CRPS M-ESTIMATION FOR MAX-STABLE MODELS WORKING MANUSCRIPT DO NOT DISTRIBUTE. By Robert A. Yuen and Stilian Stoev University of Michigan

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1 arxiv: math.pr/ CRPS M-ESTIMATION FOR MAX-STABLE MODELS WORKING MANUSCRIPT DO NOT DISTRIBUTE By Robert A. Yuen and Stilian Stoev University of Michigan Max-stable random fields provide canonical models for the dependence of multivariate extremes. Inference with such models has been challenging due to the lack of tractable likelihoods. In contrast, the finite dimensional cumulative distribution functions (CDFs) are often readily available and natural to work with. Motivated by this fact, in this work we develop an M-estimation framework for max-stable models based on the continuous ranked probability score (CRPS) of multivariate CDFs. We start by establishing conditions for the consistency and asymptotic normality of the CRPS-based estimators in a general context. We then implement them in the max-stable setting and provide readily computable expressions for their asymptotic covariance matrices. The resulting point and asymptotic confidence interval estimates are illustrated over popular simulated models. They enjoy accurate coverages and offer an alternative to likelihood based methods. The new CRPS-based estimators were used to study rainfall extremes in Switzerland. 1. Introduction. Max-stable processes are a canonical class of statistical models for multivariate extremes. They appear in a variety of applications ranging from insurance and finance (Embrechts et al., 1997; Finkenstädt and Rootzén, 4) to spatial extremes such as precipitation (Davison and Blanchet, 11; Davison et al., 1) and extreme temperature (Erhardt and Smith, 11). Max-stable processes are exactly the class of non-degenerate stochastic processes that arise from limits of independent component-wise maxima. This provides a theoretical justification for their modeling of multivariate extremes. However, many useful max-stable models suffer from intractable likelihoods, thus prohibiting standard maximum likelihood and Bayesian inference. This has motivated development of composite maximum likelihood inference for fitting max-stable models (Padoan et al., 1) as well as certain approximate Bayesian approaches (Reich and Shaby, 1), (Erhardt and Smith, 11). Support: UM Rackham Merit Fellowship and NSF-AGEP Grand DMS Support: ** AMS subject classifications: Primary 6K35, 6K35; secondary 6K35 Keywords and phrases: Max-stable, Extreme value 1

2 YUEN AND STOEV In contrast to their likelihoods, the cumulative distribution functions (CDFs) for many max-stable models are available in closed form, or they are tractable enough to approximate within arbitrary precision. This motivates statistical inference based on the minimum distance method (Wolfowitz, 1957), (Parr and Schucany, 198). In this paper, we propose an M-estimator for parametric max-stable models based on minimizing distances of the type (1.1) (F θ (x) F n (x)) µ (dx). R d where F θ is a d-dimensional CDF of a parametric model, F n is a corresponding empirical CDF and µ is a tuning measure that emphasizes various regions of the sample space R d. Using elementary manipulations it can be shown that minimizing distances of the type (1.1) is equivalent to minimizing the continuous ranked probability score (CRPS) Definition 1. (CRPS M-estimator) Let µ be a measure that can be tuned to emphasize regions of a sample space R d. Define ( ) (1.) E θ (x) = Fθ (y) 1 {x y} µ (dy) R d Then for independent random vectors { X (i)} n with common distribution i=1 function F θ we define the following CRPS M-estimator for θ. (1.3) θn = argmin θ Θ n E θ (X (i)). i=1 For simplicity, we shall assume throughout that the parameter space Θ is a compact subset of R p, for some integer p. The remainder of this paper is organized as follows. In Section we review some essential multivariate extreme value theory and provide definitions and constructions of max-stable models. In Section 3 we establish regularity conditions for consistency and asymptotic normality of the CRPS M-estimator and provide general formulae for calculating its asymptotic variance. In Section 4 we specialize these calculations to the max-stable setting. In Section 5 we conduct a simulation study to evaluate the proposed estimator and in Section 6 we provide an application to precipitation extremes over Switzerland.

3 CRPS M-ESTIMATION FOR MAX-STABLE MODELS: DO NOT DISTRIBUTE 3. Extreme values and max-stability. Let Y (i) = {Y (i) t } t T, i = 1,, be independent and identically distributed measurements of certain environmental or physical phenomena. For example, the Y (i) t s may model wave-height, temperature, precipitation, or pollutant concentration levels at a site t in a spatial region T R. If one is interested in extremes, it is natural to consider the asymptotic behavior of the point-wise maxima. Suppose that, for some a n (t) > and b n (t) R, we have (.1) { 1 } max a n (t) Y (i) t b n (t) i=1,,n t T d {X t } t T, as n, for some non-trivial limit process X, where d denotes convergence of the finite-dimensional distributions. The class of extreme value processes X = {X t } t T arising in the limit describe the statistical dependence of worst case scenaria and are therefore natural models of multivariate extremes. The limit X in (.1) is necessarily a max-stable process in the sense that for all n, there exist c n (t) > and d n (t) R, such that { 1 } max c n (t) i=1,,n X(i) t d n (t) t T d = {X t } t T, where {X (i) t } t T are independent copies of X and where = d means equality of finite-dimensional distributions (Ch.5 in Resnick (1987)).Due to the classic results of Fisher-Tippett and Gnedenko, the marginals of X are necessarily extreme value distributions (Fréchet, reversed Weibul or Gumbel). They can be described in a unified way through the generalized extreme value distribution (GEV): { } (.) G ξ,µ,σ (x) := exp (1 + ξ(x µ)/σ) 1/ξ +, σ >, where x + = max{x, }, and where µ, σ and ξ are known as the location, scale and shape parameters. The cases ξ >, ξ <, and ξ correspond to Fréchet, reverse Weibull, and Gumbel, respectively (see, e.g. Ch.3 and 6.3 in Embrechts et al. (1997) for more details). The dependence structure of the limit extreme value process X rather than its marginals is of utmost interest in practice. Arguably, the type of the marginals is unrelated to the dependence structure of X and as it is customarily done, we shall assume that the limit X has standard 1-Fréchet marginals. That is, (.3) P(X t x) = G 1,1,σt (x) = e σt/x, x >, for some scale σ t > (Ch.5 of Resnick, 1987).

4 4 YUEN AND STOEV.1. Representations of max-stable processes. Let X = {X t } t T be a max-stable process with 1-Fréchet marginals as in (.3). Then, its finitedimensional distributions are multivariate max-stable random vectors and they have the following representation: (.4) P(X ti x i, i = 1,, d) = exp { S d 1 + ( ) max w i/x i i=1,,d } H(dw), where x i >, t i T, i = 1,, d and where H = H t1,,t d is a finite measure on the positive unit sphere S d 1 + = {w = (w i) d i=1 : w i, d w i = 1} known as the spectral measure of the max-stable random vector (X ti ) d i=1 (see e.g. Proposition 5.11 in Resnick (1987)). The integral in the expression (.4) is referred to as the tail dependence function of the max-stable law. We shall often use the notation: i=1 V (x) V t1,,t d (x) := log P(X ti x i, i = 1,, d), where x = (x i ) d i=1 Rd +, for the tail dependence function of the max-stable random vector (X ti ) d i=1. It readily follows from (.4) that for all a i, i = 1,, d, the max-linear combination ξ := max a ix ti i=1,,d is 1-Fréchet random variable with scale σ ξ = d 1 S (max i=1,,d a i w i )H(dw). + with the property that all its non- Conversely, a random vector (X ti ) d i=1 negative max-linear combinations are 1-Fréchet is necessarily multivariate max-stable (de Haan (1978)). This invariance to max-linear combinations is an important feature that will be used in our estimation methodology (Section 4, below). Some max-stable models are readily expressed in terms of their spectral measures while others via tail dependence functions. These representations however are not convenient for computer simulation or in the case of random processes, where one needs a handle of all finite-dimensional distributions. The most common constructive representation of max-stable process models is based on Poisson point processes (de Haan, 1984b; Schlather, ; Kabluchko et al., 9). See also (Stoev and Taqqu, 5) for an alternative. Indeed, consider a measure space (S, S, ν) and let Π := {(ɛ i, S i )} i N be a Poisson point process on R + S with intensity measure dxdν.

5 CRPS M-ESTIMATION FOR MAX-STABLE MODELS: DO NOT DISTRIBUTE 5 Proposition 1. Let g t L 1 (S, S, ν), t T be a collection of nonnegative integrable functions and let (.5) X t := S g t dπ max i N ɛ 1 i g t (S i ), (t T ). Then, the process X = {X t } t T is max-stable with 1-Fréchet marginals and finite-dimensional distributions: { ( ) } (.6) P(X ti x i, i = 1,, d) = exp max g t i (s)/x i ν(ds). i=1,,d The proof of this result is sketched in the Appendix. Relation (.5) is known as the de Haan spectral representation of X and {g t } t T L 1 +(S, S, ν) as the spectral functions of the process. It can be shown that every separable in probability max-stable process has such a representation (Proposition 3. in Stoev and Taqqu (5)). The max-functional in (.5) has the properties of an extremal stochastic integral. Indeed, we have max-linearity: max a ix ti = i=1,,d S S ( ) max a ig ti dπ, i=1,,d for all a i. The above max-linear combination is therefore 1-Fréchet and has a scale coefficient: ( ) max a ig ti dν = max a L ig. ti i=1,,d i=1,,d 1 (ν) S One can also show that X t and X s are independent, if and only if g t (u)g s (u) =, for ν-almost all u S. That is, the extremal integrals defining X t and X s are over non-overlapping sets. This shows that for max-stable process models pairwise independence implies independence. Further, X tn converges in probability to X t if and only if g tn converges in L 1 (ν) to g t, as n. For more details, see e.g de Haan (1984a) and Stoev and Taqqu (5). Remark 1. The expressions (.4) and (.6) may be related through a change of variables (Proposition 5.11 Resnick (1987)). While the spectral measure H in (.4) is unique, a max-stable process has many different spectral function representations. Nevertheless, Relation (.5) provides a constructive and intuitive representation of X, that can be used to build interpretable models.

6 6 YUEN AND STOEV.. Max-stable models. A great variety of max-stable models can be defined by specifying the measure space (S, S, ν) and an accompanying family of spectral functions g t or equivalently through a consistent family of spectral measures or tail dependence functions. We review next several popular max-stable models and their basic features. (Multivariate logistic) Let X = (X ti ) d i=1 have the CDF ( d ) α, F X (x) = e V (x), where V (x) = σ x 1/α t i for σ > and α [, 1]. The parameter α controls the degree of dependence, where α = 1 corresponds to independence (V (x) = σ d i=1 x 1 i ), while α = to complete dependence (V (x) = σ max i=1,,d x 1 i, interpreted as a limit). This model is rather simple since the dependence is exchangeable but it provides a useful benchmark for the performance of the CRPS-based estimators since the MLE is easy to obtain in this case (see Table below). The recent works of Fougères et al. (9) and Fougères et al. (13) develop far-reaching generalizations of multivariate logistic laws by exploiting connections to sum-stable distributions. (Max-linear or spectrally discrete models) Let A = (a ij ) d k be a matrix with non-negative entries and let Z j, j = 1,, k be independent standard 1-Fréchet random variables. Define i=1 (.7) X i = max j=1,,k a ijz j, i = 1,, d. The vector X = (X i ) d i=1 is max-stable. It can be shown that the CDF of X has the form (.4) were the spectral measure (.8) H(dw) = k a j δ {a j / a j }(dw), j=1 is concentrated on the normalized column-vectors of the matrix A, i.e. on a j / a j := (a ij / a j ) d i=1, where a j = d i=1 a ij. Conversely, any max-stable random vector with discrete spectral measure H has a max-linear representation as in (.7), where the columns of the matrix A may be recovered from (.8). We shall also call such models spectrally discrete. Since any spectral measure H can be approximated arbitrarily well with a discrete one, max-linear models are dense in the class of all max-stable

7 CRPS M-ESTIMATION FOR MAX-STABLE MODELS: DO NOT DISTRIBUTE 7 models. As argued in Einmahl et al. (1), max-linear distributions arise naturally in economics and finance, as models extreme losses. The Z j s represent independent shock-factors that lead to various extreme losses in a portfolio X depending on the factor loadings a ij. Max-linear models are particularly well-suited for CRPS-based inference, since their tail dependence function has a simple closed form: (.9) V (x) = k max a ij/x i, x = (x i ) d i=1 R d +. i=1,,d j=1 See Section 5 below for a simple example of CRPS-based inference for maxlinear models and Einmahl et al. (1) for an alternative M-estimation methodology. (Moving maxima and mixed moving maxima) Let (S, S, ν) (R k, B R k, Leb) and g t (s) := g(t s), t, s R k, for some non-negative integrable function g, R g(s)ds <. Then (.5) yields the so-called moving maxima k random field: X t := R k g(t s)dπ(s) max i N g(t S i)/ɛ i, (t R k ). The choice of the kernel g as a multivariate Normal density in R yields the well-known Smith storm model, where the S i s may be interpreted as storm locations, g is the spatial storm attenuation profile and 1/ɛ i its strength. More flexible models can be obtained by taking maxima of independent moving maxima, resulting into the so-called mixed moving maxima: (.1) X t = R k U g(t s, u)dπ(s, u) max i N g(t S i, U i )/ɛ i, (t R k ) where Π is a Poisson point process on S = R k U with intensity ν(ds, du) = dsm(du), and where g is such that R k U g(s, u)dsm(du) <. Here m(du) is the mixing measure, which may be continuous or discrete, and the U i s may be viewed as different types of storms. The mixed moving maxima random fields are stationary, ergodic and, in fact, mixing (Stoev, 8; Kabluchko and Schlather, 1). By (.6), their tail dependence functions are ( ) V (x) = max g(t i s, u)/x i dsm(du), x = (x i ) d i=1 R d +. i=1,,d R k U

8 8 YUEN AND STOEV (Spectrally Gaussian models) By viewing (S, S, ν) as a probability space, in the case ν(s) = 1, the spectral functions {g t } t T in (.5) become a stochastic process. By picking g t = h(w t ) to be non-negative transformations of a Gaussian process w t on (S, S, ν), one obtains interesting and tractable max-stable models whose dependence structure is governed by the covariance structure of the underlying Gaussian process {w t } t T. The popular Smith, Schlather, and Brown-Resnick random field models are of this type (Smith, 199; Schlather, ; Brown and Resnick, 1977; Stoev, 8; Kabluchko et al., 9). (Schalther models) Let {w t } t R k be a stationary Gaussian random field with zero mean and let g t (s) := (w t (s)) +, s S. Then X t in (.5) has the following tail dependence function ( ) (.11) V (x) = E ν max (w t i ) + /x i, x = (x i ) d i=1 R d +, i=1,,d where E ν denotes integration with respect to the probability measure ν. (Brown-Resnick) Let w = {w t } t R k be a zero mean Gaussian random field with stationary increments. Set g t (s) := e wt(s) vt/, where v t = E ν (wt ) is the variance of w t. The seminal paper of (Brown and Resnick, 1977) introduced this model with w the standard Brownian motion and showed that, surprisingly, the resulting max-stable process X t in (.5) is stationary, even though w is not. The cornerstone work of Kabluchko et al. (9) showed that {X t } t R k is stationary for a general stationary increments centered Gaussian process w. It also obtained important mixed moving maxima representations of X under further conditions on w. The tail dependence function of X in this case is ( ) (.1) V (x) = E ν max i=1,,d ewt i vt i / /x i, x = (x i ) d i=1 R d +. It can be shown that the Smith model (Smith, 199) is a special case of a Brown-Resnick model with a degenerate random field {w t } d = {t Z}, t R k, k < d, where Z is a Normal random vector in R k. The above models can be deemed spectrally Gaussian since their tail dependence functions (and hence spectral measures) are expectations of functions of Gaussian laws. One can consider other stochastic process models for the underlying spectral functions g t and thus arriving at doubly stochastic max-stable processes. We comment briefly on some general probabilistic properties of these models. Remark. If {g t } t R k is a stationary process in (S, S, ν), then the maxstable process X = {X t } t R k is also stationary and non-ergodic. Thus the

9 CRPS M-ESTIMATION FOR MAX-STABLE MODELS: DO NOT DISTRIBUTE 9 Schlather models are non-ergodic. This is important in applications, since a single observation of the random field X at an expanding grid, may not yield consistent parameter estimates. Kabluchko et al. (9) have shown that Brown-Resnick random fields with non-stationary {w t } such that lim t (w t v t /s) =, almost surely, have mixed moving maxima representations as in (.1). They are therefore mixing (?) and consistent statistical inference from a single realization of such max-stable random fields is possible. Remark 3. The Poisson point process construction in (.5) involves a maximum over an infinite number of terms. As a result, computer simulations of spectrally Gaussian max-stable models necessitates truncation to a finite number. In the case of the Brown-Resnick model, the number of terms required to produce a satisfactory representation is prohibitively large. Accurate simulation of Brown-Resnick processes is an active area of study (Oesting et al., 11). Consequently, valid simulation studies regarding inference under Brown-Resnick models have yet to appear. For this reason the remaining discussion of spectrally Gaussian max-stable models including simulation and application is restricted to the Schlather model..3. Measures of dependence in max-stable models. (Co-variation) For X t as in (.5), define [X t, X s ] := g t g s dν g t dν + g s dν S S S S g t g s dν, (t, s T ). Note that S g tdν and S (g t g s )dν are the scale coefficients of the Fréchet random variables X t and X t X s. The co-variation [X t, X s ] is nonnegative and equals zero if and only if X t and X s are independent. Thus, it plays a somewhat similar role to that of the covariance for Gaussian processes. (Extremal coefficient) A popular summary measure of multivariate dependence in max-stable models is the extremal coefficient. Define ϑ (D) := log P (X t 1, t D) V (1). For a process {X t, t D} with standard 1-Fréchet marginals max t D 1 x t V (x) t D 1 x t

10 1 YUEN AND STOEV Table 1 Correlation functions for Gaussian random fields. For the Matérn covariance function, K θ is the modified Bessel function of the second kind. ρ θ (t, s), h = t s Stable exp [ (h/θ 1) θ ] θ 1 >, θ (, ] Matérn ( θ h/θ 1 ) θ Γ(θ ) θ 1 K θ ( θh/θ 1 ) θ 1 >, θ > Cauchy (1 + (h/θ 1) ) θ θ 1 >, θ > and thus 1 ϑ (D) d, where ϑ (D) = 1 corresponds to complete dependence while ϑ (D) = d implies that X t s, t D are independent. Using the fact that V t (1) + V s (1) V t,s (1, 1) = [X t, X s ], for a process with standard 1-Fréchet marginals, we obtain ϑ({t, t + h}) = [X t, X t+h ]. In the case of the Schlather model there exist and explicit formula for the bivariate extremal coefficient in terms of the correlation function which is ϑ({t, s}) = 1 + (1 ρ(t, s))/. Figure 1 displays realizations from the Schlather model for differing correlation functions given in Table 1. Note that these examples are all (spectrally) isotropic in the sense that the correlation ρ (t, s) of the underlying Gaussian process depends only on the distance h = t s between locations t and s. This however is not a requirement in general. Figure 1 also provides some visual evidence that the covariance structure of w influences the dependence structure of the resultant max-stable random field X. Consequently, it possible to parameterize the dependence structure of the max-stable random field using a large variety of covariance functions available for parameterizing Gaussian processes. 3. Consistency and asymptotic normality of CRPS M-estimators. In this section, we establish general conditions for the consistency and asymptotic normality of CRPS-based M-estimators. This is motivated by questions of inference in max-stable models, but may be of independent interest. Section 4 implements and specializes these results to the max-stable setting. Theorems here are distillations of those commonly given in general theory for M-estimators, for example see (van der Vaart, 1998). Their proofs are given in Appendix A..

11 CRPS M-ESTIMATION FOR MAX-STABLE MODELS: DO NOT DISTRIBUTE 11 Fig 1. Schlather max-stable model realizations using correlation functions of Table 1 under varying parameter settings. Top: Stable correlation function. Middle: Matérn correlation function. Bottom: Cauchy correlation function. Realizations were generated using the R package SpatialExtremes (Ribatet, 11). The circles indicate locations of observation staions in the simulation study of Section θ 1 = 1, θ = θ 1 = 5, θ = θ 1 = 1, θ = θ 1 =, θ = θ 1 = 1, θ = θ 1 = 3.5, θ =

12 1 YUEN AND STOEV Theorem. Let X, X (1), X (),... be iid random vectors with cumulative distribution function F θ. Let θ n be as in Definition 1 with θ an interior point of Θ. Suppose that the following conditions hold: (i) (identifiability) For all θ 1, θ Θ, (3.1) θ 1 θ F θ1 F θ a.e µ (ii) (integrability) For B (θ ) Θ, an open neighborhood of θ (3.) sup R d θ B(θ ) (1 F θ (x)) µ (dx) <. (iii) (continuity) The function θ R d (F θ (x) F θ (x)) µ(dx) is continuous in the compact parameter space Θ R p. Then θ n p θ, as n. Theorem 3. Assume the conditions and notation of Theorem hold so that in particular, θ p n θ. Moreover, suppose that: (i) The measurable function θ E θ (x) is differentiable at θ (for almost every x) with gradient E θ (x) := θ E θ (x). θ=θ (ii) There exists a measurable function L (x) with E (L (X)) <, such that for every θ 1 and θ in B (θ ) (3.3) E θ1 (x) E θ (x) L (x) θ 1 θ. (iii) The map θ EE θ (X) admits a second-order Taylor expansion at the point of minimum θ with non-singular second derivative matrix (3.4) H θ := θ θ EE θ (X). θ=θ Then (3.5) ) ) d n (θn θ N (, H 1 θ J θ H 1 θ, as n, where (3.6) J θ := E { E θ (X) E θ (X) }.

13 CRPS M-ESTIMATION FOR MAX-STABLE MODELS: DO NOT DISTRIBUTE 13 The following result provides explicit conditions on the family of CDFs {F θ, θ Θ} that imply conditions (i)-(iii) of Theorem 3. It also gives concrete expressions for the bread and meat matrices H θ and J θ in terms of F θ, which can be used to compute the asymptotic covariances in (3.5). Proposition. Assume the conditions and notation in Theorem. Suppose moreover that: (i) θ F θ (y) is twice continuously differentiable for all θ in B (θ ) with gradient F θ (y) := F θ (y) / θ and second derivative matrix F θ (y) := F θ (y) / θ θ. (ii) For all a R p with a > ( (3.7) a F θ (y)) µ (dy) >. R d (iii) R ( sup d θ B(θ ) F θ (y) + F θ (y) + F ) θ (y) µ (dy) <. Then (i)-(iii) of Theorem 3 are satisfied and therefore (3.5) holds, where (3.8) H θ := F θ (y) F θ (y) µ (dy) R d and (3.9) J θ := 4 β θ (y 1, y ) F θ (y 1 ) F θ (y 1 ) µ (dy 1 ) µ (dy ) R d R d where β θ (y 1, y ) = F θ (y 1 y ) F θ (y 1 ) F θ (y ). Remark 4. Practical inference utilizing the CRPS M-estimator is limited to cases where optimization of θ E θ is feasible. Likewise confidence intervals are only obtained when the matrices H 1 θ, J θ can be computed. Given the multivariate integration involved, this may require specialized methods for various models. In the max-stable setting this is achieved through judicious specification of the measure µ, discussed in the following section. Remark 5. Condition (3.7) ensures that the bread matrix H θ in (3.8) is non-singular. It is rather mild and fails only if the gradient F θ (y) lies in a lower dimensional hyper-plane for µ-alomost all y. In practice, unless the model is over-parameterized this condition typically holds. Remark 6. The expressions (3.8) and (3.9) can be used in practice to compute the asymptotic covariance matrix in (3.5). In Sections 4 and 5 we have implemented numerical and Monte Carlo based methods for calculating H θ and J θ under the models introduced in Section.

14 14 YUEN AND STOEV 4. CRPS M-estimation for max-stable models. Our goal is to implement the general CRPS method of the previous section to the case of multivariate max-stable models described in Section. Calculation of the CRPS for such models is aided by a closed form expression of the univariate CRPS for 1-Féchet random variates which is established by following Lemma. Lemma 1. Suppose the measure µ in Definition 1 of the CRPS is specified as µ (dr) = r 1/ dr for r R +. Then the univariate CRPS with respect to the 1-Féchet distribution function e v/r has the following closed form (4.1) F (m, v) := ( e v/r 1 {m r} ) r 1/ dr [ ( m = 4 e v/m 1 ) + ( v γ 1 (v/m) where γ α (z) = z tα 1 e t dt is the incomplete gamma function. )] π, See Appendix A.3 for a proof. We introduce the notation F to distinguish the univariate Fréchet CRPS from the multivariate case. The functional F is the basis for many of the calculations that follow. Now recall that the CDF of a 1-Féchet max-stable random vector X = (X i ) i=1,...,d is characterized by the tail function V (x) as follows (4.) F X (x) = P (X i x i, i = 1,..., d) = e V (x), where V exhibits a homogeneity property V (rx) = V (x) /r for all r >, x (, ] d. This means that for any u = (u i ) i=1,...,d R d +, the max-linear combination (4.3) M u := max i=1,...,d X i u i is a 1-Fréchet variable with scale V (u). Indeed, P (M u r) = P (X i ru i, i = 1,..., d) = e V (ru) = e V (u)/r. This max-linearity invariance property motivates a particular choice of the measure µ that appears in Definition 1 for the multivariate CRPS. Let (4.4) µ (dy) µ (dr, du) := r 1/ dr w U δ {w} (du), where u = y/ y, r = y = d i=1 y i and U R d +. With this choice of µ we have the following closed form expression for the multivariate CRPS in terms of the max-linear combinations {M u } u U.

15 CRPS M-ESTIMATION FOR MAX-STABLE MODELS: DO NOT DISTRIBUTE 15 Proposition 3. With µ as in (4.4), for the CRPS in (1.), we have (4.5) E θ (X) = [, ) d [ e V θ(y) 1 {X y} ] µ (dy) = u U F (M u, V θ (u)) with F as in Lemma 1. Proof. Using the substitution u = y/ y and r = y, specifying the measure µ as in (4.4) results in [ ] E θ (X) = e Vθ(y) 1 {X y} µ (dy) = u U [, ) d [e V θ(ru) 1 {X ru} ] r 1/ dr. Observe that {X ru} = {X i ru i, i = 1,..., d} is equivalent to {M u r}, where M u is as in (4.3). Therefore, using the homogeneity property V θ (ru) = V θ (u) /r, we obtain [e V θ(ur) 1 {X ru} ] r 1/ dr = Lemma 1 applied to the last integral yields (4.5). [ ] e Vθ(u)/r 1 {Mu r} r 1/ dr. In practice, given a set of independent observations X (1), X (),..., X (n) from the model F θ (x) = exp ( V θ (x)) we obtain the CRPS-based estimator of θ as follows CRPS estimation procedure 1. Construct the set U R d +. The distribution of U can be determined heuristically. In { general, finite uniform } random samples from the simplex d 1 := u (, 1) d, u = 1 work well.. Construct the max-linear combinations M u (i) all i = 1,..., n and u U. 3. Using numerical optimization, compute: θ n = arg min θ Θ n i=1 u U ( F M (i) u = max j=1,...,d X (i) j /u j, for ), V θ (u).

16 16 YUEN AND STOEV In Section 5, we illustrate this methodology over several concrete examples. The explicit construction of the set U is given in each example and the computation of the tail dependence function V θ when it is not available in closed form is discussed. The following result provides readily computable expressions for the bread and meat matrices appearing in the asymptotic covariance of the CRPS estimators. Corollary 1. Using the same specification of the measure µ as in (4.4) π ( ) H θ = (V θ (u)) 3/ Vθ (u) Vθ (u) u U and where J θ = u,w U c θ (u, w) ( ) V θ (u) Vθ (w) Vθ (u) V θ (w) { } c θ (u, w) = Cov γ 1 (V θ (u) /M u ), γ 1 (V θ (w) /M w ). Remark 7. M u and M w are dependent since in view of (4.3) they are defined as max-linear combinations of the vector X. The coefficient c θ (u, w) can be computed using Monte Carlo methods by simulating a large number of independent copies of X under the F θ model. In practice the resulting asymptotic covariance matrix estimates yield confidence intervals with close to nominal coverage (see Tables and 4). 5. Simulation. In this section we conduct simulation studies for CRPS M-estimation under 3 different max-stable models. The first example provides a comparison of CRPS M-estimation to the MLE. The second example displays the ability of CRPS M-estimation to detect dependence structure that is unidentifiable if one restricts to inference methods that rely on bivariate distributions only. The third example is applicable to the spatial setting and is used in the application to Swiss rainfall in Section Example: multivariate logistic model. The multivariate logistic is a special case that allows comparison between our CRPS based estimator and the MLE. This is because the full joint likelihood is available in this simple model. Hence, we can estimate the relative efficiency of the CRPS estimator in this idealized case. To this end, let θ = (σ, α) Θ = (, ) (, 1) and recall

17 CRPS M-ESTIMATION FOR MAX-STABLE MODELS: DO NOT DISTRIBUTE 17 Table Logistic model simulation results using 5 replications. Reported are the empirical mean and standard deviation of the CRPS and (MLE) estimates. Coverages are based on plug-in estimates of 95% asymptotic confidence intervals. In the case of the CRPS estimates, confidence intervals are generated using the expressions from Corollary 1. CRPS (MLE) n = 1 n = 1 σ(5) α(.7) σ(5) α(.7) mean 5. (5.4).7 (.699) (5.9).71 (7.) sd.519 (.319).48 (.8).158 (.1).15 (.8).95 coverage.934 (.96).94 (.91).954 (.948).95 (.956) V θ (x) = σ ( t D x 1/α t is the tail dependence function of a multivariate logistic max-stable model. We estimate the parameters for the model when D = 5 and θ = (5,.7), using samples sizes n = 1 and n = 1 with 5 replications each. Realizations were generated using the R package evd (Stephenson, ). For each realization X (i), i = 1,..., n we construct the max-linear combinations using a random uniform sample U d 1 where U = 1. Numerical optimization of the CRPS criterion in (4.5) was carried out using R s optim routine with an arbitrary starting point in the interior of Θ. Results for both the CRPS estimators and the MLE are shown in Table. Observe that we have essentially unbiased estimators. The asymptotic confidence intervals based on (3.5) were computed using the expressions in Corollary 1 and have close to nominal coverages even for moderate sample size n = 1. As expected, the CRPS is less efficient than the MLE however, the results in Table provide evidence that suggest the CRPS is a good alternative when the MLE is not available as is the case with the remaining examples. M (i) u 5.. Example: Max-linear model. Let d = 3 and k = 4 and define two (d k) matrices B = 1 1 and C = ) α Let Z 1,..., Z 4 be iid 1-Féchet random variables and define (5.1) X i = max j=1,...,k a ijz j,

18 18 YUEN AND STOEV Table 3 Error rate based on 5 replications of the CRPS estimator (5.) for max-linear model (5.1). n = 1 n = 5 n = 1 Error rate where (a ij (θ)) = A (θ) = θb + (1 θ) C, θ {, 1}. The stable tail dependence function for this model is V θ (x) = k max a ij (θ) /x i. i=1,...,d j=1 We simulated 5 replications from the max-linear model (5.1) with θ = 1. For each realization X (i), i = 1,..., n we construct the max-linear combinations M u (i) using a random uniform sample U d 1 where U = 1. We estimate θ via the CRPS estimator (5.) θn = arg min θ {,1} n i=1 u U ( F M (i) u ), V θ (u) There is no need for numerical optimization in this case since we can calculate the CRPS under θ = 1 and θ =. Results in Table 3 show that the error rate for θ n = θ decreases as the sample size n increases. Remark 8. When considering the marginal structure θ = 1 : θ = : X 1 = Z 1 Z X 1 = Z 1 Z X = Z 1 Z 3 X = Z 1 Z 3 X 3 = Z Z 3 X 3 = Z 1 Z 4 the bivariate and univariate marginals are equal under θ = or θ = 1. Hence, the parameter θ is unidentifiable from statistics based on bivariate distributions. Because the CRPS relies on the full joint distribution of X, it is able to discriminate between the two parameterizations. One can similarly construct different max-linear models that have equal k-dimensional distributions for k d.

19 CRPS M-ESTIMATION FOR MAX-STABLE MODELS: DO NOT DISTRIBUTE 19 Table 4 CRPS and MCLE estimates for Schlather model. Reported are mean and standard deviation of 1 replications using sample size n = 1 and n = 1. CRPS based confidence intervals for θ = (1, 1) were calculated using plug-in estimates for the expressions in Corollary 1 and resulting 95% coverages are reported. Coverages for MCLE estimates are based on sandwich estimators of Padoan et al. (1). CRPS (MCLE) n = 1 n = 5 θ 1 (1) θ (1) θ 1 (1) θ (1) mean (99.8) 1.5 (1.1) (1.5) 1.1 (1.) sd (14.9).63 (.18) (7.1).4 (.8).95 coverage.98 (.95).9 (.93).96 (.95).9 (.94) 5.3. Example: Schlather model. We now provide an example that is applicable in the spatial setting. Let {w t } t T be a Gaussian process on T R with standard normal margins and let ρ θ (t, s) be its associated correlation function parameterized by θ. Define { [ ] } V θ (x) = E θ max πwt t D + /x t then V θ (x) is the tail dependence function of a Schlather max-stable model with standard 1-Fréchet marginals, where the process is observed at a set of locations D. In this case V θ (x) is not available in closed form, instead we use a Monte Carlo approximation from a large sample w (i) t, i = 1,..., K under θ. For this simulation we assume a stable correlation function, i.e. ρ θ (t, s) = exp [ ( t s /σ) α ], θ = (σ, α) Θ = (, ) (, ]. The top row of Figure 1 shows realizations from this Schlather model under two different parameter settings. For our study we set θ = (1, 1) and simulated 1 replications at d = 3 uniformly sampled locations over a 5 5 grid. This corresponds to the top left panel in Figure 1. Realizations were generated using the R package SpatialExtremes (Ribatet, 11). For each realization X (i), i = 1,..., n we construct the maxlinear combinations M u (i) using a random uniform sample U d 1, where U = 1. For sample sizes n = 1 and n = 1, we numerically optimize the CRPS criterion (4.5) using R s optim routine with multiple starting points in the interior of Θ. Simulation results in Table 4 show the CRPS estimates are essentially unbiased and display close to nominal coverage.

20 YUEN AND STOEV Fig. Map of Switzerland showing the set D of 35 observation locations used in the analysis. 6. Application to Swiss rainfall. In this section we apply the CRPS estimation procedure to rainfall data measured near Zurich Switzerland. The data consist of n = 47 summer annual maxima between 196-8, measured at d = 35 locations. It is a subset of the dataset rainfall available in the SpatialExtremes R package. The observations Y (i) t (at location t for year i) were standardized to 1- Fréchet using the probability integral transform ( ) ξt Y (i) X (i) t = 1 + t µ t σ t 1/ξ t where µ t, σ t and ξ t are estimates of the GEV location, scale, and shape parameters fitted marginally at each location using maximum likelihood. This pre-processing is roughly analogous to de-trending non-stationary time series for further analysis and this step is common for analysis of multivariate extremes. We fit the Schlather model with Cauchy, Matérn, and stable covariance functions using the CRPS estimation procedure of Section 4 applied to the transformed data. We specified the set U as a random sample from d 1 with U = 1. The stable tail dependence function V θ (x) was

21 CRPS M-ESTIMATION FOR MAX-STABLE MODELS: DO NOT DISTRIBUTE 1 Table 5 CRPS Schlather model fits for Swiss rainfall data. Reported are the CRPS estimates for scale (θ 1) and shape/smoothness (θ ) parameters of the Stable, Matérn and Cauchy covariance functions. Standard errors were calculated using the formulae in Corollary 1. Covariance θ1 (S.E) θ (S.E.) CRPS Stable 1.31 (16.34) 1.17 (1.418) Matérn (56.71).4369 (1.3) Cauchy 1. (33.58) 1. (5.635) approximated by the Monte Carlo average V θ (x) 1 K K i=1 max t D { [ ] } (i) πw t /x t + where w (i), i = 1,..., K are iid standard Gaussian processes with correlation function ρ θ (t, s). The parameter estimates for each model are given in Table 5. The stable and Matérn correlation functions provide a better fit in terms of the CRPS compared to the Cauchy correlation function. The large standard errors indicate the difficulty in estimating these parameters for comparatively small (n = 47) sample sizes. Observe that for any reasonable significance level, the resulting confidence intervals include zero for both parameters in all models. In Section 7 we discuss the possibility of different specifications for the tuning measure µ that could result in more efficient estimators in order to draw concrete conclusions. 7. Discussion. We have developed a general inferential framework for max-stable models based on the continuous ranked probability score (CRPS). It is shown that CRPS M-estimators are consistent and asymptotically normal. Simulation studies across common spectrally continuous and discrete max-stable models yeild essentially unbiased estimators with close to nominal coverage. Our estimators were about half as efficient versus the MLE in the case of the simple multivariate logistic model, where a tractable likelihood exists. Overall the method displays flexibility and broad applicability in the max-stable setting. An application to Swiss rainfall yielded standard error estimates that are rather large. It is possible that efficiency for CRPS estimates can be improved through better tuning of the measure µ in the CRPS. For instance consider µ (dr, du) = r η δ {w} (du). w U

22 YUEN AND STOEV It can be shown that CRPS M-estimation remains consistent for all η (, )\{1} with little complication over the case η = 1/ (equivalent to the specification (4.4)), which was chosen for analytical simplicity. This begs the question of specifying η to maximize the expected Hessian of the CRPS, which should result in more efficient estimators. This is beyond the scope of the present paper and it will be studied in a future work. APPENDIX A: PROOFS A.1. De Haan s spectral representation. For completeness, we provide next the formal proof of the Poisson point representation due to de Haan. For more details see de Haan (1984a); Stoev and Taqqu (5); Kabluchko (9). Proof of Proposition 1. By (.5), for all x i >, i = 1,, d, where P(X ti x i, i = 1,, d) = P(Π A) = P(Π A c = ), A = {(u, s) R + S : g ti (s)/u x i, i = 1,, d}. Observe that A c = {(u, s) : max i=1,,d g ti (s)/x i > u}. Since Π is a Poisson point process on R + S with intensity duν(ds), { maxi=1, P(Π A c,d g ti (s)/x i } = ) = exp duν(ds), S which equals (.6) and completes the proof. The above argument shows that the integrability of the functions g t implies the X t s in (.5) are non-trivial random variables. A.. Proofs for Section 3. Proof of Theorem. Observe that the estimator θ n in Definition 1 trivially satisfies n 1 n i=1 Eθn (X i ) n 1 n i=1 E θ (X i ) o P (1). Therefore, by Thm. 5.7 of van der Vaart, 1998, the desired consistency follows if 1 n P (A.1) sup E θ (X i ) EE θ (X) n and θ Θ i=1 (A.) sup EE θ (X) > EE θ (X), for all ɛ >. θ: θ θ ɛ, θ Θ

23 CRPS M-ESTIMATION FOR MAX-STABLE MODELS: DO NOT DISTRIBUTE 3 We will first show (A.). By Fubini s Theorem, we have (A.3) EE θ (x) = (F θ (y) F θ (y)) µ (dy) R d + F θ (y) (1 F θ (y)) µ (dy) R d F θ (y) (1 F θ (y)) µ (dy) = EE θ (x). R d This implies (A.) because the continuity condition (iii) gaurantees the supremum therein is attained for some θ θ. We now show (A.1). Let F n (x) = n 1 n i=1 1 { X (i) x } and F = 1 F. Note that (A.4) 1 n sup E θ Θ θ (X (i)) EE θ (X) n i=1 = sup (1 F θ (x)) (F n (x) F θ (x)) µ (dx) θ Θ R d F n (x) F θ (x) µ (dx). R d Fix ɛ >. Markov s inequality and another application of Fubini gives { } (A.5) P F n (x) F θ (x) µ (dx) > ɛ R = 1 E F n (x) F θ (x) µ (dx). ɛ R d Next, using the identity a b = a + b a b we have that the RHS of (A.5) equals 1 (A.6) E { F n (x) + F θ (x) F n (x) F θ (x) } µ (dx) ɛ R d = { F θ (x) µ (dx) E [ F n (x) F θ (x) ] } µ (dx) ɛ R d R d Note that E [ F n (x) F θ (x) ] F θ (x), and by condition (ii), R F d θ (x) µ (dx) <. Thus, by the Lebesgue dominated convergence theorem lim E [ F n (x) F θ (x) ] µ (dx) = lim E [ F n (x) F θ (x) ] µ (dx). n R d R d n

24 4 YUEN AND STOEV The strong law of large numbers implies that F n (x) F θ (x) converges almost surely to F θ (x) F θ (x) F θ (x). Hence, by applying dominated convergence again, we obtain lim n E [ F n (x) F θ (x) ] = F θ (x), for all x R d. This, by (A.6) implies that the right-hand side of (A.5) vanishes as n, which in view of (A.4) yields the desired convergence in probability (A.1) and the proof is complete. Proof of Theorem 3. Since the CRPS estimator θ n minimizes the CRPS distance, we trivially have n 1 n i=1 Eθn (X i ) n 1 n i=1 E θ (X i ) ( o P n 1 ). Thus, by Thm. 5.3 of van der Vaart, 1998 the asymptotic normality in (3.5) follows, provided conditions (i)-(iii) hold. Proof of Proposition. By a standard argument using the Lebesgue DCT, condition (iii) of this proposition ensures that integration and differentiation can be interchanged in all that follows. We proceed by establishing (i)-(iii) of Theorem 3. (i) By the differentiability of θ F θ for all θ B (θ ) the function θ E θ is differentiable at θ since exchanging integration and differentiation allows E θ = (F θ (y) 1 {x y}) µ (dy) θ R d θ=θ = (F θ (y) 1 {x y}) F θ (y) µ (dy). R d (ii) Observe that E θ1 (x) E θ (x) equals { (F θ1 (y) 1 {x y}) (F θ (y) 1 {x y}) } µ (dy) R d = {[(F θ1 (y) + F θ (y)) 1 {x y}] (F θ1 (y) F θ (y))} µ (dy) R d F θ1 (y) F θ (y) µ (dy) R d where the last relation follows from the triangle inequality and fact that F θ (y) 1 {x y} max {F θ (y), 1 F θ (y)} 1. Then, by the mean value theorem and the Cauchy-Schwartz inequality F θ1 (y) F θ (y) µ (dy) θ 1 θ sup (A.7) F θ (y) µ (dy) R d R d L θ 1 θ θ B(θ )

25 CRPS M-ESTIMATION FOR MAX-STABLE MODELS: DO NOT DISTRIBUTE 5 where L := R sup d θ B(θ ) F θ (y) µ (dy). By assumption (ii) of this proposition, L is finite. Hence (ii) ( of Theorem 3 holds where L (X) L is constant (and therefore trivially E L (X) ) < ). (iii) Existence of a second order Taylor expansion for θ EE θ (X) follows from the twice continuous differentiability of θ F θ for all θ B (θ ) by θ θ EE θ (X) (A.3) = = θ θ Rd R d (F θ (y) F θ (y)) µ (dy) θ θ (F θ (y) F θ (y)) µ (dy). The above display implies that Rd H θ = θ θ (F θ (y) F θ (y)) µ (dy) θ=θ = F θ (y) F θ (y) µ (dy) = (3.8) where non-singularity of H θ follows from (ii) because for all a R p with a > [ a T H θ a = a F (y)] µ (dy) >. R Finally, we derive J θ by considering its ijth entry. Let i denote / θ i. ] (J θ ) ij = E [ i E θ (X) j E θ (X) θ=θ { = E ( ) F θ (y 1 ) 1 {X y1 } i F θ (y 1 ) µ (dy 1 ) R d ( } ) F θ (y ) 1 {X y } j F θ (y ) µ (dy ) R d θ=θ { } = 4E b θ (X, y 1, y ) i F θ (y 1 ) j F θ (y ) µ (dy 1 ) µ (dy ) R d R d R d θ=θ where b θ (X, y 1, y ) = ( 1 {X y1 } F θ (y 1 ) ) ( 1 {X y } F θ (y ) ). Expanding the integrand and applying Fubini gives (J θ ) ij = 4 β θ (y 1, y ) i F θ (y 1 ) j F θ (y ) µ (dy 1 ) µ (dy ) R d R d where β θ (y 1, y ) = Eb θ (X, y 1, y ) = F θ (y 1 y ) F θ (y 1 ) F θ (y ) which is exactly the ijth element of (3.9), as desired.

26 6 YUEN AND STOEV A.3. Proofs for Section 4. Proof of Lemma 1. Observe that e v/s 1 {x s} = e v/s for < s < x and hence the integrand in (4.1) vanishes, as s. Also, by using a Taylor series expansion of the exponential function at zero, it is easy to see that (e v/s 1 {x s} ) = (e v/s 1) v/s, as s. Therefore, the integral in (4.1) is finite. We have that F(x, v) = x e v/s s 1/ ds + x (e v/s 1) s 1/ ds =: I 1 + I. Using that ( s) = 1/ s and integration by parts in both integrals, we obtain and I 1 = xe v/x 4v I = x(e v/x 1) 4v Routine manipulations yield x x s 1/ 1 e v/s ds s 1/ 1 (e v/s e v/s )ds. (A.8) I 1 + I = ( x(e v/x ) 1) + 4v s 1/ 1 e v/s ds s 1/ 1 e v/s ds x } {{ } } {{ } =:J 1 =:J Now, by making the changes of variables y = v/s and z = v/s in the last two integrals respectively, we obtain v/x J 1 J = v 1/ y 1/ 1 e y dy (v) 1/ z 1/ 1 e z dz. This, in view of (A.8), yields the expression in terms of the incomplete gamma function in (4.1). The proof of Corollary 1 is aided by the following lemma Lemma. Let X be distributed 1-Fréchet with scale v, i.e. P (X x) = e v /x. Then

27 CRPS M-ESTIMATION FOR MAX-STABLE MODELS: DO NOT DISTRIBUTE 7 (i) (ii) (A.9) E X = πv (A.1) Eγ 1 (v/x) = π (v + v) v (iii) so that in particular Eγ 1 (v /X) = π/. (A.11) EF (X, v) = ( π v + v v ) v Proof. For (A.9) note that X is equal in distribution to a -Fréchet random variable with scale v which has finite expectation πv. For (A.1), applying Fubini s Theorem, and observing that E ( ) 1 {X v/s} = e v s/v, we get Eγ 1 (v/x) = = = e v s/v s 1/ 1 e s ds s 1/ 1 e vs/(v +v) ds πv v + v. This establishes (A.1). For (A.11), substituting the espression F (X, v) from Lemma 1 we have [ ( X EF (X, v) = 4E e v/x 1 ) + ( )] π v γ 1 (v/x) which, after substituting (A.9) and (A.1) yeilds [ (A.1) EF (X, v) = 4 E Xe v/x πv π + v v + v + πv Using the fact that X is distributed 1-Fréchet with scale v we have E Xe v/x = se v/s v e v /s s ds = v s 3/ e (v+v)/s ds. ].

28 8 YUEN AND STOEV Now the substitution t = s 1 gives (A.13) E Xe v/x = v t 1/ e (v+v)t dt = πv v + v. Plugging (A.13) into (A.1) yields [ πv EF (X, v) = 4 v + v πv [ = π = (A.11). + v v v + v v + π v + v ] πv ] v v + v v Proof of Corollary 1. Recall M u := max t D {X t /u t } and (A.14) H θ = θ θ EE θ (X). θ=θ Substituting (4.5) gives H θ = θ θ E F (M u, V θ (u)) u U = θ θ EF (M u, V θ (u)) u U θ=θ θ=θ. Note that Lemma implies EF (M u, V θ (u)) = ( π V θ (u) + V θ (u) V θ (u) ) V θ (u) from which it follows θ θ T EF (M u, V θ (u)) = θ=θ π ( (V θ (u)) 3/ Vθ (u) Vθ (u)) and Now recall H θ = π ( (V θ (u)) 3/ Vθ (u) Vθ (u)). u U

29 CRPS M-ESTIMATION FOR MAX-STABLE MODELS: DO NOT DISTRIBUTE 9 (A.15) J θ = E { E θ (X) E θ (X) }. Substituting (4.5) we have where J θ = E = u,w U u,w U Ḟ (M u, V θ (u)) Ḟ (M w, V θ (w)) V ( ) θ (u) Vθ (w) E {Ḟ (Mu, V θ (u)) Ḟ (M w, V θ (w))} Vθ (u) ( Vθ (w) π γ 1 (V θ (u) /M u ) Ḟ (M u, V θ (u)) =. V θ (u) Our goal is to calculate } (A.16) E {Ḟ (Mu, V θ (u)) Ḟ (M w, V θ (w)) where the expectation is taken under θ. Using the fact that M u is distributed 1-Fréchet with scale V θ (u), Lemma (ii) implies (A.17) Eγ 1 (V θ (u) /M u ) = and πv θ (u) Vθ (u) + V θ (u). (A.18) Eγ 1 (V θ (u) /M u ) θ=θ = π = Eγ 1 (V θ (w) /M w ) θ=θ. and so (A.16) becomes { } E γ 1 (V θ (u) /M u ) γ 1 (V θ (w) /M w ) θ=θ π V θ (u) V θ (uw) where the numerator is in fact { } Cov γ 1 (V θ (u) /M u ), γ 1 (V θ (w) /M w ) = c θ (u, w). as defined in Corollary 1. )

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