Bayesian inference with M-splines on spectral measure of bivariate extremes

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1 Methodology and Computing in Applied Probability manuscript No. (will be inserted by the editor) Bayesian inference with M-splines on spectral measure of bivariate extremes Khader Khadraoui Pierre Ribereau Received: date / Accepted: date Abstract We consider a Bayesian methodology with M-splines for the spectral measure of a bivariate extreme-value distribution. The tail of a bivariate distribution function F in the max-domain of attraction of an extreme-value distribution function G may be approximated by that of its extreme value attractor. The function G is characterized by a probability measure with expectation equal to 1/2, called the spectral measure, and two extreme-value indices. This spectral measure determining the tail dependence structure of F. The approximation of the spectral measure is proposed thanks to a nonparametric Bayesian estimator that guarantees to fulfill a moment and a shape constraints. The problem of routine calculation of posterior distributions for both coefficients and knots of M-splines is addressed using the Markov chain Monte Carlo (MCMC) simulation technique of reversible jumps. Keywords Bayesian inference M-splines Bivariate extremes Spectral measure Monotone shape MCMC Mathematics Subject Classification (21) Primary 62G32 62F15 62G5 secondary 62F3 6J22 65D7 1 Introduction Suppose that we observe a random sample (X i1, X i2 ), i = 1,..., n, from an unknown bivariate distribution F in the max-domain of attraction of a bivariate Khader Khadraoui Laval University, Department of Mathematics and Statistics, Quebec city G1V A6, Canada. Tel.: +1(418) Ext Fax: +1(418) khader.khadraoui@mat.ulaval.ca Pierre Ribereau Université Lyon 1, Institut Camille Jordan ICJ UMR 528 CNRS, 69622, Lyon, France.

2 2 Khader Khadraoui, Pierre Ribereau extreme value distribution G. It is known that the tail of F is well approximated by the tail of G except for the case of asymptotic independence. Each margin of G is characterized by three parameters. The dependence structure of G is characterized by a spectral measure which is a finite Borel measure on a compact interval. Hence, approximate inference on the tail of F can be done via inference on the six parameters and inference on the spectral measure. The estimation of the six parameters is well understood while estimation of the spectral measure remains a problem that deserves further investigation although some works appear in the literature to address this question. The literature on spectral measure usually focuses on frequentest inference within parametric approaches; see for instance Coles and Tawn (1991, 1994); Joe et al. (1992); Smith (1994); Ledford and Tawn (1996); de Haan et al. (28); Boldi and Davison (27) and non-parametric procedures; see for instance de Haan and de Rond (1998); de Haan and Sinha (1999); Einmahl et al. (26, 21); Schmidt and Stadtmüller (26); Einmahl and Segers (29). Recently, using the method of Lagrange multipliers, a constrained piecewise linear estimate has been proposed in Einmahl and Segers (29) in order to incorporate the moment constraint characterizing the class of spectral measure. A review on dependence function and on spectral measure estimators can be found in the monographs Coles (21); de Haan and Ferreira (26). Bayesian approaches on spectral measure under moment constraint are rather rare although some works appear in the literature. It is the aim of our article to propose a smooth non-parametric estimation of the spectral measure using an M-spline basis. This estimation is proposed via a Bayesian approach that guarantees to satisfy some desirable constraints (concerning the expectation and the shape of the spectral measure) thanks to the prior distribution. The coherence of the Bayesian paradigm with inference on univariate and multivariate extremes has been argued in literature (Aitchidson and Dunsmore, 1975; Coles and Tawn, 1996b,a; Guillotte et al., 211). The contribution of this paper are twofold: first, we propose a nonparametric Bayesian estimator for the spectral measure which fulfills both the moment and the monotone restrictions; second, we use an M-spline basis in the construction of a monotone smooth estimator which to our knowledge this is the only example of a constrained estimator with M-splines in the spectral measure estimation frameworks. Splines are of importance for the reason that they are used wherever curves are to be fit: being polynomial, they can be evaluated quickly; being picewise polynomial, they are very flexible; their representation in terms of M-splines or B-splines provides geometric information and insight. Our methodology is presented in the following by constructing the estimator and selecting a prior distribution for the spectral measure. Then, the spectral measure is estimated by the posterior mode as it necessarily fulfills the moment and the shape constraints. Note that this not necessarily the case for the posterior mean in the presence of shape constraints (Abraham and Khadraoui, 215; Khadraoui, 217a). The posterior mode is computed using simulations from the posterior distribution. The problem of routine simulations from posterior distribution for both coefficients and knots of M-splines is

3 Bayesian inference with M-splines on spectral measure of bivariate extremes 3 addressed using the Markov chain Monte Carlo (MCMC) simulation technique of reversible jumps (Green, 1995). The paper is organized as follows. In Section 2, we review the general theoretical results for spectral measures. The construction of the subspace of spectral measures and selection of prior distribution are done in Section 3. Section 4 is devoted to the Bayesian inference. In Section 5, we provide a simulation study to compare the performances of the Bayes M-splines estimator for the spectral measure introduced in this paper with three estimators proposed recently in the literature. Section 6 discusses further research. 2 Bivariate tail approximation In this Section, we describe the dependence structure of a bivariate tail in various equivalent ways: from the spectral measure Φ introduced in de Haan and Resnick (1977) to the spectral measure H in Coles and Tawn (1991). We consider an observed vector (X 1, X 2 ) of one realization from continuous distribution function F and marginal distributions functions F 1 and F 2. We define D = [, ] 2 \ {(, )} and, for l = 1, 2, we put: Z l = 1 1 F l (X l ). (2.1) For every continuous γ : D R with compact support and assuming that, for ξ v ξp[ξ 1 (Z 1, Z 2 ) ] ν( ), (2.2) where v stands for vague convergence of measures (in D), we have lim ξ ξe[γ(ξ 1 (Z 1, Z 2 ))] = γdν. We precise that the exponent measure ν enjoys two crucial properties: homogeneity, and standardized marginals, D ν(c ) = c 1 ν( ), < c <, (2.3) ν([z, ] [, ]) = ν([, ] [z, ]) = 1/z, < z. (2.4) From (2.4) it is easy to remark that ν([, ] 2 \ [, ) 2 ) =. Let be an arbitrary norm on R 2. Consider the following polar coordinates, (r, φ), of (z 1, z 2 ) [, ) 2 \ {(, )}: r = (z 1, z 2 ) (, ), φ = arctan(z 1 /z 2 ) [, π/2]. (2.5)

4 4 Khader Khadraoui, Pierre Ribereau Now, using polar coordinates (r, φ) given in (2.5), we define a Borel measure Φ on [, π/2] by ( ) Φ( ) = ν {(z 1, z 2 ) [, ) 2 : r 1, φ }. (2.6) We can interpret the spectral measure defined in (2.6) as follows: [ ] v ξp (Z 1, Z 2 ) ξ, arctan(z 1 /Z 2 ) Φ( ), ξ. (2.7) We set z 1 (r, φ) = r sin φ/ (sin φ, cos φ) and z 2 (r, φ) = r cos φ/ (sin φ, cos φ). For every ν-integrable γ : D R and using the previous crucial property (2.3) of homogeneity, we have [,π/2] D γdν = [,π/2] γ(z 1 (r, φ), z 2 (r, φ))r 2 drφ(dφ), (2.8) where the exponent measure ν is, in the polar coordinate system (r, φ), a product measure r 2 drφ(dφ) and, in particular, is completely determined by its spectral measure Φ. The previous crucial property (2.4) of standardization restrictions on the exponent measure ν translate into a moment constraint on Φ: cos φ (sin φ, cos φ) Φ(dφ) = sin φ Φ(dφ) = 1. (2.9) [,π/2] (sin φ, cos φ) The bivariate tail dependence structure is analyzed as follows: X 1 and X 2 are completely tail dependent, that is, ξp[z 1 ξ, Z 2 ξ] 1 as ξ, if and only if Φ is concentrated on {φ = π/4} (equivalently ν is concentrated on the main diagonal); Similarly, X 1 and X 2 are tail independent, that is, ξp[z 1 ξ, Z 2 ξ] as ξ, if and only if Φ is concentrated on {φ =, φ = π/2} (equivalently ν is concentrated on the coordinate axes). Note that in case of completely tail dependence Φ({π/4}) = (1, 1) while in case of tail independence Φ({}) = Φ({π/2}) = 1. Up to now, we shall make no assumptions on the marginal distributions F 1 and F 2 except for continuity. However, if in addition to (2.2) we consider F on quadrants of the form [u 1, ) [u 2, ) (where u 1 and u 2 are a high thresholds), then the domain of attraction condition on a bivariate distribution function G yields good approximation for F. We note that F is well approximated only on a subset of its support. Precisely, if (2.2) holds and if there exist real sequences a nl > and b nl, for l {1, 2}, and a bivariate cumulative distribution function G with non degenerate margins, then, for all x, y R F n( ) n a n1 x + b n1, a n2 y + b n2 G(x, y), [ ] G(x, y) = exp l{ log G 1 (x), log G 2 (y)}, (2.1)

5 Bayesian inference with M-splines on spectral measure of bivariate extremes 5 where the stable tail dependence function l (Drees and Huang, 1998; Huang, 1992) can be expressed in terms of the spectral measures ν, Φ or H through ( ) l(x 1, x 2 ) = ν {(z 1, z 2 ) [, ] 2 : z 1 x 1 1 or z 2 x 1 2 } max(x 1 sin φ, x 2 cos φ) = Φ(dφ) [,π/2] sin φ, cos φ = 2 max ( ) wx 1, (1 w)x 2 dh(w), [,1] for (x 1, x 2 ) [, ) 2. The spectral measure H being a probability measure on [, 1] with mean equal to 1/2. The stable tail dependence function l was introduced by DM Mason in an unpublished 1991 manuscript and Huang (1992). Drees and Huang (1998) showed that the following tail empirical dependence function l attains the optimal rate of convergence for estimators of the stable tail dependence function, l(x1,..., x d ) = 1 n 1 { j=1,...,d:rij>n+1 kx k j}, i=1 where 1 A denotes the indicator function of the set A, r ij denotes the rank of X ij among X 1j,..., X nj (more precisely r ij = n s=1 1 (X sj X ij)) with k = k n and k/n. For more details about the connections between the stable tail dependence function at the first sight and the exponent and spectral measures on the other hand we refer the interested reader to the references Beirlant et al. (24); de Haan and Ferreira (26). Given a large thresholds, u 1 and u 2, the marginal cumulative distribution functions can be given, for l {1, 2}, by ( x l u ) 1/ηl l log{g l (x l δ l )} = ζ l 1 + η l, (2.11) σ l for x l such that η l (x l u l )+σ l >, where η l is a shape parameter (the extreme value index), σ l is a scale parameter and δ l = (ζ l, η l, σ l ). We note that < ζ l = log{g l (u l δ l )} and, as u l is large, we approximate ζ l by the marginal probability of exceeding the threshold (ζ l 1 G l (u l δ l )). Therefore, the aim from now on consists in studied the domain of attraction of the bivariate maxstable distribution G which might be characterized by its marginal parameters (δ 1, δ 2 ) and its spectral measure H with expectation equal to 1/2. 3 M-spline estimator and prior distribution In this section we use an M-spline basis to propose an estimator for the spectral measure. As we have seen in Section 2, the dependence structure of a bivariate tail is described in various equivalent ways. For the sake of simplicity, we propose in this paper to estimate H. From now on, we consider the same symbol to denote the spectral measure and its cumulative distribution, i.e., H(w) = H([, w]) for w [, 1].

6 6 Khader Khadraoui, Pierre Ribereau 3.1 M-spline estimator for spectral measure In this subsection, we shall construct a class H of smooth spectral measures whose only atoms, if any, are at and at 1. Our class H will be supported on the set of smooth functions constructed from a spline basis. Fix some order q, a natural number such that we assume in this paper q 3, and let K 2 be another natural number, which will increase with n, and partition the open unit interval (, 1) into K subintervals ((k 1)/K, k/k) for k = 1,..., K. Consider the linear space of splines of order q relative to this partition, that is, all functions s : (, 1) R which are piecewise polynomial of degree < q and which are, in the case that q 2, q 2 times continuously differentiable. A thorough presentation of splines is given in de Boor (1987). It can be shown that this is a J = (q + K 1)-dimensional vector space. A convenient basis in our study is the set of M-splines. Let t = (t 1,..., t K 1+2q ) be a nondecreasing sequence of knots, such that ( (q 1) t := K,..., 1 K,, 1 K, 2 K,..., K 1 K, 1, (K + 1) K K + (q 1) ),...,, K and let M 1,q,..., M J,q be the M-spline functions of order q and with complete knot vector t (interior and exterior knots). In this connection, it is worthwhile to stress that the term M-spline refers to a certain normalized B-spline on its minimal support [t j, t j+q ), i.e., M j,q := (q/(t j+q t j ))B j,q, (3.1) where B j,q denotes the usual jth B-spline function of order q. This brings us to an immediate propriety; + M j,q = t j+q t j M j,q = 1. We precise that the spectral measure H is not absolutely continuous with respect to the Lebesgue measure on [, 1], because it gives positive probabilities a and a 1 to the boundary points and 1. The Lebesgue decomposition of H into absolutely continuous H c and singular H s parts reads H(dw) = a δ (dw) + h β J (w)dw + a 1δ 1 (dw), (3.2) where δ z is the Dirac mass at state z and dw is the Lebesgue measure on (, 1). Since H is a probability measure for any w [, 1], then a + a 1 = 1 1 h β J (w)dw = 1 Hβ J (1), where, for β R J, we define the continuous part of the spectral measure H β J : (, 1) R1 by a linear combination of the M-splines, i.e., a function of the form: J H β J (w) = β j M j,q (w), (3.3) j=1

7 Bayesian inference with M-splines on spectral measure of bivariate extremes 7 where it will be clear in Proposition 1 that the function (spline) H β J ( ) is monotone by controlling in some way the values of the coefficients (β 1,..., β J ). Thus, the spectral measure H is absolutely continuous with respect to the reference measure µ on [, 1] given by µ(dw) = δ (dw) + dw + δ 1 (dw), (3.4) with density a, if w =, h(w) = h β J (w), if w (, 1), a 1, while if w = 1. (3.5) Remark 1 The class of spectral measures satisfying the Lebesgue decomposition (3.2) seems somewhat restrictive since it does not contain the atomless (has no atoms) continuous measures which are not absolutely continuous. An example of such measures is the Cantor distribution. For q = 1 the linear space of splines consists of piecewise constant functions with cell boundaries k/k for k =, 1,..., K. Our construction (3.3) therefore contains Cantor spectral measure constructed on piecewise constant functions as a special case (H J j=1 β jm j,1 ). Now, as the moment constraint is 1 wdh(w) = 1/2, then, using (3.3) and the derivative formula of M-spline functions as given in (de Boor, 1987, Ch. X), we can write 1 1 ( J ) wdh β J (w) = wd β j M j,q (w) = K = = 1 [ w J j=3 J j=1 j=1 ( J ) w (β j β j 1 )M j,q 1 (w) dw j=2 ] 1 β j M j,q (w) 1 J (q 1) β j M j,q (1) 1 J j=q j=j (q 2) J β j M j,q (w)dw (3.6) j=1 1 q 1 β j β j M j,q (w)dw (3.7) j=1 β j M j,q (w)dw 1 1 = Q(β 3:J, M 3:J,q ) β 1 M 1,q (w)dw β 2 M 2,q (w)dw = 1/2 a 1, where M 1,q (1) = M 2,q (1) = as K 2. We precise that we obtain the transition from (3.6) to (3.7) thanks to the M-splines property of normalization ( M j,q = 1) and in (3.7), the function M j,q (1) = w j,q (1)M j,q 1 (1) +

8 8 Khader Khadraoui, Pierre Ribereau {1 w j+1,q (1)}M j+1,q 1 (1) where w j,q (w) = w tj t j+q 1 t j if t j < t j+q 1 and otherwise. We recall the reader that the M-spline function of order 1 is M j,1 (w) = K1 [tj,t j+1)(w). Then, in order to introduce the moment and the monotone shape constraints, it will be convenient to work with the following parameterization. Suppose that we have a countable collection of candidate M-spline bases {M J, J J } where J = {4,..., J sup } and J sup N \. Model with M J basis has vector of unknown parameters, assumed to lie in ΘJ a Rd J, where the dimension d J = J + 2 may vary from model to model. Generally, all the parameters here vary over Θ = J J ( {J} Θ a J where ΘJ a is defined as follows: There exists a constant Cβa M that depends on β a = (a, β 1,..., β J, a 1 ) and M, where M = (M 1,q,..., M J,q ) denotes the M-spline basis, such that ΘJ a = {(a, β 1,..., β J 1, a 1 ) D : < a, a 1 < 1 } 2 and β 1 β J, (3.8) where D is a compact set of R J+1 and the Jth coefficient β J being a function of β 1,..., β J 1, a 1 and the M-spline basis via the mean restriction (3.7): tq+1 tq+2 β J = (1/2 a 1 + β 1 M 1,q (w)dw + β 2 M 2,q (w)dw ), )/( Q(β 3:J 1, M 3:J 1,q ) M J,q (1) 1 ) M J,q (w)dw t K+q 1 = C βa M. (3.9) Example 1 Take K = 2 and q = 2. Then, we obtain of course the sequence of knots t = (.5,,.5, 1, 1.5). In Figure 1, we plot the M-spline basis considered here in this example. It is easily seen from (3.7) that β 3 should satisfy in the M-spline estimator β 3 = β 1/2 1 1 M 1,2 (w)dw + β 2 M 2,2(w)dw a 1 M 3,2 (1) 1 1/2 M 3,2(w)dw = 1 2 β 1 + β a Remark 2 Arguably, although in the parameter subspace ΘJ a defined by (3.8) the atoms are restricted to (a, a 1 ) (, 1 2 )2, the Bernoulli( 1 2 ) spectral measure remains approximated closely here when (a, a 1 ) is close to ( 1 2, 1 2 ). As well, the Cantor spectral measure still be approximated closely when one may take (a, a 1 ) close to (, ) and q = 1 in the M-spline basis.

9 Bayesian inference with M-splines on spectral measure of bivariate extremes 9 K = 2 M 1,2 M 2,2 M 3,2 t 1 = 1/2 t 2 = t 3 = 1/2 t 4 = 1 t 5 = 1.5 Fig. 1 The linear M-spline basis considered in example 1; here q = 2. Consider µ the finite positive measure on [, 1] (its use was specified in (3.4)) and H the space of smooth spectral measures H, whose only atoms, if any, are at and at 1 where we mean by smooth all measure having a Lebesgue density h β J belongs to the Hölder space Cα (, 1). (This is the set of all functions that have α derivatives, for α the largest integer strictly smaller than α, with the α th derivative being Lipschitz of order α α.) From (3.2) and (3.3), we consider H a random function from a probability space (Θ, A, P) into (H, B) where B is the Borel σ field. Clearly, for (J, β a ) Θ, the σ field on H is the smallest such that the map (J, β a ) H is measurable. More precisely, a set E H is measurable if the set of β a J J R J+2 fulfilling constraints (3.8)-(3.9) and such that H E is a Borel set. For every H H, let β a be the vector of coordinates of the orthogonal projection from H onto the vector subspace C generated by the M-spline bases {M J, J J }, which is the approximating subspace. Thus, β is the unique β such that inf H c H β J J,β a ΘJ a J = Hc H β J, where β in ΘJ a satisfies the shape constraint β 1 β J with β J = C βa M. We turn now to the control of the monotone shape restriction and we explain how one may do this through controlling (simply) the vector of coefficients β. For this reason, we present in the following some statements concerning the link between the shape of H β J given by (3.3) and its coefficients (the monotonicity of H β J can be read off from its coefficients). These statements are useful in order to take into account the geometric prior information known from extreme value theory on H. Proposition 1 For w (, 1), let H β J (w) = J j=1 β jm j,q (w) where the coefficients (β 1,..., β J ) ΘJ a, β J = C βa M and (M 1,q,..., M J,q ) is an M-spline basis with order q > 2. If the coefficients verify β 1 β 2 β J, then the absolutely continuous part H β J of the spectral measure is monotone increasing: ( ) (1) H β J (w) for every w (, 1).

10 1 Khader Khadraoui, Pierre Ribereau Proof Using derivative formula of splines given in (de Boor, 1987, p.138) and the knot sequence t (equidistant knots), we deduce that for q > 1 ( J ) (1) J β j M j,q (w) = K (β j β j 1 )M j,q 1 (w), (3.1) j=1 j=2 where M j,q 1 is again an M-spline with the same knot sequence that M j,q but of one order lower. Since M j,q 1 (w) for every w (, 1) and if (β j β j 1 ) ( J (1) holds for j = 2,..., J, then the derivative j=1 β jm j,q (w)). We note that Proposition 1 provides a sufficient (but not necessary) condition under which H β J is a monotone increasing smooth function whose function and derivative values at and 1 may be prescribed to be. Remark 3 From (3.1), it is straightforward to explicit a convex (or concave) shape constraint using the M-spline basis in terms of the combination ( J (2) (β j + β j 2 2β j 1 ) for j = 3,..., J. Note that j=1 β jm j,q (w)) = K 2 J j=3 (β j 2β j 1 + β j 2 )M j,q 2 (w) for q > 2 and w (, 1). As a consequence, we are able to provide a sufficient condition under which the spline is a convex smooth function (if β j + β j 2 2β j 1 for j = 3,..., J) or a concave smooth function (if β j + β j 2 2β j 1 holds for j = 3,..., J). The convex shape is useful to estimate the Pikands dependence measure. A thorough analysis of the Pikands dependence measure is beyond the scope of the present paper. Clearly, controlling the continuous part shape of the spectral measure reduces to controlling the shape of a finite sequence of coefficients β 1,..., β J (Khadraoui, 217b). Thus, for a monotonicity constraint, it is straightforward to construct a set ΘJ a RJ+2 such that H β J necessarily fulfills the shape and the moment constraints. The following Proposition 2 provides Schonenberg s spline approximation for the absolutely continuous part of the spectral measure under the monotone shape constraint. Proposition 2 Let consider the subsets S J and S defined respectively by J } S J := {H βj = β j M j,q : β j R, β a ΘJ a, j=1 and { S := b i S i : S i i J J } S J and b i for i = 1, 2,.... Then, the closure of S in uniform norm is precisely the set of increasing continuous part of spectral measures on (, 1).

11 Bayesian inference with M-splines on spectral measure of bivariate extremes 11 Proof For a given function g on (, 1), the Schonenberg s spline approximation is defined by V g (w) = 1 K J g(t j )M j,q (w), (3.11) j=1 where t j := (t j t j+q 1 )/(q 1). It is obvious to remark that the closure S of S is contained in the set of increasing continuous part of spectral measures on (, 1). To prove the converse, it is sufficient to take H c be an increasing continuous part of a spectral measure and thus from the spline construction (3.11) it is easily seen that V H c is in S. From de Boor (1987), if H c is twice continuously differentiable, there exists a constant const H c,q (that may depend on H c and q) such that: sup H c (w) V H c(w) const Hc,q t 2, (3.12) w (,1) where t = max j t j t j 1. It follows from (3.12) that V H c converges to H c uniformly if t is small enough. Then, this shows that H c is in S which completes the proof. In other words, Proposition 2 means that any twice continuously differentiable function H c in S can be approximated by a spline in S generated from an M-spline basis. 3.2 Prior distribution The model approximating the tail of the unknown bivariate distribution function F is specified thanks to a spectral measure H H and marginal parameters (δ 1, δ 2 ) T 2 where δ l = (ζ l, η l, σ l ), l {1, 2} and T = (, ) (, ) 2. Since the parameter space for H is given by Θ, thus the complete parameter space is Ω = Θ T 2. In the sequel, we put θ = (J, β a ) and the model is parametrized by (θ, δ 1, δ 2 ) which defines F through the factorization: ({ } F : J J {J} ΘJ a (, ) 2 (, ) 4) F (θ, δ 1, δ 2 ) F (x, y), where F denotes the set of bivariate extreme value distributions and we put [ ] F (x, y) = exp l{ log F 1 (x), log F 2 (y)}, for (x, y) R 2, { ( ) 1/ηl x u F l (x) = exp ζ l 1 + η l l σ l }, for l {1, 2}, l(x 1, x 2 ) = 2 [,1] max(wx 1, (1 w)x 2 )dh(w), for (x 1, x 2 ) [, ) 2. The model is completed with a non-parametric prior distribution π for (θ, δ 1, δ 2 ) ({ J J {J} Θ a J } (, )2 (, ) 4 ). We precise that there should be no confusion whether π refers to either the constant or the prior

12 12 Khader Khadraoui, Pierre Ribereau distribution in the rest of the article. In the following we construct in particular a probability measure on the parameter space Θ where the map (J, β) H β J induces the absolutely continuous part of a probability measure on H which we specify as the prior for H c. We use π to denote a generic notation for some probability density. Proposition 3 Assume that π J (J) > for J = 4, 5,..., J sup together with π a (a) >, and the conditional density π J,β a(j, β a ) of π( {J} ΘJ a ) has support Θ. Let H c be a given continuous part of a spectral measure (increasing and continuous function on (, 1) with expectation equal to 1/2 a 1 ). Then, for every ɛ > π J,β a {(J, β a ) } ({J} ΘJ) a : H β J Hc < ɛ >. (3.13) J J Proof At first, we put VH J c(w) = J j=1 Hc (t j )M j,q(w) for every w (, 1). The application of the Schonenberg s spline approximation, by choosing K l sufficiently large ( t small enough), enables us to write V J l H c Hc ɛ/2, π J (J l ) > and π( {J l } Θ a J ) > has support Θ a l J. Then, using the fact l that H β J V H J c max β j H c (t j ), for β 1 β J and β J = C β j=1,...,j we can write { } π (J, β a ) Θ : H β J Hc < ɛ π {(J l, β al ) Θ : H βj V J l l H c < ɛ } { 2 π (J l, β al ) {J l } Θ a J : max β l l j=1,...,j l j H c (t j ) < ɛ } 2 >, which completes the proof. Proposition 3 shows that the support of the M-spline prior for the spectral measure can be quite large. Under the parameterization described previously, the prior distribution π is expressed as a trans-dimensional prior distribution on the random vector (θ, δ 1, δ 2 ), which, for convenience, factorizes as π J (J)π a (a)π β (β J J, a)π δ1 (δ 1 )π δ2 (δ 2 ) where β J = (β 1,..., β J 1 ). The prior distributions, with respect to the Lebesgue measure or the counting measure, for the spectral measure are specified as follows: (β J J, a, τ 2 ) N Θ J J 1 (m, τ 2 V) with density proportional to { exp β J V 1 β J 2τ }1 2 {β J Θ J }, τ 2 IG(τ 1, τ 2 ) with density equal to τ τ 1 2 Γ (τ (τ 2 1) ) τ1+1 exp{ τ2 τ }, 2 a U (,1/2) 2 with density proportional to 1 (,1/2) 2(a, a 1 ), J P(λ 1 ) with density equal to exp(λ 1 ) (λ1)j J!, a M,

13 Bayesian inference with M-splines on spectral measure of bivariate extremes 13 where Θ J = { } (β 1,..., β J 1 ) : (a, β 1,..., β J 1, a 1 ) ΘJ a, (3.14) m = (,..., ) is the prior expectation and V is the (J 1) (J 1) variance covariance matrix that will be specified in the following Proposition 4 in terms of the mean and the shape restrictions imposed on the coefficients. Concerning the marginal parameters, we consider independent priors for both margins given by π δl (δ l ) exp{ ζ l 2 } exp{ λ 2η l }σ l exp{ σ l λ 3 }, for l {1, 2}, where ζ l, η l and σ l respectively follow independent normal, exponential and gamma distributions. Concerning the adjustment of the prior-hyperparameters τ 1, τ 2, λ 1, λ 2 and λ 3, we set τ 1 = τ 2 =.1 in the prior of τ 2 (the variance of the truncated Gaussian prior π β ) such that the mode of the inverse gamma density is situated at τ 2 /(τ 1 + 1). We avoid to assign τ 1 and τ 2 to zero for not having an improper prior and avoid large values also to reduce the sensitivity of posterior inference to τ 1 and τ 2. In the same spirit, we assign λ 1 = λ 2 = 5 and λ 3 = 2. 4 Bayesian inference We introduce in this section the explicit likelihood used to develop the Bayesian inference. In this context, as Ledford and Tawn (1996), we adopt a censoring approach in the following manner: Let (X 1, X 2 ) = (X 1 u 1, X 2 u 2 ), we set I = (1 [u1, )(X 1 ), 1 [u2, )(X 2 )) and define: f (X1, X2 θ, τ 2, δ 1, δ 2 ) := F (u 1, u 2 θ, τ 2, δ 1, δ 2 ), if I = (, ), X 1 F (X 1, u 2 θ, τ 2, δ 1, δ 2 ), if I = (1, ), X 2 F (u 1, X 2 θ, τ 2, δ 1, δ 2 ), if I = (, 1), 2 X 1 X 2 F (X 1, X 2 θ, τ 2, δ 1, δ 2 ), if I = (1, 1), where an explicit expression for f will be obtained in the sequel. For σ l + η l (X l u l ) >, we consider that f l (X l ) = d F l (X l ) = ζ ( l X l u ) ( 1 l η +1) 1 + η l l Fl (X l ), for l {1, 2}, dx l σ l σ l where we write H({}) = a, H({1}) = a 1 and assume that H is absolutely continuous on (, 1) with Radon-Nikodym derivative h. It is straightforward

14 14 Khader Khadraoui, Pierre Ribereau that { 1 } l(x 1, x 2 ) = 2 a 1 + wh(w)dw, x 1 x 2 x 1 +x 2 { l(x 1, x 2 ) = 2 a + x 2 x 2 x 1 +x 2 } (1 w)h(w)dw, 2 x 1 x ( 2 l(x 1, x 2 ) = 2 x 1 x 2 (x 1 + x 2 ) 3 h x ) 2, x 1 + x 2 where l(x 1, x 2 ) can be writing l(x 1, x 2 ) = x 1 + x 2 + x1 x2 2 x l(x 1, x 2)dx 1dx 2, (x 1, x 2 ) [, ) 2. 1 x 2 We indicated previously that if H({}) > and H({1}) > (i.e., if the spectral measure has atoms at and 1), then f is positive on the set {(X1, X2 ) : σ l + η l (Xl u l) >, l = 1, 2}. We are now able to give an exact expression for f, by putting θ = (θ, τ 2, δ 1, δ 2 ), as follows: F (u 1, u 2 θ), if I = (, ), f 1(X 1) f (X1, X2 F 1(X 1) x 1 l(x 1, log{f 2 (u 2 )})F (X 1, u 2 θ), if I = (1, ), θ) = f 2(X 2) F 2(X 2) x 2 l( log{f 1 (u 1 )}, x 2 )F (u 1, X 2 θ), if I = (, 1), } { 2 l=1 f l (X l ) F l (X l ) l (x 1, x 2 )F (X 1, X 2 θ), if I = (1, 1), where x 1 = log{f 1 (X 1 )} and x 2 = log{f 2 (X 2 )} and l (x 1, x 2 ) = l(x 1, x 2 ) l(x 1, x 2 ) 2 l(x 1, x 2 ). x 1 x 2 x 1 x 2 Let X = {(X i1, X i2 ) : i = 1,..., n} stands for an observed sample from F and X = {(Xi1, X i2 ) : i = 1,..., n} stands for the corresponding censored sample. Then, the censored likelihood is given by n L(X θ) = f (Xi1, Xi2 θ). (4.1) i=1 It is natural that the censored likelihood (4.1) depends on the thresholds u 1 and u 2. The join posterior density for θ is computed from the Bayes theorem, up to a normalizing constant, as follows π(β J, a, τ 2, J, δ 1, δ 2 X ) L(X θ) π β (β J τ 2, a, J) π τ (τ 2 ) π J (J) π a (a) π δ1 (δ 1 ) π δ2 (δ 2 ). (4.2) Simulations from the posterior distribution (4.2) can be obtained by a reversible jumps Metropolis-Hastings algorithm (similarly to Khadraoui (217a)). Because the prior distribution π is expressed as a trans-dimensional prior distribution, implementation of the MCMC algorithm requires exact knowledge of π β (β J τ 2, a, J) (i.e., requires knowledge of the normalizing constant of the truncated Gaussian density N Θ J J 1 (, )).

15 Bayesian inference with M-splines on spectral measure of bivariate extremes 15 Proposition 4 Under the moment and the monotone constraints and for K 2, the truncated prior density of the coefficients (β J τ 2, a, J) with respect to the Lebesgue measure on R J 1 is given exactly by { π β (β J τ 2, a, J) = C βa M 1 J 1 exp( w2 exp } (J 1) 2τ )dw 2 { β J V 1 β } J 2τ 2 1 {β J Θ J }, (4.3) where V 1 = (v i,j ) 1 i,j J 1 is the (J 1) (J 1) variance-covariance matrix given by 2, if i = j = 1,..., J 2, 1, if i = j = J 1, v i,j = 1, if j = i ± 1, j = 1,..., J 1,, otherwise. Proof There exists a subset S J = {(β 1,..., β J 1 ) : β 1 β J 1 1}, (4.4) such that we can write Θ J = C βa M S J. We use λ(θ J ) and λ(s J ) to denote the normalizing constants corresponding respectively to Θ J and S J. Then, we can write (J 1)λ(SJ λ(θ J ) = {CM} βa ). (4.5) For the sake of simplicity and in order to obtain an expression for λ(s J ), we put ω = (ω 1,..., ω J 1 ) a random vector with Gaussian distribution where 1 [,1/J 1] each component is truncated to [, J 1 ] such that ω N J 1 (, τ 2 I J ) where I J is the (J 1) (J 1) identity matrix. Thus, the density of ω is given by ( π(ω τ 2, J) = λ exp ω ω where λ is a constant given by λ 1 = 1 J 1 ) J 1 2τ 2 j=1 1 { ωj 1 J 1 }, ( ) 1 exp ω2 J 1 ( ) 1 2τ 2 dω 1 exp ω2 J 2τ 2 dω J 1. Now, using the random vector ω, we can easily construct a vector β J with Gaussian distribution and satisfying the monotone constraint as follows β J = T 1 1 J ω, where TJ = (t i,j ) 1 i,j J 1 is the (J 1) (J 1) matrix given by { t 1, if i j, i,j =, else.

16 16 Khader Khadraoui, Pierre Ribereau By inverting the matrix T 1 J, we can also explicit ω in terms of β J in the following way: ω = T J β J, where T J = (t i,j ) 1 i,j J 1 is the matrix inverse of T 1 J given by 1, if i = j, t i,j = 1, if j = i 1, i = 2,..., J 1,, else. Therefore, the probability density of β J can be deduced from that of ω as follows: π(ω τ 2, J)dω = π(t J β J τ 2, J) dω dβ J dβ J ( = λ1 { TJ β J 1} exp (T Jβ J ) (T J β J ) ) dω dβ J 2τ 2 dβ J ( = λ1 {β1 β J 1 1} exp β J (T J T ) J)β J 2τ 2 dβ J, where the Jacobian dω dβ J = 1. Finally, we deduce the normalizing constant λ(θ J ) = } (J 1) { {C 1 βa J 1 ( ) (J 1), M exp dw} w2 (4.6) 2τ 2 which completes the proof. In the following section, we explore the numerical performances of our methodology to better qualify the contribution and provide some validation. 5 Numerical results In this section, we compare the performances of the spectral measure Bayes- M-splines estimator introduced in this paper with three estimators proposed recently in the literature in Einmahl et al. (21), Einmahl and Segers (29) and de Carvalhoc et al. (213). To be complete in this section, let us recall briefly these three methods that we aim to compare with. For this, let consider some i.i.d. observations x = {(x i,1, x i,2 ), i = 1,..., n} sampled from a distribution in the domain of attraction of a bivariate extreme value distribution. We start with fixed some threshold k n >. The choice of the threshold is both important and delicate where it determines the observations that will be used in the statistical inference. It is well known that the threshold must be lower than n and more precisely it must verify k n together with kn n. We return to this aspect in more details later. Once the threshold is established we use again r i,j the rank of x i,j in {x 1,j,..., x n,j } for i = 1,..., n and j {1, 2}. Let z i,j = n n+1 r i,j. For i = 1,..., n we denote by s i = z i,1 +z i,2 and ω i = zi,1 s i. Let put I n,kn the set of indices i = 1,..., n such that s i > n k n and I is the cardinal of I n,kn. Then, we can see that (ω i, i I n,kn ) as a sample from the spectral measure. It is therefore possible to do inference on H from

17 Bayesian inference with M-splines on spectral measure of bivariate extremes 17 (ω i, i I n,kn ). However, it must be signaled that a transformation such as the one we have just described creates dependence between ω i even when there is none between (X i,1, X i,2 ), i = 1,..., n. We are now in position to describe the three methods considered in the comparison: Empirical Spectral Measure (ESM): The authors in Einmahl et al. (21) proposed from the previous transformation an empirical estimator ĤESM of the spectral measure defined by Ĥ ESM (w) = 1 I i I n,kn 1 [ωi,1](w), w [, 1]. (5.1) This simple estimator poses a problem where it does not respect the mean constraint. Spectral measure based on the empirical maximum likelihood (MELE): The authors in Einmahl and Segers (29) proposed an estimator ĤMELE that satisfy the mean constraint and defined by Ĥ MELE (w) = 1 I i I n,kn p i 1 [ωi,1](w), w [, 1], (5.2) where p = (p 1,..., p I ) is solution to the following constrained optimization problem max p=(p1,...,p I ) R I i I + n,kn log(p i ), i I n,kn p i = 1, (5.3) i I n,kn ω i p i = 1 2. The solution is given by p i = 1 I 1 1+µ (ω i 1 2 ), i I n,k n where µ R is the Lagrange multiplier that solve i I n,kn ω i µ (ω i 1 2 ) =. The Euclidean maximum likelihood estimate (EMSM): The estimator denoted ĤEMSM and proposed in de Carvalhoc et al. (213) uses the same principle as ĤMELE but with the difference that the optimization problem that it solves is significantly simpler. It is defined by Ĥ EMSM (w) = 1 I i I n,kn p i 1 [ωi,1](w), ω [, 1], (5.4) where p is solution to the following constrained optimization problem 1 max p=(p1,...,p I ) R I 2 i I + n,kn ( I p i 1) 2, i I n,kn p i = 1, (5.5) i I n,kn ω i p i = 1 2. The solution is given by p i = 1 I (1 ( ω 1 2 )S 2 (ω i ω)) with ω = 1 I i I n,k ω i and S 2 = 1 I i I n,kn (ω i ω) 2.

18 18 Khader Khadraoui, Pierre Ribereau As we have just seen, the previous three methods are based on the rank transformed sample (ω i, i I n,kn ) whereas our Bayes M-spline methodology is based on the censored sample x. Now we proceed to compare our method with these three previously described methods. For each of the estimators, we draw 1 samples (each of size n = 1) from three well known models; two models are logistic and one asymmetric (more details about these models will be given in the following). For the methods based on rank sample, we consider 1 different thresholds k n = n α with α [.55,.7] is chosen in an equidistant grid. For the Bayes method, the two thresholds u 1 and u 2 are chosen in a manner that we obtain the same number of observations in the tail region determined thanks to the threshold k n (for each α and each sample). For each value of the threshold, we assess the performance of the estimators via the mean integrated square error MISE(Ĥ) = E [ 1 ] (Ĥ(w) H(w))2 dw. (5.6) The first model used is the logistic model since it is very useful in the theory of extreme values because of its great adaptability. Indeed, for e > 1, it is given by H e (w) = 1 ( ) 1 ((1 w) e 1 w e 1 )((1 w) e + w e ) 1+ 1 e. (5.7) 2 In this model (5.7) when e we obtain the case of independence and the spectral measure is concentrated at 1 2. Conversely, H puts a mass 1 2 at and at 1 when e 1 which coincide with the case of total dependence. For the simulations, we take e = 2 which corresponds to a moderate dependence and e = 4 which corresponds to a very marked independence. The second model used is the asymmetric logistic model. It generalizes the logistic model by allowing an asymmetry of the marginal distributions by the introduction of two additional parameters. For e > 1, it is given by H e,φ1,φ 2 (w) = 1 ( 1 + φ 1 φ 2 (φ e 2 1(1 w) e 1 φ e 2w e 1 ) ) (5.8) (φ e 1(1 w) e + φ e 2w e ) 1+ 1 e. In the simulations we take e = 2.5, φ 1 =.4 and φ 2 =.6 which enable us to obtain a spectral measure that presents two atoms at and at 1 given respectively by H e,φ1,φ 2 ({}) = 1 φ2 2 =.2 and H r,φ1,φ 2 ({1}) = 1+φ1 2 =.7. It is known that the three estimators presented above are unable to detect the atoms (we have always ω i ], 1[) and this example is a good illustration to measure the extent to which the estimate is deteriorated by the presence of atoms concerning the methods based on the rank sample. The simulation results are summarized in Figures 2, 3 and 4. One remarkable aspect of this numerical study is how our estimator performed to take into account the presence of atoms and the monotone with the mean constraints. For different value of α (different thresholds) the mean integrated square error

19 Bayesian inference with M-splines on spectral measure of bivariate extremes 19 MISE Bayes EMSM MELE ESM Spectral measure (a) MISE 1 3. α (b) Spectral measure estimate, here α =.575 w Posterior probability Number of interior knots (J q) (c) Posterior probability of knots, here α =.575. Fig. 2 Figures shown the simulation results for the logistic model with e = 2. (a) MISE of each estimator as a function of the threshold for α [.55,.7] and 1 samples with n = 1. (b) Bayes M-splines spectral measure estimate with 95% credible intervals with α =.575. (c) Posterior distribution of the number of interior knots with α =.575. (MISE) is sensitive to the choice of the threshold. The sensitivity of the error of estimation with α is illustrated by Figures 2(a), 3(a) and 4(a). Usually we note that the differences between the error of ESM-method, MELE-method and EMSM-method are not very significant. For the logistic model with e = 2 and e = 4 (Figures 2 and 3) there are no atoms and the error of estimation for the three methods based on rank sample is greater than those of our method. This feature can be explained by the fact that our estimate is smooth since it is a spline (whereas the others are piecewise linear) together with the fact that the free-knot approach considered here detects better the high and low variability regions of the data and facilitates the insertion of more coefficients in the high variability region. For the asymmetric model (see Figure 4) the presence of atoms at and at 1 deteriorates the error of estimation for the

20 2 Khader Khadraoui, Pierre Ribereau MISE Bayes EMSM MELE ESM Spectral measure (a) MISE 1 3. α (b) Spectral measure estimate, here α =.65. w Posterior probability Number of interior knots (J q) (c) Posterior probability of knots, here α =.65. Fig. 3 Figures shown the simulation results for the logistic model with e = 4. (a) MISE of each estimator as a function of the threshold for α [.55,.7] and 1 samples with n = 1. (b) Bayes M-splines spectral measure estimate with 95% credible intervals with α =.65. (c) Posterior distribution of the number of interior knots with α =.65. three methods (ESM, MELE and EMSM) contrariwise our estimate remains consistent on [, 1]. 6 Discussion The simulation results of Section 6 indicate that our Bayes M-spline method does considerably better than the three methods based on rank sample for the logistic and the asymmetric models studied in this paper. On the basis of the numerical study evidence, it appears that our method is a good robust c hoice since it always competitive with the others estimates of spectral measure and does considerably better for spectral measures with atoms, such as the

21 Bayesian inference with M-splines on spectral measure of bivariate extremes 21 MISE Bayes EMSM MELE ESM Spectral measure (a) MISE 1 1. α (b) Spectral measure estimate, here α =.575. w Posterior probability Number of interior knots (J q) (c) Posterior probability of knots, here α =.575. Fig. 4 Figures shown the simulation results for the asymmetric model with e = 2.5, φ 1 =.4 and φ 2 =.6. (a) MISE of each estimator as a function of the threshold for α [.55,.7] and 1 samples with n = 1. (b) Bayes M-splines spectral measure estimate with 95% credible intervals with α =.575. (c) Posterior distribution of the number of interior knots with α =.575. asymmetric model. In all cases, it is preferable to choose the Bayes M-spline estimator that gives estimator of the spectral measure with better estimation properties and smaller error of estimation. Of course, these better results are at the expense of additional model complexity. Such numerical efficiency improvements have been pointed out also in the paper Guillotte et al. (211). We precise here that B-splines and M-splines modeling of the tail dependence were introduced first by K. Khadraoui and P. Ribereau in an unpublished 213 manuscript (Khadraoui and Ribereau, 213) and second in Topyurek et al. (213). The methodology developed in this article open a door to other complex problems such as the extension to multivariate spectral measure estimation

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