Asymptotic behavior of an intrinsic rank-based estimator of the Pickands dependence function constructed from B-splines

A bivariate extreme-value copula is characterized by its Pickands dependence function, i.e., a convex function defined on the unit interval satisfying boundary conditions. This paper investigates the large-sample behavior of a nonparametric estimator of this function due to Cormier et al. (Extremes 17:633–659, 2014). These authors showed how to construct this estimator through constrained quadratic median B-spline smoothing of pairs of pseudo-observations derived from a random sample. Their estimator is shown here to exist whatever the order \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m \ge 3$$\end{document}m≥3 of the B-spline basis, and its consistency is established under minimal conditions. The large-sample distribution of this estimator is also determined under the additional assumption that the underlying Pickands dependence function is a B-spline of given order with a known set of knots.


Introduction
Let (X, Y) be a continuous random pair with joint distribution function H. Let F and G denote the margins of X and Y, respectively. The unique copula C characterizing the dependence between X and Y is then the joint distribution of the pair (U, V) = (F(X), G(Y)).
Of frequent interest in applications, e.g., in quantitative risk management, are situations where the underlying copula C is unknown but assumed to belong to the class C of extreme-value copulas. It was shown by Pickands (1981) that C ∈ C if and only if, for all u, v ∈ (0, 1), for a convex map A ∶ [0, 1] → [1∕2, 1] such that, for all t ∈ [0, 1] , max(t, 1 − t) ≤ A(t) ≤ 1 . The map A is called a Pickands dependence function and the class of such functions is hereafter denoted A.
Given a random sample from H, one can estimate C readily by plugging into (1) a nonparametric estimator A n of A. This leads to a bona-fide extreme-value copula so long as A n is intrinsic, i.e., A n ∈ A . Estimation of A has already been considered, both when the margins F and G are known, and in the more realistic case where they are not. See, e.g., Gudendorf and Segers (2010) or Genest and Nešlehová (2012) for surveys of the early literature.
The purpose of this paper is to study, for the first time, the large-sample behavior of a rank-based, intrinsic estimator of A based on constrained B-spline smoothing (Cormier et al. 2014). From the work of these authors and Vettori et al. (2018), this estimator is known to perform well in finite-sample settings when compared, e.g., to the madogram-based procedure of Naveau et al. (2009) and to the more traditional Pickands, CFG, and Hall-Tajvidi estimators made intrinsic via the technique of Fils-Villetard et al. (2008).
An extended version of the estimator proposed by Cormier et al. (2014) is introduced in Sect. 2. Its existence is established for B-splines of any order m ≥ 3 in Sect. 3, and conditions under which it is consistent are detailed in Sect. 4. The proof of this result, Theorem 1, is then given in Sect. 5.
It seems difficult to determine the asymptotic distribution of the estimator of Cormier et al. (2014) in full generality. As described in Sect. 6, however, it is possible to achieve this goal in the much more restrictive-but nonetheless instructivecase where the unknown underlying Pickands dependence function A is a B-spline of given order m with k known internal knots, i.e., A can be expressed as a linear combination (1) C(u, v) = exp ln(uv)A ln(v) ln (uv) (2) A = 1 1,m + ⋯ + m+k m+k,m , 1 3 Asymptotic behavior of an intrinsic rank-based estimator… of m + k B-spline basis elements that are piece-wise polynomials of degree m − 1 on a partition of (0, 1) induced by k fixed points called internal knots.
The large-sample distribution of the estimator ̂n of is given in Sect. 6 for B-splines of degree m ∈ {3, 4} . The proof of this convergence result, Theorem 2, is detailed in Sect. 7. The large-sample distribution of ̂n is then used in Sect. 8 to establish the asymptotic behavior of Â n =̂⊤ n and of its first and second derivatives. As explained there, the latter lead to consistent and asymptotically unbiased estimators of the spectral distribution and density.
The assumption that a Pickands dependence function is of the form (2) is fairly mild, given that any convex function can be approximated with any desired accuracy by increasing the number of internal knots (de Boor 1978). What is restrictive is the assumption that the knots can be identified in advance. While weakening this assumption is beyond the scope of the present paper, Theorem 1 suggests that this should be possible and Theorem 2 provides useful information on the properties and shape that the limit will take.

Construction of an intrinsic B-spline estimator of A
Let (X 1 , Y 1 ), … , (X n , Y n ) be a random sample from the unknown joint continuous distribution H with underlying copula C and, for every real t ∈ ℝ, set where (E) denotes the indicator function of the set E. For each integer i ∈ {1, … , n} , write and let Ĉ n be the empirical copula defined, for all real u, v ∈ [0, 1] , by Suppose that C is of the form (1) for some unknown Pickands dependence function A. To construct an intrinsic nonparametric estimator of A using B-splines, first observe as in Cormier et al. (2014) that, for any real u, v ∈ (0, 1), Thus if one sets, for each integer i ∈ {1, … , n}, (4) , 1 3 the pairs (T 1 ,Ẑ 1 ), … , (T n ,Ẑ n ) should lie on, or near, the curve t ↦ A(t) , given that Ĉ n is a uniformly strongly consistent estimator of C (Gänßler and Stute 1987;Genest et al. 2017). A graph of these pairs is termed an A-plot in Cormier et al. (2014).
To estimate A, Cormier et al. (2014) propose to fit through the pairs (T 1 ,Ẑ 1 ), … , (T n ,Ẑ n ) a linear combination Â n of B-splines of order 3 under shape constraints which ensure that Â n ∈ A . Their approach, which extends readily to B-splines of arbitrary order m ≥ 3 , is described below.

Definition of B-splines
For any given integer m ≥ 2 , consider a partition on the interval (0, 1) induced by k points, viz. called internal knots. For notational convenience, further set The entire sequence ( 1 , … , 2m+k ) is then denoted .
A B-spline of order m on the interval [0, 1] with knot sequence is a continuous function which is a polynomial of degree m − 1 when restricted to any interval of the form ( m+j , m+j+1 ) with j ∈ {0, … , k} . Any B-spline can be written as a linear combination of m + k orthogonal functions 1,m , … , m+k,m . Each of these so-called B-spline basis functions is a piece-wise polynomial of degree m − 1 constructed in such a way that its m − 2 first derivatives are continuous. Let be the vector of B-spline basis functions corresponding to the set of knots (5). The set of B-splines of order m is then given by { ⊤ ∶ ∈ ℝ m+k }.

Construction of the B-spline basis
The B-spline basis 1,m , … , m+k,m of order m with k distinct internal knots (5) is built iteratively as follows via the Cox-de Boor recursion formula; see, e.g., de Boor (1978). For every integer j ∈ {1, … , k + 2m − 1}, let Then for each integer ∈ {2, … , m} , define recursively, for every integer j ∈ {1, … , k + 2m − } and real t ∈ [0, 1], The result of this construction is illustrated in Fig. 1 in the special case when m = 3 and k = 4 internal knots. The basis consists of m + k = 7 piece-wise
In general, the first basis function 1,m takes the value 1 at 0 and decreases monotonically to zero. Analogously, the last basis function m+k,m starts at 0 and rises monotonically to 1, which it reaches at 1. In contrast, each interior basis function 2,m , … , m+k−1,m is zero left of a certain internal knot, at which point it rises monotonically to a peak before falling back monotonically to zero, where it remains thereafter. By construction, B-spline basis functions of order m are strictly positive over at most m adjacent intervals.

Construction of the B-spline estimator
In the spirit of Cormier et al. (2014), who considered only B-splines of order 3, fix an integer m ≥ 3 and a set of k distinct internal knots (5) with corresponding B-spline basis of order m as defined in (6). Further pick an arbitrary smoothing constant n ∈ (0, ∞) , where n is the sample size.
A B-spline estimator of unknown underlying Pickands dependence function A is then given by where ̂n = (̂1, … ,̂m +k ) ⊤ is any minimizer of the objective function based on the vectors T = (T 1 , … ,T n ) ⊤ and Ẑ = (Ẑ 1 , … ,Ẑ n ) ⊤ of pseudo-observations with components defined in Eq. (4). Here, ‖ ⋅ ‖ 1 and ‖ ⋅ ‖ ∞ refer to the 1 (taxicab) and maximum norm, respectively. Cormier et al. (2014) favored the ‖ ⋅ ‖ 1 norm over the more standard ‖ ⋅ ‖ 2 norm for added robustness, in keeping with the median smoothing approach of Koenker et al. (1994) and its implementation in the R package cobs along with the constrained optimization method described by He and Ng (1999).
To ensure shape constraints, the function L is minimized over the set of vectors = ( 1 , … , m+k ) ⊤ in ℝ m+k for which the B-spline is a Pickands dependence function. From an operational viewpoint the set B consists of vectors ∈ ℝ m+k that satisfy the following three conditions: (C1) ⊤ (0) = ⊤ (1) = 1 or, equivalently, 1 = m+k = 1.
(C2) ⊤ �� (t) ≥ 0 for every real t ∈ [0, 1] , with the convention that when the second derivative fails to exist, the inequality holds for either choice of meaningful one-sided derivative.
Condition (C1) is the same as in Cormier et al. (2014). When m ∈ {3, 4} , Condition (C2) is equivalent to the requirement that which ensures that ⊤ is convex and meets the end-point constraints because the second derivative ⊤ ′′ is then linear between the knots, and hence non-negativity at the knots guarantees non-negativity everywhere on [0, 1]. When (C1) and (C2) hold, (C3) guarantees that Accordingly, the estimator is intrinsic, i.e., Â n ∈ A. Note that (C3) is a better and more economical choice than the approximate condition of Cormier et al. (2014), which stated that for some large, unspecified integer N and all j ∈ {1, … , N − 1}, The second summand in Eq. (8) is a penalization term. It plays an important role when the number of knots and their locations are unknown, as is often the case in practice. The minimization procedure is then typically performed over a large number of equally-spaced quantiles of the empirical distribution of the pseudo-sample T 1 , … ,T n . It is well known that this penalization can be expressed equivalently as a set of constraints on the coefficients, viz.
which must hold for some * and every integer j ∈ {m, … , m + k} . The second derivative of A is then prevented from changing abruptly between successive knots. Penalized B-spline estimation is also referred to as P-spline estimation.

Existence of the B-spline estimator
The procedure described in Sect. 2 is applicable to any order m ≥ 3 and choice (5) of k interior knots, as well as for any smoothing parameter n ∈ (0, ∞) . The result below implies that this always leads to at least one minimizer of the map L ∶ B → [0, ∞) in Eq. (8).

Proposition 1
The set B is a non-empty, convex, compact subset of ℝ m+k .
Proof To see that B is non-empty, let = (1, … , 1) be a vector of ones and observe that the map A = ⊤ is identically equal to 1 on the interval [0, 1] because the components of form a partition of unity. It is clear that A ∈ A as it corresponds to the independence copula. Therefore, ∈ B.
That B is convex and closed is obvious from Conditions (C1)-(C3). To show that B is also bounded, let ‖ ⋅ ‖ be any fixed norm on ℝ m+k . Because the map ↦ ‖ ⊤ ‖ ∞ is continuous, its infimum over the set { ∈ ℝ m+k ∶ ‖ ‖ = 1} is achieved at some * therein. As the B-spline basis functions are linearly independent, one has ‖ ⊤ * ‖ ∞ = c > 0 . Now for any ∈ B , one has ‖ ⊤ ‖ ∞ ≤ 1 but also and hence ‖ ‖ ≤ 1∕c . Therefore, B is bounded. □ The following result, which will be used in Sect. 4, highlights the fact that the upper bound on the norm of the elements in B depends on the choice of order m ≥ 3 but not on the number k or location of the interior knots in (5).
Proof It follows from the proof of Proposition 1 that, for each ∈ B , one has ‖ ‖ ∞ ≤ 1∕c , where Because the B-spline basis forms a partition of unity, one has Indeed, for any vector ∈ ℝ m+k with ‖ ‖ ∞ = 1 and any real t ∈ [0, 1], because the basis functions are non-negative and form a partition of unity. This upper bound is attained at ∈ ℝ m+k . As a consequence, sup ∈B ‖ ‖ ∞ ≤ m,∞ , where m,∞ is the so-called condition number given by where the second supremum is taken over all sequences of knots with distinct interior knots. It was shown by de Boor (1972) that the condition number is at most 2m 9 m−1 , whence the result. □

Consistency of the B-spline estimator
Conditions will now be described under which the B-spline estimator defined in Eq. (7) is consistent. To distinguish the true underlying Pickands dependence function from an arbitrary element in the set A , the former will henceforth be denoted A 0 and the corresponding extreme-value copula by C 0 . Other relevant Asymptotic behavior of an intrinsic rank-based estimator… notation is set in Sect. 4.1 and the main result, Theorem 1, is stated in Sect. 4.2. The proof of Theorem 1 per se is relegated to Sect. 5.

Notation
Fix an order m ≥ 3 and for each sample size n, let n = ( 1 , … , 2m+k n ) be an ordered sequence of knots such that The notation for the k n interior knots is consistent with Eq. (5) and the corresponding B-spline basis defined in Eq. (6) can be denoted to emphasize its dependence on n.
Similarly, let B n denote the set of vectors in ℝ m+k n corresponding to the knots (10) and order m, as per Eq. (9). Recall that B n is a non-empty convex and compact set by Proposition 1, and define K n ⊂ A by For each integer n ∈ ℕ , now consider the map L 0,n ∶ A → ℝ defined, for each A ∈ A such that ‖A �� ‖ ∞ < ∞ , by where A(T) = (A(T 1 ), … , A(T n )) ⊤ and the components of the vectors T = (T 1 , … ,T n ) ⊤ and Ẑ = (Ẑ 1 , … ,Ẑ n ) ⊤ are as defined in Eq. (4). This map is welldefined on K n because as stated in Remark 1 in Sect. 5.2, Clearly, the minimization of L 0,n over K n results in the same estimator as the minimization over B n of the objective function L in Eq. (8). Therefore, the consistency of the B-spline estimator Â n can be established by looking at the largesample behavior of any and all elements in the set Note at the outset that M n ≠ ∅ for every integer n ∈ ℕ . To this end, endow the set A with the topology induced by the norm ‖ ⋅ ‖ ∞ . The objective function L 0,n is then continuous and convex on K n , and the latter is a compact set given that it is the image of B n with respect to the continuous map ↦ ⊤ n . Therefore, there exists at least one ̂n ∈ B n such that 1 3

Statement of the theorem
Assume the following conditions on the sequence n of knots and on the large-sample behavior of the sequence n of smoothing constants.

Condition (K)
The sequence n of knots is such that for some integer N ∈ ℕ and all integers n ≥ N , there exists 0,n ∈ B n with Condition (S) For a given sequence n of knots with k n distinct interior knots, n = o(ns 2 n ) , where This paper's first major finding is then the following.
Theorem 1 Fix an order m ≥ 3 and assume that n is a sequence of knots fulfilling Condition (K). Suppose also that n is a sequence of smoothing constants satisfying Condition (S). For each integer n ∈ ℕ , let M n be the set defined in Eq. (12). Then, as n → ∞, where A 0 is the true underlying Pickands dependence function.
When m ∈ {3, 4} , it is shown in the Appendix that Condition (K) on the knots is automatically verified for any sequence n of knots whose mesh size tends to 0 as n → ∞ . The construction described therein could possibly be extended to splines of any order m ≥ 5 . However, the issue is not pursued here, not only because the argument seems involved, but also because the convexity constraints are more difficult to enforce when m ≥ 5 and this limits the practical use of such spline estimators at present. Condition (S) regulates the size of the penalty term as n → ∞ . For example, s n = 1∕(k n + 1) for equidistant interior knots so that if the number k n of interior knots is of the order of √ n , say, this condition is then tantamount to requiring that n = o(1) . It will be shown in Lemma 4 that the penalty term vanishes asymptotically when Condition (S) holds. In their paper, Cormier et al. (2014) adopted the common practice of placing the interior knots on a grid of equally spaced empirical quantiles. Their investigation revealed that taking k n ≈ √ n yielded the best performance for the sample sizes in their study. They also considered the use of Schwarz's information criterion and cross validation techniques to select the penalty term and found that the former was preferable both in terms of performance and computational effort.
These observations motivate the use of the penalty term but as a careful review of the arguments described in Sect. 5 reveals, the conclusions of Theorem 1 remain valid even when n = 0 for every sample size n. That is, the unpenalized version of the estimator is consistent under Condition (K), given that Condition (S) then holds trivially.

Proof of the consistency result
The argument leading to Theorem 1, which is rather involved, relies on preliminary results reported in Sect. 5.1. Theorem 1 per se is proved in Sect. 5.2.

Technical preliminaries
Let C[0, 1] be the set of continuous functions f ∶ [0, 1] → ℝ , and endow this space with the uniform norm topology. Let D 0,n be the first summand in the definition of the objective function L 0,n in Eq. (11). More formally, define the map D 0,n ∶ A → ℝ by setting, for every A ∈ A, where A(T) = (A(T 1 ), … , A(T n )) ⊤ and the components of the vectors T = (T 1 , … ,T n ) ⊤ and Ẑ = (Ẑ 1 , … ,Ẑ n ) ⊤ are as defined in Eq. (4).
Clearly, the map D 0,n is convex. However, it is also Lipschitz with constant 1 given that, for any constant a ∈ ℝ , the map x ↦ |a − x| − |a| is itself Lipschitz with the same constant.
It will now be shown that viewed as a function of the random vectors T and Ẑ , as n → ∞ , D 0,n converges in probability, denoted where T denotes the random variable defined as in terms of a random pair (U, V) distributed as the extreme-value copula C 0 with Pickands dependence function A 0 .

Lemma 1 One has, as
Proof Fix an arbitrary ∈ (0, ∞) . It follows from the Arzelà-Ascoli theorem that the convex set A is a relatively compact subset of C[0, 1]. Given that the set A is closed, it is actually a compact subset of C[0, 1]. One can thus choose a finite set From the triangle inequality one has, for any A ∈ A, The first and third terms on the right-hand side are each bounded above by because D 0,n and D 0,∞ are both Lipschitz with constant 1. Furthermore, the middle term is bounded above by Given that ∈ (0, ∞) is arbitrary, the proof of Lemma 1 will be complete if one can show that, as n → ∞, which holds true if, for any fixed A ∈ A , one has, as n → ∞, To establish claim (15), define, for any integer > 2 and all integers n ≥ ,  (16) and (17), one gets

3
Asymptotic behavior of an intrinsic rank-based estimator… Next, define and observe that Together with the triangle inequality, the above considerations imply that, for any fixed integer > 2 and every integer n ≥ , one has It only remains to prove that, as n → ∞, Because the empirical copula Ĉ n is not strictly positive on its entire domain, it will be convenient to use a slight variant Č n thereof advocated by Bücher et al. (2011), which depends on a constant ∈ (1, ∞) . The exact value of this constant will not have any influence on the proceedings.
For every real u, v ∈ (0, 1) , let and introduce The second equality follows from the mean value theorem, which guarantees the existence of the scalar c * u,v between C 0 (u, v) and Č n (u, v).
Now for all u, v ∈ I , one has because C 0 is an extreme-value copula and hence it is positive quadrant dependent; see, e.g., Genest and Nešlehová (2012). This leads to an upper bound for the righthand side in the previous display, and hence one gets Finally, note that ‖Č n − C 0 ‖ ∞ → 0 almost surely, as n → ∞ , by an application of the Glivenko-Cantelli theorem; see, e.g., p. 51 in the monograph by Gänßler and Stute (1987). Consequently, the term in the square brackets converges to 1 almost surely, and, in turn, O n ( ) → 0 almost surely, as n → ∞ . It thus follows that for any integer > 2 , one has almost surely, as n → ∞, Furthermore, D * 0,n, (A) is a linear rank statistic with a bounded score function. As such, it converges in probability to D 0,∞, (A) , as n → ∞ ; see, e.g., Genest et al. (2013). Thus claim (19) is established; claim (15) then follows from (18). This concludes the proof of Lemma 1. ◻ The following property of the random variable T will prove useful. (14)

Lemma 2 If a random pair (U, V) is distributed according to copula (1) with Pickands dependence function A, then the support of the random variable T defined in
Proof Let F be the distribution function of T. It is known from Proposition 1 in Ghoudi et al. (1998) that for all t ∈ (0, 1) , where A � (t) denotes the right-hand derivative of A at t. Now suppose that F(t) = c ∈ (0, 1) on some interval I ⊂ [0, 1] . As mentioned, e.g., by Capéraà et al. (1997), one must then have, for all u, v ∈ I with u ≤ v, which is easily seen to be a non-convex function of v for any fixed u. This is a contradiction, given that A is convex. Therefore, F is strictly increasing on the interval The following observation concerning D 0,∞ will also be needed in the proof of Theorem 1 given in Sect. 5.2.

Lemma 3 The unique minimizer of the map
Proof The map D 0,∞ is obviously minimized, over A , at the point A = A 0 . To show that the minimizer is unique, first assume that t ∈ [0, 1] is in the support of the random variable T = ln(V)∕ ln(UV) with (u, v) distributed as the extreme-value copula C 0 with Pickands dependence function A 0 . If A(t) ≠ A 0 (t) , one then has |A(t) − A 0 (t)| > 0 on some open neighborhood of t, which implies that D 0,∞ (A) > D 0,∞ (A 0 ) . Now it was shown in Lemma 2 that the support of T is an interval [a, b] with 0 ≤ a ≤ 1∕2 ≤ b ≤ 1 and that for any t ∈ [0, 1] ⧵ [a, b] , one has A(t) = max(t, 1 − t) . Therefore, any A ∈ A which agrees with A 0 on [a, b] must also agree with A 0 on all of [0, 1]. This concludes the proof of Lemma 3. ◻

Proof of Theorem 1
For each integer n ∈ ℕ , let M n be the set defined in Eq. (12). The fact that this set is nonempty has already been argued at the end of Sect. 4.1. Its convexity follows readily from the convexity of the map L 0,n . What must be shown is that, as n → ∞, where A 0 is the true underlying Pickands dependence function.
Recall from Lemma 3 that A 0 is the unique minimizer of D 0,∞ over A . Thus if A n is any element in the set M n of minimizers of L 0,n , one has where Q n is as in Lemma 1 and A 0,n = ⊤ 0,n n is as per Condition (K). The first inequality exploits the fact that A 0,n ∈ K n , while the second holds because A 0,n ∈ A.
Given that A 0 is the minimizer of D 0,∞ and the latter is Lipschitz with constant 1, one also has where n = ‖A 0,n − A 0 ‖ ∞ . By combining this inequality with the chain of inequalities (22), one deduces that Again, as A 0 is the minimizer of D 0,∞ , one can conclude that and hence It will now be shown that the third summand in this upper bound, i.e., the penalty term, is asymptotically negligible.
Lemma 4 Let m ≥ 3 be a given order and n be a sequence of knots with k n distinct interior knots. If n satisfies Condition (S), then, as n → ∞, Proof In view of Condition (S), it suffices to show that sup ∈B n ‖ ⊤ �� n ‖ ∞ ≤ m ∕s 2 n , where m is a constant that depends on m but not on n. As mentioned on p. 117 of de Boor (2001), one has, for every real t ∈ (0, 1), with the convention that when m = 3 , the right-hand second derivative is taken. In the above, (3) 1 = (3) 2 = 0 and, for any integer j ∈ {3, … , m + k}, Given that the basis functions n,1,m−2 , … , n,m+k,m−2 form a partition of unity, an argument similar to the proof of Corollary 1 implies that with m = 8m(m − 1)(m − 2)9 m−1 , where the last inequality follows from Corollary 1 as ∈ B n by assumption. This concludes the proof of Lemma 4. □

Remark 1
The argument developed in the proof of Lemma 4 makes it clear, as already stated in Sect. 4.1, that A ∈ K n ⇒ ‖A �� ‖ ∞ < ∞.
In view of Lemmas 1 and 4, inequality (23) implies Now fix an arbitrary ∈ (0, ∞) . Given that D 0,∞ is continuous and that its unique minimizer over A is A 0 , one has that, for any fixed integer n ∈ ℕ, Indeed, if this were not the case, one could find a sequence A k in K n with Because K n is compact, however, one could then extract a convergent subsequence converging to A * ∈ K n with the property that D 0,∞ (A * ) = D 0,∞ (A 0 ) (by continuity of D 0,∞ ) while at the same time ‖A * − A 0 ‖ ∞ ≥ . This would then contradict the fact that A 0 is the unique minimizer of D 0,∞ . In view of (25), there exists ∈ (0, ∞) so that D 0,∞ (A) − > D 0,∞ (A 0 ) for all A ∈ K n with the property that ‖A − A 0 ‖ ∞ > . Hence, for any A ∈ K n ,

It follows that
The probability on the right-hand side converges to 0, as n → ∞ , by Eq. (24). Because was arbitrary, the proof of Theorem 1 is complete.

Asymptotic behavior of ̂n
Having established the existence and consistency of the B-spline estimator (7), the next milestone one would hope to reach is the determination of the asymptotic distribution of this estimator in the broadest possible conditions. At this point, however, this goal remains elusive.
As a step towards a full resolution of this issue, this section describes conditions under which one can identify the limiting distribution of any sequence ̂n of minimizers of Eq. (8) involved in the construction of the B-spline estimator of A. This result will then be used in Sect. 8 to identify the large-sample distribution of the B-spline estimator Â n . More specifically, it will be assumed henceforth that the unknown underlying Pickands dependence function A 0 is itself a B-spline with a fixed order and a given set of knots. This condition is spelled out below for easy reference.

Condition (A) The Pickands dependence function is of the form
for known knot sequence with k internal knots as in (5) and some m ∈ {3, 4} . Moreover, the vector 0 is in the relative interior of B.
Note that a vector ∈ ℝ m+k belongs to the relative interior of B if and only if (C1) holds and the following conditions, which are stricter than (C2) and (C3), are fulfilled: Given the richness of the space spanned by B-splines of orders 3 and 4, including complete freedom in the number and location of internal knots, the assumption that a Pickands dependence function can be written in the form (2) is not a serious limitation in practice, although it implies that the extreme-value copula C 0 induced by A 0 through Eq. (1) is neither the product copula nor the Fréchet-Hoeffding upper bound, respectively induced by Pickands dependence functions defined, for all t ∈ [0, 1] , by A(t) = 1 and A(t) = max(t, 1 − t) . What is restrictive, however, is the requirement that the set of knots should be known in advance. While this is unrealistic in practice, Theorem 2 below, which is proved in Sect. 7, does represent an essential intermediate step on the way to determining the limiting distribution of Â n . Before stating the result, observe that when Condition (A) holds, C 0 fulfills Condition 2.1 of Segers (2012). It thus follows from Proposition 3.1 therein that, as n → ∞ , the empirical copula process converges weakly in the space ∞ [0, 1] 2 of bounded functions on [0, 1] 2 equipped with the uniform norm to a C 0 -pinned centered Gaussian process Ĉ defined, for all u, v ∈ (0, 1) , by where, for all real u, v ∈ (0, 1), while ℂ is a Brownian bridge with covariance given, for all u, v, s, t ∈ [0, 1] , bŷ

3
Asymptotic behavior of an intrinsic rank-based estimator… with a ∧ b = min(a, b) for arbitrary reals a, b ∈ ℝ.

Furthermore, the argmin in Eq. (26) is almost surely a single point.
Note that the integral in Eq. (26) is finite. Indeed, the elementary inequality ||x − y| − |x|| ≤ |y| implies that, for all real numbers u, v ∈ (0, 1), where Moreover, it is interesting to note that the limiting distribution defined in Eq. (26) is symmetric about zero. This is formally stated and proved below.

Proposition 2 The law of B is symmetric about zero.
Proof Define, for any measurable function H on the unit square and vector b ∈ ℝ m+k , the functional Now consider the centered Gaussian process defined, for all real numbers u, v ∈ (0, 1), by .

Then because −ℍ is another Gaussian process with the same law, B(ℍ) and B(−ℍ) are identically distributed, so −B(ℍ) has the same law as B(ℍ) . This shows that the law of B = B(ℍ) is symmetric about zero. ◻
Finally, note that Theorem 2 implies that Â n =̂⊤ n is a consistent estimator of A 0 = ⊤ 0 . However, consistency is actually guaranteed by Theorem 1 under milder assumptions on the tuning parameter n . Indeed, it is immediate that when Condition (A) holds, Condition (K) is satisfied if n = for every sample size n ∈ ℕ . Given that s n in Condition (S) is then constant, the latter condition is fulfilled whenever n = o(n) , and this is weaker than n = o( √ n). As with Theorem 1 the conclusions of Theorem 2 remain valid when the smoothing constant n is taken to be equal to zero for every sample size n.

Proof of the limiting distribution of ̂n
The proof of Theorem 2 relies on several technical lemmas. An important result about the empirical copula process is first stated in Sect. 7.1; the proof of Theorem 2 is then given in Sect. 7.2.

Preliminary results concerning the empirical copula process
As in the proof of Lemma 1, the variant Č n of the empirical copula with some tuning constant ∈ (1, ∞) defined in Eq. (20) will be used. Then, as shown by Bücher et al. (2011), the process has the same weak limit Ĉ as ℂ n under Condition 2.1 of Segers (2012), which is met when A 0 is continuously differentiable on (0, 1); see Example 5.3 therein. Further recall, as already noted in the proof of Lemma 1, that for every integer i ∈ {1, … , n} , one has Č n (Û i ,V i ) =Ĉ n (Û i ,V i ) so that the spline estimators of A 0 based on Č n and Ĉ n coincide.
The following lemma plays an important part in the proof of Theorem 2.

Lemma 5 For every integer
. Then for any extremevalue copula C whose Pickands dependence function A is continuously differentiable on (0, 1), one has, for Č n = √ n (Č n − C), Proof As already noted, if C is as specified in the statement of the lemma, the empirical process Č n = √ n (Č n − C) converges weakly, as n → ∞ . Next, from Taylor's theorem, ln(1 + x) = x − x 2 ∕(2x 2 0 ) for some x 0 between 1 and 1 + x . Thus if for arbitrary reals u, v ∈ (0, 1) one sets x =Č n (u, v)∕C (u, v)

3
Asymptotic behavior of an intrinsic rank-based estimator… where and c * u,v = x 0 C(u, v) is a scalar between C(u, v) and Č n (u, v) . Therefore, and hence As already argued in the proof of Lemma 1, one has for all real numbers u, v ∈ I . Moreover, given that the sequence ||Č n || ∞ converges weakly, as n → ∞ , it is uniformly tight and hence Finally, note that for all real numbers u, v ∈ (0, 1), The right-hand side of (28) converges almost surely to 1, as n → ∞ , for any integer > 2 , as discussed in the proof of Lemma 1. Thus which concludes the proof of Lemma 5. □

Proof of Theorem 2
To prove Theorem 2, first introduce, for each integer i ∈ {1, … , n} , the unobservable random variable where B * n is the set of vectors b ∈ ℝ m+k such that = 0 + b∕ √ n ∈ B. Finally, let P n denote a convex function with values in [−∞, +∞] defined, for all b ∈ ℝ m+k , by With this notation, it is then immediate that Therefore, in view of Theorem 3.2 of Geyer (1996), Theorem 2 holds true provided that (i) P n epiconverges in law as n → ∞ to the map D given, at any b ∈ ℝ m+k , by where C 0 is the extreme-value copula induced by A 0 ; (ii) the minimum of D is achieved at a unique point with probability 1.
Each of these conditions will be checked in turn.

Epiconvergence in law of P n to D
Given that n = o( √ n) by assumption, the (deterministic) second summand on the right-hand side of Eq. (29) is asymptotically negligible so that, for any fixed vector b ∈ ℝ m+k , By Lemma 3.1 of Geyer (1996) and its preceding discussion, it suffices to show that, for any integer p ∈ ℕ , the p-dimensional distributions of D n converge in law to those of D in the space [−∞, ∞] p .

Uniqueness of the minimizer of D
Clearly, the set of all minimizers of D is convex; it will now be shown that it is nonempty and bounded. Consider the recession function D + 0 of the proper and closed convex function D which, by Theorem 8.5 of Rockafellar (1970) and Lebesgue's dominated convergence theorem, can be written, for b ∈ R, as For b ∉ R , one has (D + 0 )(b) = +∞. To prove that (D + 0 )(b) is strictly positive for any non-zero b ∈ R , note that, for all vectors b ∈ R, In view of Corollary 13.3.4(c) in Rockafellar (1970), it thus follows that 0 is an element of the interior of the domain of the convex conjugate D * of D. By Theorem 27.1(d) in that reference, the minimum set of D, which is the subgradient D * (0) of D * at 0, is non-empty and bounded as asserted.
It will now be argued that D * (0) is almost surely unique. To this end, consider the process defined, for all real numbers u, v ∈ (0, 1) , by whose sample paths are continuous on (0, 1) 2 almost surely. For any such continuous path G and vectors b, ∈ ℝ m+k , define and It will now be shown that there exists a set G of continuous paths G such that, for each vector b ∈ R and vector ∈ ℝ m+k , ≠ 0 , the map is strictly convex on ℝ . From this it then clearly follows that whenever G ∈ G , the map b ↦ K(G, b) is strictly convex on ℝ m+k , so that K(G, b) has a unique minimizer in R . It will also be proved hat Pr( ∈ G) = 1 , and hence that K( , b) has a unique minimizer in R almost surely, as claimed. This will rely on the following two technical lemmas.
Lemma 6 Suppose that (W 1 , W 2 ) is an arbitrary random pair with support S = I × ℝ , where I is a closed set such that I ≠ {0} . Then the map {(w 1 , w 2 ) ∶ (w 2 − 1 w 1 )(w 2 − 2 w 1 ) < 0} and the support S is relatively open and non-empty, the probability that (W 1 , W 2 ) lies in this intersection is strictly positive. □ Lemma 7 Let G be the set of all continuous functions on (0, 1) 2 with the property that for every rational ∈ (0, ∞) , the map u ↦ G(u, u ) is a surjection from (0, 1) to the real line. Then Pr( ∈ G) = 1.
Proof Fix some rational ∈ (0, ∞) and define the map ∶ (0, 1) → (0, 1) by setting, for every real number u ∈ (0, 1), for every real number t ∈ (0, 1) . Consequently, It suffices to show that, almost surely, Let Ċ 0,1 and Ċ 0,2 denote the partial derivatives of C 0 with respect to its first and second argument, respectively. For every real u ∈ (0, e −1∕ ( ) ) , write where with and Because (u) → +∞ as u ↓ 0 , it suffices to investigate the limiting behavior of N 1 , N 2 , and N 3 . Each of these terms is discussed in turn.
(i) Limiting behavior of N 1 . Given that is strictly increasing, one has so that ℂ(u, u ) = { (u)} , where is a Brownian bridge. Hence almost surely by the law of the iterated logarithm. (ii) Limiting behavior of N 2 . A straightforward calculation shows that Moreover, ℂ(u, 1) is a Brownian bridge. By the law of the iterated logarithm and the fact that (u)∕u = u ( )−1 → 0 as u → 0 by (35), one then finds that (iii) Limiting behavior of N 3 . In this case, note that and that ℂ(1, u) is a Brownian bridge. By the law of the iterated logarithm and the fact that (u)∕u = u ( )− → 0 as u → 0 by Eq. (35), one finds that Put together, and almost surely, as claimed. It follows that this property holds simultaneously across all rational values of ∈ (0, ∞) , given that this is a countable intersection. This concludes the proof of Lemma 7. □ To complete the proof of Theorem 2, let G be as in Lemma 7. By the same result, Pr( ∈ G) = 1 . Now pick any element G ∈ G . The claim follows from Lemma 6 if the support of (X, Y) given in Eq. (34) First note that from Lemma 2, the support of T is [0, 1] and that one cannot have supp (X) = I = {0} . For if it were, then ⊤ (t) would be identically 0 in t on the interval [0, 1]. But this is only possible if = 0 because the elements of are linearly independent, and = 0 is ruled out by assumption.
It remains to check that the support of the pair (X, Y) is I × ℝ . Let O = (x 1 , x 2 ) × (y 1 , y 2 ) be a relatively open rectangle in I × ℝ . The inverse image of (x 1 , x 2 ) under the continuous map t ↦ ⊤ (t) is open and non-empty so it contains some rational number t ∈ (0, 1) . Let be the rational number = t∕(1 − t) . Then the range of G(u, u ) is all of ℝ so there is a point u ∈ (0, 1) such that y = G(u, u ) − b ⊤ (t) ∈ (y 1 , y 2 ) . Given that the map is continuous, the inverse image of O under this map is open in (0, 1) 2 ; in particular, it is non-empty.
By Condition (A) and Corollary 4 in Trutschnig et al. (2016), the support of (U, V) is the unit square and hence every open set has positive measure for the copula C 0 . Thus Therefore, the support of (X, Y) satisfies the conditions of Lemma 1. The proof of Theorem 2 is thus complete.
Remark 2 Note that the strict inequality constraints imposed in the statement of Theorem 2 on the parameter vector 0 can be relaxed. Indeed, the arguments given above extend to the more general case where 0 satisfies Conditions (C1)-(C3) provided that R is redefined as This set does not have the property that b ∈ B implies that −b ∈ B , so the symmetry conclusion asserted in Proposition 2 then no longer holds. As an example consider the Pickands dependence function A given by If both 1/4 and 3/4 are knots, then this map A is a B-spline of order 3. In this specific case, the support of T is [1/4, 3/4] and the support of the induced extreme-value copula C is {(u, v) ∈ [0, 1] 2 ∶ u 3 ≤ v ≤ u 1∕3 } . Small modifications are then required to Lemmas 2 and 7 but the conclusions all hold as given, with R modified.

3
Asymptotic behavior of an intrinsic rank-based estimator…

Consequences of Theorem 2
As a first application of Theorem 2, one can now establish the limiting behavior of the empirical process defined, for all t ∈ [0, 1] , by where Â n is as defined in Eq. (7). Because ‖ ‖ ∞ < ∞ , the map ℝ m+k → ∞ [0, 1] given by ↦ ⊤ is continuous. Theorem 2, the continuous mapping theorem, and Proposition 2 thus together imply the following result.

Corollary 2 Suppose that Condition (A) holds and that
In particular, the estimator Â n is consistent and asymptotically unbiased.
Beyond being intrinsic, the estimator Â n has the advantage of being smooth and its derivatives are easy to compute. The latter are of interest as they can be used to construct estimators of the spectral distribution S associated with A and its density s, whenever it exits. The spectral density is particularly appealing in practice, because it provides greater visual insight than A into the nature of the dependence between X and Y.
Given a Pickands dependence function A, the corresponding unique spectral distribution function S on the interval [0, 1] is implicitly defined via which is valid for every real number t ∈ [0, 1] ; see Einmahl and Segers (2009). From Eq. (36), one can easily deduce that, for every real number t ∈ [0, 1), where A ′ denotes the right-hand derivative of A, while of course S(1) = 1 . If the map A ′ is absolutely continuous on (0, 1), then S = S 1 + S 2 , where S 1 is discrete with support {0, 1} and S 2 is absolutely continuous with density given, for almost all real numbers t ∈ (0, 1) , by s(t) = A �� (t)∕2.
When A = ⊤ 0 and m ∈ {3, 4} , the map A � = ⊤ 0 � is absolutely continuous on (0, 1) with derivative ⊤ 0 ′′ wherever it exists. Hence, for every real number t ∈ [0, 1) , one has while for any real t ∈ (0, 1), where the ′′ is understood as the right-hand derivative when m = 3. One advantage of working with B-splines is that the derivatives of the basis functions are readily available through a recursive formula valid for every integer j ∈ {1, … , k + 2m − } and real t ∈ [0, 1] , viz.
Consequently, A ′ and A ′′ are linear combinations of B-splines of order m − 1 and m − 2 , respectively, on the original knot vector with a new set of weights.

Corollary 4 Suppose that Condition (A) holds and that
It follows from Corollary 3, Corollary 4, and Proposition 2 that the estimators Ŝ n and ŝ n are consistent and asymptotically unbiased. Note also that because Â n is intrinsic, Ŝ n is intrinsic as well, i.e., it satisfies the moment constraint that the expected value of the random variable with distribution function Ŝ n equals 1/2. To illustrate, consider a random sample of size n = 400 from Gumbel's asymmetric logistic copula model with = .7 , = .3 , and = 6 (Tawn 1988). Figure 2 displays the A-plot of the sample along with the true Pickands dependence function in black. Also plotted on this graph are: (a) the rank-based, end-point corrected, Pickands (dotted) and so-called CFG (dashed) estimators from Genest and Segers (2009); (b) the rank-based B-spline estimators of degree m = 3 (in red) and m = 4 (in blue) investigated herein. (38) The B-spline estimators were computed using 40 internal knots; the penalties were chosen to be = 15 when m = 3 and = 70 when m = 4 , based on the cross-validation procedure described by Cormier et al. (2014). Looking at Fig. 2, one can see that although the Pickands and Capéraà-Fougères-Genest (CFG) estimators meet the end-point constraints, they are not convex, and hence not intrinsic. In contrast, the B-spline estimators are valid Pickands dependence functions by design. The graphs of the curves corresponding to B-splines of degree 3 and 4 are nearly superposed, showing little improvement when the degree is increased from 3 to 4.
While the degree of the B-splines may not matter much when estimating A, it has a perceptible effect when estimating S, and even more so when estimating s. This is portrayed in Fig. 3, which is based on the same data as Fig. 2. The left panel shows the estimators Ŝ n of S in Eq. (39) corresponding to B-splines estimators of degree 3 (in red) and 4 (in blue). The latter is distinctly superior to the former, although both perform very similarly in estimating the point masses at 0 and 1, whose theoretical values are .15 and .35, respectively.
The right panel of Fig. 3 shows the corresponding estimators of s. Clearly, the estimator ŝ n in Eq. (39) based on B-splines of degree 4 is far superior to the analogous estimator constructed from B-splines of degree 3. The horizontal axis of this graph also reports the position of the 40 internal knots, which correspond to quantiles of the empirical distribution of the set T 1 , … , T 400 , as recommended by Cormier et al. (2014).

Conclusion
This paper examined the large-sample behavior of a rank-based procedure proposed by Cormier et al. (2014) for the intrinsic estimation of the Pickands dependence function characterizing a bivariate extreme-value copula. The estimator, which involves a linear combination of B-splines of order m ≥ 3 , was shown to be consistent under Conditions (K) and (S), which specify the large-sample behavior of the set of knots and of the penalty term, respectively. This consistency result, Theorem 1, suggests that it should be possible to determine the asymptotic distribution of this estimator under appropriate conditions. This seems difficult, however. As a partial solution to this problem, the large-sample distribution of the estimator was computed under the assumption that the underlying Pickands dependence function, A, can be expressed as a linear combination of B-spline basis elements of a given order m ∈ {3, 4} with a fixed and known set of knots. Given the richness of quadratic and cubic B-splines, the condition on m is not much of a limitation, but knowledge of the number and position of the interior knots is a practically unrealistic assumption which will hopefully be lifted in subsequent work.
In the future, it would also be worth investigating how one could relax the requirement that m ∈ {3, 4} . A visual motivation for this extension is provided in the bottom panel of Fig. 3. The difficulty in this endeavor is the convexity constraint, which is no longer simple to formulate when m > 4.
Finally, there would be merit in comparing the performance of the estimators of S and s proposed herein to other nonparametric estimators of the same quantities considered in the growing literature on the subject. Relevant references include Einmahl et al. (2001), Einmahl and Segers (2009), Guillotte et al. (2011), de Carvalho et al. (2013, as well as the Bayesian estimation method based on normalized B-splines of Khadraoui and Ribereau (2019).
When A 0,n satisfies (i)-(iii), the Curry-Schoenberg Theorem (Curry and Schoenberg 1966), also reported as Theorem (44) by de Boor (2001), then implies the existence of 0,n ∈ B n such that A 0,n = ⊤ 0,n n . Property (iv) then yields that (40) holds and therefore the knot sequence n fulfills Condition (K) as soon as n → 0 as n → ∞.
Before examining the cases m = 3 and m = 4 separately, recall that the right-hand derivative A ′ 0 of A 0 ∈ A is monotone increasing given that A 0 is convex by assumption.
It is easy to see that, for every real t ∈ (0, 1) , one has This string of inequalities may be integrated to get The monotonicity of A ′ 0 then implies that In view of the central inequalities in the chain (41), there must exist a real number ∈ [0, 1] with where = − + (1 − ) + . Notice that is a continuous, monotone increasing, piece-wise linear spline with the given set of knots. It follows that the map defined, for every real t ∈ [0, 1] , by − ( m+i ) = A � 0 ( m+i−1 ) and A 0,n (t) = A 0 (0) + ∫ t 0 (x)dx = 1 + ∫ t 0 (x)dx fulfills conditions (i) and (ii). The fact that (iii) holds is an immediate consequence of Theorem 1 in Trutschnig et al. (2016). Finally, it follows from the string of inequalities (41) that, for every real t ∈ [0, 1] , one has which establishes (iv) and thus the required claim.
Case m = 4 First, focus temporarily on a subsequence of knots given by when k n is odd and by when k n is even. Now proceed as in the case m = 3 to construct continuous piecewise linear maps − and + on this subsequence of knots. This means that for each integer i ∈ {0, … , ⌊k n ∕2⌋ + 1}, As when m = 3 , extend the definition of A ′ 0 by putting A � 0 (t) = A � 0 (0) for every real t ∈ (−∞, 0) and A � 0 (t) = A � 0 (1) for every real t ∈ (1, ∞) . One then has, for every real t ∈ (0, 1), Next, construct quadratic splines − and + defined on the original set of knots which are monotone and interpolate − and + at the subset of knots in (42) or (43), as follows.
These constraints are satisfied by and When k n is even, define + (t) on [ m+2i , m+2(i+1) ] for any integer i ∈ {0, … , k n ∕2 − 1} the same way as when k n is odd. Then for any real t ∈ [ m+k n , m+k n +2 ] , set The resulting map + is again continuously differentiable on (0, 1).
In view of this construction and an entirely analogous one for − , one has, for every real t ∈ (0, 1), Now, proceeding along similar lines as in the case m = 3 , one finds that Therefore, one can find a scalar ∈ [0, 1] such that where the map = − + (1 − ) + is a continuously differentiable, monotone increasing, piece-wise quadratic spline with the given set of knots. Arguing analogously as in the case m = 3 and using Theorem 1 in Trutschnig et al. (2016), one finds that the map defined, for every real t ∈ [0, 1] , by satisfies (i)-(iii). Finally, the inequalities (44) yield so that (iv) holds as well, and this concludes the argument. .