dimension 2 Dim M . The rank of the geodesic γ is the dimension of the space of Jacobi fields such that t → J(t) is a constant function on R. The rank of a geodesic γ is at least one because of the trivial t → γ(t) which describes the variation by sliding the geodesic along itself. The rank of the manifold M is the smallest rank of geodesics in M . Using rank rigidity theorem ( [B1], [BS]), we reduce in section 2 the proof of Theorem 1.1 to proving that if (M, g) is rank one, equality in (2) implies that ( M , g) is a symmetric space. For this, we show in section 3 that equality in (2) implies that ( M , g) is asymptotically harmonic (see the definition below). This uses the Dirichlet property at infinity (Ballmann [B2]). Finally, it was recently observed by A. Zimmer ([Z]) that asymptotically harmonic universal covers of rank one manifolds are indeed symmetric spaces.

Generalities and reduction of Theorem 1.1

We recall the notations and results from Ballmann's monograph [B3] about the Hadamard manifold ( M , g) that we use. The space M is homeomorphic to a ball. The covering group G := π 1 (M ) satisfies the duality condition ([B3] page 45).

2.1. Boundary at infinity. Two geodesic rays γ, γ ′ in M are said to be asymptotic if sup t≥0 d(γ(t), γ ′ (t)) < ∞. The set of classes of asymptotic unit speed geodesic rays is called the boundary at infinity M (∞). M ∪ M (∞) is endowed with the topology of a compact space where M (∞) is a sphere and where, for each unit speed geodesic ray γ, γ(t) → [γ] as t → ∞. The action of the group G on M (∞) is the continuous extension of its action on M . For any x, ξ ∈ M × M (∞), there is a unique unit speed geodesic γ x,ξ such that γ x,ξ (0) = x and [γ x,ξ ] = ξ. The mapping ξ → γx,ξ (0) is a homeomorphism π -1

x between M (∞) and the unit sphere S x M in the tangent space at x to M . We will identify S M with M × M (∞) by (x, v) → (x, π x v). Then the quotient SM is identified with the quotient of M × M (∞) under the diagonal action of G.

Fix x 0 ∈ M and ξ ∈ M (∞). The Busemann function b ξ is the function on M given by:

b ξ (x) = lim y→ξ d(y, x) -d(y, x 0 ). Clearly, b gξ (gx) = b ξ (x) + b gξ (gx 0 ). Moreover, the function x → b ξ (x) is of class C 2 ([HI]). It follows that the fonction ∆ x b ξ satisfies ∆ gx b gξ = ∆ x b ξ and therefore defines a function B on G \ ( M × M (∞)) = SM .
It follows from the argument of [HI] that the function B is continuous on SM (see [B3], Proposition 2.8, page 69).

2.2. Jacobi fields. Let (x, v) be a point in T M . Tangent vectors in T x,v T M correspond to variations of geodesics and can be represented by Jacobi fields along the unique geodesic γ x,v with initial value γ(0) = x, γ(0) = v. A Jacobi field J(t), t ∈ R along γ x,v is uniquely determined by the values of J(0) and J ′ (0). We describe tangent vectors in T x,v T M by the associated pair (J(0),

J ′ (0)) of vectors in T x M . The metric on T x,v T M is given by (J 0 , J ′ 0 ) 2 = J 0 2 + J ′ 0 2 . Assume (x, v) ∈ SM . A vertical vector in T x,v S M is a vector tangent to S x M .
It corresponds to a pair (0, J ′ (0)), with J ′ (0) orthogonal to v. Horizontal vectors correspond to pairs (J(0), 0). In particular, let X be the vector field on S M such that the integral flow of X is the geodesic flow. The geodesic spray X x,v is the horizontal vector associated to (v, 0). The orthogonal space to X is preserved by the differential Dg t of the geodesic flow. More generally, the Jacobi fields representation of T T M satisfies D x,v g t (J(0), J ′ (0)) = (J(t), J ′ (t)).

For any vector Y ∈ T x M , there is a unique vector Z = S x,v Y such that the Jacobi field J with J(0) = Y, J ′ (0) = Z satisfies J(t) ≤ C for t ≥ 0 ([B3] Proposition 2.8 (i)). The mapping S x,v : T x M → T x M is linear and selfadjoint. The vectors (Y, SY ) describe variations of asymptotic geodesics and the subspace E s

x,v ⊂ T x,v T M they generate corresponds to T W s x,v , where W s x,v , the set of initial vectors of geodesics asymptotic to

γ x,v , is identified with M × π x (v) in M × M (∞). Observe that S x,v γx,v (0) = 0 and that the operator S x,v preserves ( γx,v (0)) ⊥ . Recall from [B3], Proposition 3.2 page 71, that, for Y ∈ ( γx,v (0)) ⊥ , with π x v = ξ, D Y (∇b ξ ) = -S x,v Y,
and therefore ∆ x b ξ = -Tr S x,v with π x (v) = ξ.
Similarly, there is a selfadjoint linear operator U x,v :

T x M → T x M such that the Jacobi field J with J(0) = Y, J ′ (0) = U Y satisfies J(t) ≤ C for t ≤ 0. The subspace E u x,v ⊂ T x,v T M they generate corresponds to T W u x,v
, where W u x,v is the set of opposite vectors to vectors in W s

x,-v . By definition, S γx,(0) = -U γx,-v (0) , so that we also have:

B(x, v) := -Tr S x,v = Tr U x,-v .
We have Ker S = Ker U and Y ∈ Ker S if, and only if, the Jacobi field J(t) with J(0) = Y, J ′ (0) = 0 is bounded for all t ∈ R. The rank of the geodesic γ x,v therefore is κ = Dim Ker S and the geodesic γ x,v is of rank one only if Det((U -S)| ( γx,v (0)) ⊥ ) = 0. Recall that SM is identified with the quotient of M × M (∞) under the diagonal action of G. Clearly, for g ∈ G, g(W s

x,v ) = W s Dg(x,v) so that the W s define a foliation W s on SM . The leaves of the foliation W s are quotient of M , they are naturally endowed with the Riemannian metric induced from g.

2.

3. Proof of Theorem 1.1. We continue assuming that ( M , g) has nonpositive curvature. By the Rank Rigidity Theorem (see [B3]), ( M , g) is of the form

( M 0 × M 1 × • • • × M j × M j+1 × • • • × M k , g) 1 ,
where g is the product metric

g 2 = ( g 0 ) 2 + ( g 1 ) 2 + • • • + ( g j ) 2 + ( g j+1 ) 2 + • • • + ( g k ) 2 , ( M 0 , g 0 ) is Euclidean, ( M i , g i ) is an irreducible symmetric space of rank at least two for i = 1, • • • , j and a rank-one manifold for i = j + 1, • • • , k. If the ( M i , g i ), i = j + 1, • • • k,
are all symmetric spaces of rank one, then ( M , g) is a symmetric space. Moreover in that case, all inequalities in (1) are equalities: this is the case for irreducible symmetric spaces (all numbers are 0 for Euclidean space; for the other spaces, 4λ 0 and v 2 are classically known to coincide ( [O]) and we have:

4λ 0 ( M ) = i 4λ 0 ( M i ), v 2 ( M ) = i v 2 ( M i ).
To prove Theorem 1.1, it suffices to prove that if ℓ 2 = h, all M i in the decomposition are symmetric spaces. This is already true for i = 0, 1, • • • j. It remains to show that ( M i , g i ) are symmetric spaces for i = j + 1, • • • k. Eberlein showed that each one of the spaces ( M i , g i ) admits a cocompact discrete group of isometries (see [Kn], Theorem 3.3). This shows that the linear drifts ℓ i and the stochastic entropies h i exist for each one of the spaces ( M i , g i ).

Moreover, we clearly have

ℓ 2 = i ℓ 2 i , h = h i .
Therefore Theorem 1.1 follows from Theorem 2.1. Assume (M, g) is a closed connected rank one manifold of nonpositive curvature and that ℓ 2 = h. Then ( M , g) is a symmetric space.

A Hadamard manifold M is called asymptotically harmonic if the function B(= ∆ x b) is constant on S M . Theorem 2.1 directly follows from two propositions: Proposition 2.2. Assume (M, g) is a closed connected rank one manifold of nonpositive curvature and that ℓ 2 = h. Then ( M , g) is asymptotically harmonic.

Proposition 2.3. [[Z], Theorem 1.1] Assume (M, g) is a closed connected rank one manifold of nonpositive curvature such that ( M , g) is asymptotically harmonic. Then, ( M , g) is a symmetric space.

Proof of Proposition 2.2

We consider the foliation W of subsection 2.2. Recall that the leaves are endowed with a natural Riemannian metric. We write ∆ W for the associated Laplace operator on functions which are of class C 2 along the leaves of W. A probability measure m on SM is called harmonic if it satisfies, for any C 2 function f , we have:

SM ∆ W f dm = 0.
Let M be a closed connected manifold such that ℓ 2 = h. In [L3] it is shown that then, there exists a harmonic probability measure m on SM such that, at m-a.e. (x, v), B(x, v) = ℓ. Since B is a continuous function, Proposition 2.2 follows from Theorem 3.1. Let (M, g) be a closed connected rank one manifold of nonpositive curvature, W the stable foliation on SM endowed with the natural metric as above. Then, there is only one harmonic probability measure m and the support of m is the whole space SM .

Proof. Let m be a W harmonic probability measure on SM . Then, there is a unique G-invariant measure m on S M which coincide with m locally. Seen as a measure on M × M (∞), we claim that m is given, for any f continuous with compact support, by:

(3) f (x, ξ)d m(x, ξ) = 1 VolM M M (∞) f (x, ξ)dν x (ξ) dx,
where the family x → ν x is a family of probability measures on M (∞) such that, for all ϕ continuous on M (∞), x → ϕ(ξ)dν x (ξ) is a harmonic function on M and the measure dx is the Riemannian volume on M . The claim follows from [Ga]. For convenience, let us reprove it: on the one hand, the measure m projects on M as a G-invariant measure satisfying ∆f dm = 0. The projection of m on M is proportional to Volume, gives measure 1 to fundamental domains and formula (3) is the desintegration formula. On the other hand, if one projects m first on M (∞), there is a probability measure ν on M (∞) such that

f (x, ξ)d m(x, ξ) = M(∞) M f (x, ξ)dm ξ (dx) dν(ξ).
For ν-a.e. ξ, the measure m ξ is a harmonic measure on M ; therefore, for ν-a.e. ξ, there is a positive harmonic function k ξ (x) such that m ξ = k ξ (x)Vol. Comparing the two expressions for f d m, we see that the measure ν x is given by

ν x = k ξ (x)ν and x → M (∞) ϕ(ξ)dν x (ξ) is indeed a harmonic function.
The G-invariance of m implies that, for all g ∈ G, g * ν x = ν gx . In particular, the support of ν is G-invariant. By [E] (see [B3], page 48), the support of ν is the whole M (∞) and therefore the support of m is the whole SM . This result would be sufficient for proving Proposition 2.2, but using discretization, we are going to identify the measure ν x on M (∞) as the hitting measure of the Brownian motion on M starting from x. This shows Theorem 3.1. Fix x 0 ∈ M . The discretization procedure of Lyons and Sullivan ([LS]) associates to the Brownian motion on M a probability measure µ on G such that µ(g) > 0 for all g and that any bounded harmonic function F on M satisfies F (x 0 ) = g∈G F (gx 0 )µ(g).

Recall that for all ϕ continuous on M (∞), x → ν x (ϕ) is a harmonic function and that ν gx = g * ν x . It follows that the measure ν x 0 is stationary for µ, i.e. it satisfies:

ν x 0 = g∈G g * ν x 0 µ(g).
Since the support of µ generates G as a semigroup (actually, it is already the whole G), there is only one stationary probability measure on M (∞) (see [B3], Theorem 4.11 page 58). We know one already: the hitting measure m x 0 of the Brownian motion on M starting from x 0 . This shows that ν x 0 = m x 0 . Since x 0 was arbitrary in the above reasoning, we have ν x = m x for all x ∈ M and the measure m is given by:

f (x, ξ)d m(x, ξ) = 1 VolM M M (∞)
f (x, ξ)dm x (ξ) dx.

  lim

			t	-	1 t	ln p(t, x, x)
	ℓ = lim t	1 t	d(x, y)p(t, x, y)dVol(y)
	h = lim

t -1 t p(t, x, y) ln p(t, x, y)dVol(y) v = lim t 1 t ln VolB M (x, t),

With a clear convention for the cases when Dim M0 = 0, j = 0 or k = j.

Acknowledgements I am very grateful to Gerhard Knieper for his interest and his comments, in particular for having attracted my attention to [Z]. I also acknowledge partial support of NSF grant DMS-0811127.


G G Mastropasqua

MD Giuseppe Cancello
email: giuseppe.cancello@ieo.it

P Maisonneuve

N Rotmensz

G Viale

M G Mastropasqua

G Pruneri

E Montagna

S Dellapasqua

M Iorfida

A Cardillo

P Veronesi

A Luini

M Intra

O Gentilini

E Scarano

A Goldhirsch

M Colleoni

Prognosis in women with small (T1mic,T1a,T1b) node-negative operable breast cancer by immunohistochemically selected subtypes

Keywords: breast cancer, prognosis, tumour size, immunohistochemistry, subtypes

Prognosis in women with small (T1mic,T1a,T1b) node-negative operable breast cancer by immunohistochemically selected subtypes

Introduction

In the last decades, the use of screening mammography increased the incidence of small size invasive tumours, including tumours with diameters less than 1 cm [START_REF] Berry | Effect of screening and adjuvant therapy on mortality from breast cancer[END_REF][START_REF] Fracheboud | National Evaluation Team for Breast Cancer Screening (NETB): Decreased rates of advanced breast cancer due to mammography screening in the Netherlands[END_REF].

Uncertainties exist concerning the biology, prognosis and management of early breast cancers with size less than 1 cm and node-negative status (or pT1mic,a,bN0M0).

Several classical risk factors might have a prognostic role within the subgroup of patients with small tumours and node negative disease [START_REF] Hanrahan | Prognosis and management of patients with node-negative invasive breast carcinoma that is 1 cm or smaller in size (stage 1; T1a,bN0M0): a review of the literature[END_REF]. Features which might indicate increased risk of relapse in such patients include peritumoural vascular invasion if extensive [START_REF] Quiet | Natural history of node-negative breast cancer: a study of 826 patients with long-term follow-up[END_REF]. Also the presence of high grade and/or high proliferation indexes might select a subgroup of patients with higher risk of relapse [START_REF] Rosner | Predicting recurrence in axillary-node negative breast cancer patients[END_REF][START_REF] Colleoni | Minimal and small size invasive breast cancer with no axillary lymph node involvement: the need for tailored adjuvant therapies[END_REF][START_REF] Hanrahan | Overall survival and causespecific mortality of patients with stage T1a,bN0M0 breast carcinoma[END_REF]. Amplification or overexpression of HER2 should each be considered to elevate risk category [8][START_REF] Curigliano | Clinical relevance of HER2 overexpression/amplification in patients with small tumor size and node-negative breast cancer[END_REF].

Finally, young age should also be taken into consideration due to its prognostic role and its related different responsiveness to endocrine therapy if compared with older premenopausal women with breast cancer [START_REF] Rosner | Predicting recurrence in axillary-node negative breast cancer patients[END_REF][START_REF] Colleoni | Minimal and small size invasive breast cancer with no axillary lymph node involvement: the need for tailored adjuvant therapies[END_REF]. However, some of these studies suffered from small sample size, information on biological features were based on old methodologies and there have been few clinical studies with large number of patients followed long enough for an adequate evaluation of the prognosis.

Care for patients with breast cancer tend to be more selective interventions to minimize acute and late toxicity without compromising efficacy. In particular, for the subgroup of patients classified as pT1a,b N0 appropriate adjuvant systemic therapy involves choosing treatments tailored to individual patients.

Breast cancer is in fact a heterogeneous disease and gene expression studies have identified molecularly distinct subtypes with prognostic implications across multiple treatment settings [START_REF] Sorlie | Repeated observation of breast tumour subtypes in independent gene expression data sets[END_REF][START_REF] Oh | Estrogen-regulated genes predict survival in hormone receptor-positive breast cancers[END_REF][START_REF] Rouzier | Breast cancer molecular subtypes respond differently to preoperative chemotherapy[END_REF]. These subtypes include Oestrogen receptor (ER)-positive -Luminal A (Luminal A), ERpositive -Luminal B (Luminal B), HER2-enriched (i.e. tumours that overexpress HER2-associated 4 genes but do not express genes that define the luminal subtype), Basal-like, and normal breast-like.

HER2-enriched and Basal-like subtypes are hormone receptor negative and have poor prognosis [START_REF] Sorlie | Repeated observation of breast tumour subtypes in independent gene expression data sets[END_REF][START_REF] Perou | Molecular portraits of human breast tumours[END_REF][START_REF] Sorlie | Gene expression patterns of breast carcinomas distinguish tumor subclasses with clinical implications[END_REF].

Immunohistochemical evaluation of ER, progesterone receptor (PgR), Ki-67 and HER2 may be considered a surrogate means for identifying the molecular subtypes of breast cancer [START_REF] Cheang | Ki67 Index, HER2 Status, and Prognosis of Patients With Luminal B Breast Cancer[END_REF]. The classification according to four subtypes (Luminal A, Luminal B, HER2 and Triple Negative) appeared useful to define different prognostic subgroups with different relationship with adjuvant treatment received [START_REF] Hugh | Breast Cancer Subtypes and Response to Docetaxel in Node-Positive Breast Cancer: Use of an Immunohistochemical Definition in the BCIRG 001 Trial[END_REF]17].

Whether the use of this classification for tumours less than 1 cm might be useful in clinical practice in order to accurately predict outcome is an hypothesis that has not been tested.

The aim of this study was to evaluate the overall outcome of women with lymph node negative invasive carcinoma of the breast of ≤ 1cm in size, including microinvasive carcinoma (T1mic) according to immunohistochemically defined subtypes.

Patients and methods

We collected information on all consecutive breast cancer patients operated at the European Institute of Oncology between January 1997 and December 2005. Data on each patient's medical history, concurrent diseases, surgery, pathological evaluation and results of staging procedures (blood chemistry, haematological values, bone scan, chest film and upper abdominal ultrasound examination) were retrieved. The surgically removed breast lesions were thoroughly sampled for pathological examination.

Tissue sections from all previous needle biopsies (at least three sections/core, cut at 110-200mm intervals) and from all surgical resections performed elsewhere were reviewed. Tumours were classified histologically according to the World Health Organization Histological Classification of Breast Tumours, as modified by Rosen and Obermann [START_REF] Rosen | Tumors of the Mammary Gland[END_REF]. Tumour grading was assessed according to Elston and Ellis [START_REF] Elston | Pathological prognostic factors in breast cancer. I. The value of histological grade in breast cancer: experience from a large study with long-term follow-up[END_REF]. We looked for peritumoural vascular invasion as recommended by Rosen and Obermann [START_REF] Rosen | Tumors of the Mammary Gland[END_REF]. Microinvasive breast cancer was diagnosed according to the TNM classification and following the criteria of Rosen and Obermann [START_REF] Rosen | Tumors of the Mammary Gland[END_REF]. ER, PgR status, Ki-67 labelling index determined with the MIB1 monoclonal antibody, and HER2/neu overexpression were evaluated immunohistochemically as previously reported [START_REF] Viale | Predictive value of tumor Ki-67 expression in two randomized trials of adjuvant chemoendocrine therapy for node-negative breast cancer[END_REF]. In particular, HER2/neu overexpression was evaluated using a 1/100 dilution of a polyclonal antiserum (Dako, Glostrup, Denmark) and considering only complete and intense membrane staining of at least 10% neoplastic cells as evidence of overexpression (3+). For evaluation of ER and PgR status and Ki-67 labelling index, the percentage of cells exhibiting definite nuclear staining over up to 2000 neoplastic cells examined at 400x magnification was recorded. The stained slides were evaluated independently by two of the authors. Only nuclear immunoreactivity was evaluated for ER, PgR and MIB1. The threshold for ER and PgR positivity was 1% [START_REF] Viale | Predictive value of tumor Ki-67 expression in two randomized trials of adjuvant chemoendocrine therapy for node-negative breast cancer[END_REF]. Histological grade and biological features were evaluated on the invasive component of the tumour.

Statistics:

The Fisher exact test and the Mantel-Haenszel Chi-Square test for trend were used to assess the association between categorical and ordinal variables respectively. The primary endpoints were the incidence of loco-regional relapse (LRR), distant metastasis (DM), breast cancer related survival (BCS) and overall survival (OS). Loco-Regional Relapse (LRR) included ipsilateral breast cancer, breast cancer recurrence in the axilla, regional lymph nodes, chest wall, skin of ipsilateral breast. Distant Metastasis (DM) included all sites of recurrence except loco-regional relapses and contralateral breast cancer as first of subsequent event. Breast cancer related Survival (BCS) included loco-regional relapses, distant metastases, contralateral breast cancer or death from breast cancer. Overall Survival OS was determined as the time from surgery until the date of death (from any cause) or was censored at the date of last follow-up. Cumulative incidence and survival plots were drawn using the Kaplan-Meier method. The log-rank test was used to assess the difference between strata. Multivariate Cox proportional hazard regression analysis was used to assess the independent prognostic significance of various clinical and histopathological characteristics of the tumour on event free or overall survival. Factors included in multiple regression analyses included age, tumour size (pT), multifocality, peritumoural vascular invasion and immunohistochemical classification as follows:

•

Luminal A (ER>0 or PgR>0) and (Ki-67<14%) and (HER2 0/+/++),

• Luminal B (ER>0 or PgR>0) and ((Ki-67≥14%) or (HER2 +++)),

• HER2-positive (ER=0 and PgR=0) and (HER2 +++), and

• Triple Negative (ER=0 and PgR=0) and (HER2 0/+/++).

Other factors, including menopausal status, presence of in-situ component, ER and PgR expression.

Ki-67 labelling index, HER2 over expression, tumour grade and histotype were not included into the final multivariate model, due to lack of association at univariate analysis or presence of collinearity with immunohistochemical classification.

All analyses were performed with the SAS software, version 8.2 (Cary, NC).

Results

We included in the analysis 1691 women with pT1mic/pT1a/pT1b N0M0 breast cancer treated at the European Institute of Oncology during 1997-2005. Male patients and females with previous cancer, non-invasive breast cancers or bilateral tumours were excluded.

The characteristics of the evaluable patients are given in Table 1.

A higher proportion of pT1mic had the HER2 and triple negative immunohistochemical subtypes compared with pT1a or pT1b breast cancers.

HER2 and Triple Negative subtypes were more frequent in young patients (aged less than 35 years) than in older patients. Poor differentiation was strictly correlated to the presence of HER2 and Triple Negative subtype.

As shown in Table 2, a higher proportion of patients with small tumours classified as HER2 subtype were treated with mastectomy without radiotherapy. Moreover, about the 99% of patients with Luminal A tumours didn't receive adjuvant chemotherapy, while about 70% of patients with Triple Negative tumours received chemotherapy. Finally, only 8% and 6.5% of patients with Luminal A and Luminal B tumours, respectively, did not receive any hormonal treatment.

In our study the median follow-up was 6.3 years for events and 7 years for overall survival.

Figure 1 shows the incidence of locoregional relapse and distant metastases and survival curves according to immunohistochemical classification.

We found a statistical significant difference about LRR, BCS, OS and a borderline difference about the incidence of DM between the four subtypes. Increased rate of LRR and DM and a reduced BCS and OS are observed in the HER2 and Triple-negative subtypes (Figure 1).

We performed a multivariate analysis including significant clinical and biological features at univariate analysis (data not shown); moreover we analysed data subdividing Luminal B subtype according to HER2 overexpression status, too.

The presence of peritumoural vascular invasion was associated with a significant risk of DM and reduced OS.

Very young patients (aged<35 years) were at increased risk of LRR plus DM and had a statistically significant reduced BCS and OS compared to older patients.

pT1a and pT1b tumours had an increased risk of LRR, DM, breast cancer-events and death compared with pT1mic, though these results were not statistically significant at multivariate analyses (Table 3).

HER2 subtype was the only subtype associated with a statistical significant increased risk of LRR, DM and a reduced BCS and OS compared with Luminal A subtype, at multivariate analysis (

Discussion

Treatment of small tumours still represents an area of controversy. In fact, although several reports focused on this issue in the past years, limited information are still available on their prognosis.

In this study, performed on a large group of patients submitted to surgery in the recent years and with biological features evaluated with new techniques and by the same team of pathologists, a limited but not negligible risk of recurrence was detected confirming the importance of the identification of subgroup of patients at higher risk of recurrence.

An immunohistochemical profile, based on the degree of expression of ER, PgR, Ki-67 and HER2, was used to identify subgroups of breast cancer patients with different outcome and who will respond to different systemic adjuvant treatments [START_REF] Hugh | Breast Cancer Subtypes and Response to Docetaxel in Node-Positive Breast Cancer: Use of an Immunohistochemical Definition in the BCIRG 001 Trial[END_REF]17].

In our study the Triple Negative subtype was correlated with a significantly increased risk of LRR and breast cancer related events if compared with the Luminal A subtype. These results are in line with those already presented in two previous studies [8,[START_REF] Kwon | Triple negativity and young age as prognostic factors in lymph node-negative invasive ductal carcinoma of 1cm or less[END_REF] In a Korean analysis, patients with Triple Negative disease had a significantly lower Relapse-Free Survival rate compared to hormone receptor-positive and/or HER2-positive patients (p=0.016); in the multivariate analysis, triple negativity was an independent prognostic factors of recurrence (HR 4.96; 95% CI: 1.00-23.1; p=0.018) [START_REF] Kwon | Triple negativity and young age as prognostic factors in lymph node-negative invasive ductal carcinoma of 1cm or less[END_REF].

In Although the number of events was low in the present experience related to the fact that we are focusing on a population with a lower intrinsic metastatic potential, the observation of a worse 1 0 breast cancer survival indicated that new adjuvant strategies should be explored in this subgroup of patients.

In the present study the HER2 subtype (HER2 positive and steroid hormone receptor negative disease) was associated with worse outcome and increased incidence of LRR and DM.

Approximately 6%-10% of small node-negative tumours are HER2 positive and there is increasing evidence from some retrospective analyses for a worse outcome of patients with HER2-positive tumours.

In a study that analysed 965 T1a,bN0M0 breast cancers, patients with HER2-positive disease (including both hormone receptor positive and hormone receptor negative tumour) had higher risks of recurrence (HR, 2.68; 95% CI: 1.44-5.00; P=0.002) and distant recurrence (HR, 5.30; 95% CI, 2.23-12.62; P=0.001) than those with HER2-negative tumours [8].

In a second study, conducted at European Institute of Oncology in Milan, after a median follow-up of 4.6 years, in the hormone receptor-positive group, 5-year DFS rates were 99% for HER2negative disease and 92% for HER2-positive disease (P=0.013), while no difference was found in the group of hormone receptor-negative tumours [START_REF] Curigliano | Clinical relevance of HER2 overexpression/amplification in patients with small tumor size and node-negative breast cancer[END_REF].

Our results showed that HER2 is an independent prognostic factor, mainly when associated with negative hormone receptors (HER2 subtype). However, in the Luminal B, HER2 positive subset (ER positive and HER2 positive disease) a significantly worse BCS was observed (HR, 2.18; 95% CI: 1.03-4.60) when compared with Luminal A. The different follow-up, the different study design as well as the limited number of events registered in the subgroup of patients with tumour less than 1 cm in size might explained the partially different results observed.

A limited number of studies that evaluated age as prognostic factor in patients with small size breast cancer is available in the literature. A recently published study evaluated the outcome of 378

Korean women with small breast cancers. After a median follow-up of 60 months, authors showed that age younger than 35 years was significantly associated with a higher rate of recurrence HR Current study adds important information about the prognosis of very young women with breast cancer; also when tumour size is less than 1 centimetre age less than 35 is a significant risk factor for a reduced BCS and overall survival compared to older patients.

In conclusion, the present study confirms that the prognosis of patients with node negative small breast cancer depends on variable features. We show that risk of relapse significantly correlates with tumour subtypes identified by immunohistochemistry as well as classical prognostic features such as young age. Patients whose tumours are classified as pT1a,b pN0 include a spectrum from patients at very low risk for whom there is little evidence supporting the use of adjuvant therapy, to those with higher risk disease where an approach including chemotherapy plus targeted therapy appears clearly justified.

It should however be emphasised that the tumour subtypes identified in the current study still include heterogeneous groups of tumours, and that the identification of further tumour subtypes amenable to targeted treatments represents a research priority.

1 2 1 5

Table 3a )

 3a , while patients with Triple Negative subtype had an increased risk of LRR (Hazards Ratio (HR)

3.58; 95%CI: 1.40-9.13) and breast cancer related events (HR 2.18; 95%CI: 1.04-4.57) (Table

3a

).

No difference was found between Luminal B and Luminal A subtypes for any of the outcomes analysed (Table

3a

). When we analysed outcomes of patients with Luminal B according HER2 status (101 patients with Luminal B and HER2 overexpression), we found a reduced BCS (HR 2.15; 95%CI: 1.02-4.53) for Luminal B HER2-positive and a borderline increased risk of LRR (HR 2.70; 95%CI: 0.96-7.55) compared to Luminal A.

We reported number and rate of events and mortality in a Supplementary table (on line Supplemetary Table

)

  the study by Gonzalez-Angulo et al., patients who had Triple Negative breast cancer had 3.89

	times (95% CI: 2.56-10.14; P .0001) the risk of recurrence and 2.84 times (95% CI: 0.99-8.14; P
	.053) the risk of distant recurrence if compared with patients who had hormone receptor-positive
	breast cancer [8].

Table 1 :

 1 Patients characteristics

	Characteristics	All	Luminal A	Luminal B		HER2	Triple Negative P-value
		N		N (%)		N (%)		N (%)		N (%)
	All	1691 881	(100)	633	(100)	82	(100)	95	(100)
	Tumour size
	pT1mic	68	17	( 1.9)	22	( 3.5)	22	(26.8)	7	( 7.4)
	pT1a	362 197	(22.4)	114	(18.0)	30	(36.6)	21	(22.1)
	pT1b	1261 667	(75.7)	497	(78.5)	30	(36.6)	67	(70.5) <.0001
	Age
	<35	27	7	( 0.8)	10	( 1.6)	3	( 3.7)	7	( 7.4)
	35-49	507 239	(27.1)	219	(34.6)	22	(26.8)	27	(28.4)
	50-69 1023 576	(65.4)	342	(54.0)	50	(61.0)	55	(57.9)
	70+	134	59	( 6.7)	62	( 9.8)	7	( 8.5)	6	( 6.3) <.0001
	Menopausal status
	Pre-menopausal	581 283	(32.1)	243	(38.4)	23	(28.0)	32	(33.7)
	Peri-menopausal 1040 562	(63.8)	363	(57.3)	56	(68.3)	59	(62.1)
	Post-menopausal	70	36	( 4.1)	27	( 4.3)	3	( 3.7)	4	( 4.2)	0.21
	Histology
	Ductal 1276 596	(67.7)	523	(82.6)	77	(93.9)	80	(84.2)
	Lobular	164 102	(11.6)	59	( 9.3)	0	( 0.0)	3	( 3.2)
	Ductal+Lobular	214 162	(18.4)	35	( 5.5)	5	( 6.1)	12	(12.6)
	Other	37	21	( 2.4)	16	( 2.5)	0	( 0.0)	0	( 0.0) <.0001
	Multifocality
	Monofocal 1517 794	(90.1)	564	(89.1)	73	(89.0)	86	(90.5)
	Multifocal	174	87	( 9.9)	69	(10.9)	9	(11.0)	9	( 9.5)	0.91
	In-situ component
	Absent 1231 685	(77.8)	451	(71.2)	33	(40.2)	62	(65.3)
	Present	460 196	(22.2)	182	(28.8)	49	(59.8)	33	(34.7) <.0001
	Tumour Grade
	G1	688 568	(64.5)	113	(17.9)	1	( 1.2)	6	( 6.3)
	G2	739 279	(31.7)	406	(64.1)	29	(35.4)	25	(26.3)
	G3	205	6	( 0.7)	90	(14.2)	48	(58.5)	61	(64.2) <.0001
	Unknown	83	28	( 3.2)	24	( 3.8)	4	( 4.9)	3	( 3.2)
	Peritumoural vascular
	invasion
	Absent 1611 853	(96.8)	590	(93.2)	79	(96.3)	89	(93.7)
	Present	80	28	( 3.2)	43	( 6.8)	3	( 3.7)	6	( 6.3) 0.008

Table 2 :

 2 Treatments

	Characteristics		Luminal A	Luminal B		HER2	Triple Negative P-value
	Type of surgery
	Quadrantectomy 824	(93.5%)	572 (90.4%) 52 (63.4%) 87 (91.6%)
	Mastectomy 57	( 6.5%)	61 ( 9.6%) 30 (36.6%)	8	( 8.4%)	<.0001
	Sentinel lymph node biopsy
	No 63	( 7.1%)	54 ( 8.5%) 12 (14.6%) 18 (19.0%)
	Yes 818	(92.9%)	579 (91.5%) 70 (85.4%) 77 (81.0%)	0.003
	Chemotherapy
	Antracycline	4	( 0.5%)	16 ( 2.5%) 20 (24.4%)	4	( 4.2%)
	CMF	4	( 0.5%)	31 ( 4.9%) 11 (13.4%) 60 (63.2%)
	Other/NOS	3	( 0.3%)	2	( 0.3%)	5	( 6.1%)	2	( 2.1%)
	NONE 870	(98.8%)	584 (92.3%) 46 (56.1%) 29 (30.5%)	<.0001
	Endocrine therapy
	LHRH 25	( 2.8%)	16 ( 2.5%)	0		0
	TAM 515	(58.5%)	322 (50.9%)	1	( 1.2%)	2	( 2.1%)
	LHRH+TAM 177	(20.1%)	169 (26.7%)	0		1	( 1.1%)
	Other/NOS 95	(10.8%)	86 (13.6%)	0		0
	NONE 69	( 7.8%)	40 ( 6.3%) 81 (98.8%) 92 (96.8%)	<.0001
	Radiotherapy
	NONE 61	( 6.9%)	68 (10.7%) 28 (34.2%)	9	(9.5%)
	ELIOT 203	(23.0%)	125 (19.8%)	8	( 9.8%) 15 (15.8%)
	External RT 617	(70.0%)	440 (69.5%) 46 (56.1%) 71 (74.7%)	<.0001

Table 3a

 3a Multivariate analysisHazards ratio (HR) and 95% confidence intervals (CI) derived from univariate Cox proportional hazard regression model NOTE: Tumour grade was not included in the final multivariate model due to strong colinearity with immunihistochemical subtype.

		Loco-regional	Distant	Local + distant	Breast cancer	Overall
		relapse	metastasis	metastasis	survival	survival
	Age
	<35 1.60 (0.35-7.23) 3.34 (0.68-16.4) 2.78 (0.93-8.34) 2.94 (1.11-7.81) 4.72 (0.92-24.2)
	35-49	1.00	1.00	1.00	1.00	1.00
	50-69 0.73 (0.39-1.36) 0.90 (0.39-2.11) 0.89 (0.53-1.51) 0.79 (0.50-1.26) 2.74 (1.12-6.70)
	70+ 0.26 (0.04-1.97) 1.04 (0.22-4.84) 0.59 (0.18-1.97) 1.05 (0.46-2.40) 4.86 (1.55-15.2)
	pT
	pT1mic	1.00	1.00	1.00	1.00	1.00
	pT1a 1.37 (0.29-6.42) 0.58 (0.10-3.31) 1.37 (0.38-4.89) 1.83 (0.53-6.26) 2.39 (0.29-19.6)
	pT1b 1.29 (0.29-5.77) 0.80 (0.17-3.81) 1.44 (0.42-4.91) 1.58 (0.47-5.24) 2.34 (0.30-18.2)
	Multifocality
	Monofocal	1.00	1.00	1.00	1.00	1.00
	Multifocal 2.64 (1.35-5.16) 0.95 (0.28-3.18) 1.95 (1.07-3.53) 1.56 (0.90-2.70) 1.03 (0.40-2.65)
	Peritumoural vascular
	invasion
	Absent	1.00	1.00	1.00	1.00	1.00
	Present 1.68 (0.58-4.84) 3.73 (1.35-10.3) 1.92 (0.86-4.28) 1.67 (0.79-3.53) 3.53 (1.43-8.71)
	Immunohistochemical
	subtype
	Luminal A	1.00	1.00	1.00	1.00	1.00
	Luminal B 1.59 (0.80-3.18) 1.99 (0.81-4.85) 1.78 (1.02-3.12) 1.45 (0.90-2.33) 0.74 (0.36-1.50)
	HER2+++ 4.53 (1.56-13.1) 4.37 (1.06-17.9) 5.15 (2.20-12.1) 3.22 (1.44-7.18) 2.87 (1.05-7.89)
	Triple Negative 3.58 (1.40-9.13) 2.46 (0.61-9.92) 3.04 (1.35-6.84) 2.18 (1.04-4.57) 1.94 (0.71-5.31)

Table 3b :

 3b Multivariate analysis .28-6.17) 0.56 (0.10-3.22) 1.32 (0.37-4.73) 1.80 (0.52-6.17) 2.27 (0.28-18.6) pT1b 1.36 (0.30-6.09) 0.84 (0.17-4.01) 1.52 (0.44-5.19) 1.65 (0.49-5.48) 2.44 (0.31-19.1) Multifocal 2.56 (1.30-5.07) 0.87 (0.25-2.97) 1.92 (1.05-3.50) 1.55 (0.89-2.68) 0.91 (0.34-2.39) Present 1.48 (0.49-4.48) 3.25 (1.12-9.41) 1.80 (0.79-4.14) 1.60 (0.74-3.44) 2.88 (1.10-7.59) HER2 -/+/++) 1.31 (0.61-2.82) 1.69 (0.65-4.42) 1.49 (0.81-2.75) 1.30 (0.78-2.16) 0.52 (0.22-1.22) Luminal B (HER2 +++) 2.70 (0.96-7.55) 3.02 (0.78-11.7) 3.12 (1.37-7.09) 2.15 (1.02-4.53) 1.30 (0.38-4.38) HER2+++ 4.64 (1.60-13.4) 4.41 (1.08-18.1) 5.27 (2.25-12.3) 3.27 (1.47-7.30) 2.95 (1.07-8.10) Triple Negative 3.62 (1.42-9.24) 2.50 (0.62-10.1) 3.05 (1.35-6.86) 2.18 (1.04-4.57) 2.01 (0.73-5.49) Hazards ratio (HR) and 95% confidence intervals (CI) derived from univariate Cox proportional hazard regression model NOTE: Tumour grade was not included in the final multivariate model due to strong colinearity with immunihistochemical subtype.

		Loco-regional	Distant	Local + distant	Breast cancer	Overall
		relapse	metastasis	metastasis	survival	survival
	Age
	<35 1.69 (0.37-7.69) 3.80 (0.76-19.1) 2.93 (0.97-8.82) 3.06 (1.14-8.14) 6.71 (1.22-36.8)
	35-49	1.00	1.00	1.00	1.00	1.00
	50-69 0.75 (0.40-1.41) 0.98 (0.41-2.37) 0.92 (0.54-1.56) 0.80 (0.51-1.28) 3.66 (1.33-10.0)
	70+ 0.28 (0.04-2.09) 1.12 (0.24-5.32) 0.62 (0.18-2.06) 1.08 (0.47-2.46) 6.65 (1.93-22.9)
	pT
	pT1mic	1.00	1.00	1.00	1.00	1.00
	pT1a 1.31 (0Multifocality
	Monofocal	1.00	1.00	1.00	1.00	1.00
	Peritumoural vascular
	invasion
	Absent	1.00	1.00	1.00	1.00	1.00
	Immunohistochemical
	subtype
	Luminal A	1.00	1.00	1.00	1.00	1.00
	Luminal B (

1 8