In this section, we recall the concept of copula. We will also define the family of elliptical copulas through a brief reminder of elliptical distributions—see Appendix A for an overview of other families.

Let us first introduce the concept of ϕ-divergence.

Let f be a density on ℝ^d. We assume there exists d non-null linearly independent vectors a_j, with 1 ≤ j ≤ d, of ℝ^d, such that (3.1) f ( x ) = n ( a j + 1 ⊤ x , … , a d ⊤ x ) h ( a 1 ⊤ x , … , a j ⊤ x )with j < d, n being an elliptical density on ℝ^d−j and with h being a density on ℝ^j, which does not belong to the same family as n. Let X = (X₁, …, X_d) be a vector with f as density

We define g as an elliptical distribution with the same mean and variance as f.

For simplicity, let us assume that the family {a_j}_1≤j≤d is the canonical basis of ℝ^d:

The very definition of f implies that (X_j+1, …, X_d) is independent from (X₁, …, X_j). Hence, the density of (X_j+1, …, X_d) given (X₁, …, X_j) is n.

Let us assume that D_ϕ(g^(j), f) = 0, for some j ≤ d. We then get f ( x ) f a 1 f a 2 … f a j = g ( x ) g a 1 ( 1 − 1 ) g a 2 ( 2 − 1 ) … g a j ( j − 1 ), since, by induction, we have g ( j ) ( x ) = g ( x ) f a 1 g a 1 ( 1 − 1 ) f a 2 g a 2 ( 2 − 1 ) … f a j g a j ( j − 1 ).

Consequently, lemma C.1 and the fact that the conditional densities with elliptical distributions are also elliptical, as well as the above relationship, lead us to infer that n ( a j + 1 ⊤ x , . , a d ⊤ x ) = f ( . / a i ⊤ x , 1 ≤ i ≤ j ) = g ( . / a i ⊤ x , 1 ≤ i ≤ j ). In other words, f coincides with g on the complement of the vector subspace generated by the family {a_i}_i=1,…,j.

Now, if the family {a_j}_1≤j≤d is no longer the canonical basis of ℝ^d, then this family is again a basis of ℝ^d. Hence, lemma C.2 implies that (3.2) g ( . / a 1 ⊤ x , … , a j ⊤ x ) = n ( a j + 1 ⊤ x , … , a d ⊤ x ) = g ( . / a 1 ⊤ x , … , a j ⊤ x )which is equivalent to D_ϕ(g^(j), f) = 0, since by induction g ( j ) = g f a 1 g a 1 ( 1 − 1 ) f a 2 g a 2 ( 2 − 1 ) … f a j g a j ( j − 1 ).

The end of our algorithm implies that f coincides with g on the complement of the vector subspace generated by the family {a_i}_i=1,…,j. Therefore, the nullity of the ϕ-divergence provides us with information on the density structure.

In summary, the following proposition clarifies our choice of g which depends on the family of distribution one wants to find in f :

Let X₁, X₂,‥,X_m (resp. Y₁, Y₂,‥,Y_m) be a sequence of m independent random vectors with the same density f (resp. g). As customary in nonparametric ϕ-divergence optimizations, all estimates of f and f_a, as well as all uses of Monte Carlo methods are being performed using subsamples X₁, X₂,‥,X_n and Y₁, Y₂,‥,Y_n—extracted respectively from X₁, X₂,‥,X_m and Y₁, Y₂,‥,Y_m—since the estimates are bounded below by some positive deterministic sequence θ_m—see Appendix D.

Let ℙ_n be the empirical measure based on the subsample X₁, X₂,.,X_n. Let f_n (resp. f_a,n for any a in ℝ ∗ d be the kernel estimate of f (resp. f_a), which is built from X₁, X₂,‥,X_n (resp. a^⊤X₁, a^⊤X₂,‥,a^⊤X_n).

As defined in Section 2.2, we consider the following sequences (a_k)_k≥1 and (g^(k))_k≥1 such that (3.3) a k is a non null vector of ℝ d defined by a k = arg min a ∈ ℝ ∗ d D ϕ ( g ( k − 1 ) f a g a ( k − 1 ) , f ) g ( k ) is the density defined by g ( k ) = g ( k − 1 ) f a k g a k ( k − 1 ) with g ( 0 ) = g

The stochastic setting up of the algorithm uses f_n and g n ( 0 ) = g instead of f and g⁽⁰⁾ = g—since g is known. Thus, at the first step, we build the vector ǎ₁ which minimizes the ϕ-divergence between f_n and g f a , n g a and which estimates a₁. First, since proposition D.1 and lemma C.4 show how the infimum of the criteria (or index) D ˇ ϕ ( g f a , n g a , f n ) = 1 n ∑ i = 1 n φ ( g ( X i ) f a , n ( a ⊤ X i ) f n ( X i ) g a ( a ⊤ X i ) )is reached, we are then able to minimize the ϕ-divergence between f_n and g f a , n g a. Second, defining ǎ₁ as the argument of this minimization, proposition 4.3 infers that this vector tends to a₁. Finally, we define the density g ˇ n ( 1 ) as g ˇ n ( 1 ) = g f a ˇ 1 , n g a ˇ 1 which estimates g⁽¹⁾ through theorem 4.1.

Now, from the second step and as defined in Section 2.2, we derive the fact that the density g^(2–1) is unknown. Consequently, once again, the samples have to be truncated.

All estimates of f and f_a (resp. g⁽¹⁾ and g a ( 1 )) are being performed using a subsample X₁, X₂,…,X_n (resp. Y 1 ( 1 ) , Y 2 ( 1 ) , … , Y n ( 1 )) extracted from X₁, X₂,…,X_m (resp. Y 1 ( 1 ) , Y 2 ( 1 ) , … , Y m ( 1 ), which is a sequence of m independent random vectors with same density g⁽¹⁾) such that the estimates are bounded below by some positive deterministic sequence θ_m—see Appendix D.

Let ℙ_n be the empirical measure of the subsample X₁, X₂,…,X_n. Let f_n (resp. g n ( 1 ) , f a , n , g a , n ( 1 ) for any a in ℝ ∗ d) be the kernel estimate of f (resp. g⁽¹⁾ and f_a as well as g a ( 1 )) which is built from X₁, X₂,…,X_n (resp. Y 1 ( 1 ) , Y 2 ( 1 ) , … , Y n ( 1 ) and a^⊤X₁, a^⊤X₂,…,a^⊤X_n as well as a ⊤ Y 1 ( 1 ) , a ⊤ Y 2 ( 1 ) , … , a ⊤ Y n ( 1 )).

The stochastic setting up of the algorithm uses f_n and g n ( 1 ) instead of f and g⁽¹⁾. Thus, we build the vector ǎ₂, which minimizes the ϕ-divergence between f_n and g n ( 1 ) f a , n g a , n ( 1 ), since g⁽¹⁾ and g a ( 1 ) are unknown—and which estimates a₂. First, since proposition D.1 and lemma C.4 show how the infimum of the criteria (or index) D ˇ ϕ ( g n ( 1 ) f a , n g a , n ( 1 ) , f n ) = 1 n ∑ i = 1 n φ ( g n ( 1 ) ( X i ) f a , n ( a ⊤ X i ) f n ( X i ) g a , n ( 1 ) ( a ⊤ X i ) )is reached, we are then able to minimize the ϕ-divergence between f_n and g n ( 1 ) f a , n g a , n ( 1 ). Second, defining ǎ₂ as the argument of this minimization, proposition 4.3 infers that this vector tends, to a₂. Finally, we define the density g ˇ n ( 2 ) as g ˇ n ( 2 ) = g n ( 1 ) f a ˇ 2 , n g a ˇ 2 , n ( 1 ) which estimates g⁽²⁾ through theorem 4.1.

And so on, we end up obtaining a sequence (ǎ₁, ǎ₂, …) of vectors in ℝ ∗ d estimating the co-vectors of f and a sequence of densities ( g ˇ n ( k ) ) k such that g ˇ n ( k ) estimates g^(k) through theorem 4.1.

Let us now summarize the main steps of the stochastic implementation of our algorithm (the dual representation of the estimators will be further detailed in Table 2 below).

In this paragraph, we define the set of hypotheses on f which could possibly be of use in our work. Discussion on several of these hypotheses can be found in Appendix E.

In the remaining of this section, for legibility reasons, we replace g with g^(k−1). Let Θ = ℝ d , Θ D ϕ = { b ∈ Θ | ∫ φ ∗ ( φ ′ ( g ( x ) f ( x ) f b ( b ⊤ x ) g b ( b ⊤ x ) ) ) d P < ∞ } M ( b , a , x ) = ∫ φ ′ ( g ( x ) f ( x ) f b ( b ⊤ x ) g b ( b ⊤ x ) ) g ( x ) f a ( a ⊤ x ) g a ( a ⊤ x ) d x − φ ∗ ( φ ′ ( g ( x ) f ( x ) f b ( b ⊤ x ) g b ( b ⊤ x ) ) ) ℙ n M ( b , a ) = ∫ M ( b , a , x ) d ℙ n , P M ( b , a ) = ∫ M ( b , a , x ) d Pwhere P is the probability measure presenting f as density.

Similarly as in chapter V of Van der Vaart [22], let us define :

(A1) : For all ε > 0, there is η > 0, such that for all c ∈ Θ^D_ϕ verifying ‖c − a_k‖ ≥ ε, we have PM(c, a) − η > PM(a_k, a), with a ∈ Θ.

Putting I a k = ∂ 2 ∂ a 2 D ϕ ( g f a k g a k , f ), let us consider now four new hypotheses:

(A5) : P ‖ ∂ ∂ b M ( a k , a k ) ‖ 2 and P ‖ ∂ ∂ a M ( a k , a k ) ‖ 2 are finite and the expressions P ∂ 2 ∂ b i ∂ b j M ( a k , a k ) and I_ak exist and are invertible.

Let f be a density defined on ℝ². Let us also consider g, a known elliptical density with the same mean and variance as f. Let us also assume that the family (a_i) is the canonical basis of ℝ² and that D_ϕ(g⁽²⁾, f) = 0.

Hence, since lemma C.1 page 110 implies that g a j ( j − 1 ) = g a j if j ≤ d, we then have g ( 2 ) ( x ) = g ( x ) f 1 g 1 f 2 g 2 ( 1 ) = g ( x ) f 1 g 1 f 2 g 2. Moreover, we get f with g⁽²⁾ = f, as derived from property B.1 page 110.

Consequently, f = g ( x ) f 1 g 1 f 2 g 2, i.e., f f 1 f 2 = g g 1 g 2, and then ∂ 2 ∂ x ∂ y C f = ∂ 2 ∂ x ∂ y C g where C_f (resp. C_g) is the copula of f (resp. g).

More generally, if f is defined on ℝ^d, then the family (a_i) is once again free (see lemma C.5), i.e., the family (a_i) is once again a basis of ℝ^d. The relationship D_ϕ(g^(d), f) = 0 therefore implies that g^(d) = f, i.e., for any x ∈ ℝ^d, f ( x ) = g ( d ) ( x ) = g ( x ) Π k = 1 d f a k ( a k ⊤ x ) [ g ( k − 1 ) ] a k ( a k ⊤ x ) = g ( x ) Π k = 1 d f a k ( a k ⊤ x ) g a k ( a k ⊤ x ) since lemma C.1 page 110 implies that g a k ( k − 1 ) = g a k if k ≤ d. In other words, for any x ∈ ℝ^d, it holds (5.1) g ( x ) Π k = 1 d g a k ( a k ⊤ x ) = f ( x ) Π k = 1 d f a k ( a k ⊤ x )

Finally, putting A = (a₁, …, a_d) and defining vector y (resp. density f̃, copula C̃_f of f̃, density g̃, copula C̃_g of g̃) as the expression of vector x (resp. density f, copula C_f of f, density g, copula C_g of g) in basis A, then, the following proposition provides us with the density associated with the copula of f as being equal to the density associated with the copula of g in basis A :

Let f be an unknown density defined on ℝ^d. The objective of the present section is to determine whether the copula of f is elliptical. We thus define an instrumental elliptical density g with the same mean and variance as f, and we follow the procedure of Section 3.2. As explained in Section 5.1, we infer from proposition 5.1 that the copula of f equals the copula of g when D_ϕ(g^(d), f) = 0, i.e., when a_d is the last vector generated from the algorithm and when (a_i) is the canonical basis of ℝ^d. Thus, in order to verify this assertion, corollary 4.1 page 96 provides us with a α-level confidence ellipsoid around this vector, namely ℰ d = { b ∈ ℝ d ; n ( Var P ( M ( b , b ) ) ) − 1 / 2 ℙ n M ( b , b ) ≤ q α N ( 0 , 1 ) }where q α N ( 0 , 1 ) is the quantile of a α-level reduced centered normal distribution, where ℙ_n is the empirical measure arising from a realization of the sequences (X₁, …, X_n) and (Y₁, …, Y_n)—see Appendix D—and where M is a known function of d, f_n and g n ( d − 1 ) —see Section 4.1.

Consequently, keeping the notations introduced in Section 5.1, we perform a statistical test of the null hypothesis ( H 0 ) : ∂ d ∂ x 1 … ∂ x d C f = ∂ d ∂ x 1 … ∂ x d C g versus ( H 1 ) : ∂ d ∂ x 1 … ∂ x d C f ≠ ∂ d ∂ x 1 … ∂ x d C g

Since, under (H₀), we have D_ϕ(g^(d), f) = 0, then the following theorem provides us with a confidence region for this test.

Let f be a density on ℝ^d and let X be a random vector with f as density. The objective of this section is to determine whether f is the product of its margins, i.e., whether the copula of f is the independent copula. Let g be an instrumental product of univariate Gaussian density—with diag(Var(X₁),…, Var(X_d)) as covariance matrix and with the same mean as f. As explained at Section 5.2, we follow the procedure described at Section 3.2, i.e., proposition 5.1 infers that the copula of f is the independent copula when D_ϕ(g^(d), f) = 0, we then perform a statistical test of the null hypothesis: ( H 0 ) : f = Π i = 1 d f i versus the alternative ( H 1 ) : f ≠ Π i = 1 d f i

Since, under (H₀), we have D_ϕ(g^(d), f) = 0, the following theorem provides us with a confidence region for our test.

Let be the set of non-negative integers defined by Q = { k i ′ ; k 1 ′ = 1 , k q ′ = d , k i ′ < k i + 1 ′ }, where q—such that q ≤ d—is its cardinal. In the present section, our goal is to study the subsequence (g^(k′)) of the sequence (g^(k))_k=1‥d defined by D_ϕ(g^(k′), f) = 0 for any k′ belonging to .

First, we have:

D_ϕ(g^(d), f) = 0 ⇔ g^(d) = f, through property B.1

Moreover, defining k ∼ i ′ as the previous integer k i ′, in the space {1, …, d}, with i > 1, and as explained in Section 3.1, the relationship D_ϕ(g^(k′), f) = 0 implies that f ∼ ( y i , … , y k ∼ i + 1 ′ / y i , … , y k ∼ i ′ , y k ∼ i + 1 ′ , … , y d ) = f ∼ i , i + 1 ( y i , … , y k ∼ i + 1 ′ )where f̃_i,i+1 is the density of vector ( a i ⊤ X , … , a k ∼ i + 1 ′ ⊤ X ) in the A = (a₁,…,a_d) basis. Consequently, f ∼ ( y ) = f ∼ 1 , 2 ( y 1 , … y k ∼ 2 ′ ) ⋅ f ∼ 2 , 3 ( y k 2 ′ , … , y k ∼ 3 ′ ) … f ∼ q − 1 , d ( y k q − 1 ′ , … , y k ∼ d ′ ).

Hence, we can infer that (5.2) f ∼ ( y ) Π k = 1 d f ∼ k ( y k ) = f ∼ 1 , 2 ( y i , … , y k ∼ 2 ′ ) Π k = 1 k ∼ 2 ′ f ∼ k ( y k ) . f ∼ 2 , 3 ( y k ∼ 2 ′ , … , y k ∼ 3 ′ ) Π k = k 2 ′ k ∼ 3 ′ f ∼ k ( y k ) … f ∼ q − 1 , d ( y k ∼ q − 1 ′ , … , y k ∼ d ′ ) Π k = k ∼ q − 1 ′ d f ∼ k ( y k )

The following theorem explicitly describes the form of the f copula in the A = (a₁, …, a_d) basis:

We are in dimension 2(=d), and we use the χ² divergence to perform our optimisations. Let us consider a sample of 50(=n) values of a random variable X with a density law f defined by : f ( x ) = c ρ ( F Gumbel ( x 1 ) , F Exponential ( x 2 ) ) . Gumbel ( x 1 ) . Exponential ( x 2 )where c is the Gaussian copula with correlation coefficient ρ = 0.5, and where the Gumbel distribution parameters are −1 and 1 and the exponential density parameter is 2.

Let us generate then a Gaussian random variable Y with a density—that we will name as g—presenting the same mean and variance as f.

We theoretically obtain k = 2 and (a₁, a₂) = ((1, 0), (0, 1)).

To get this result, we perform the following test: ( H 0 ) : ( a 1 , a 2 ) = ( ( 1 , 0 ) , ( 0 , 1 ) ) versus ( H 1 ) : ( a 1 , a 2 ) ≠ ( ( 1 , 0 ) , ( 0 , 1 ) )

Then, theorem 5.1 enables us to verify (H₀) by the following 0.9(=α) level confidence ellipsoid ℰ 2 = { b ∈ ℝ 2 ; ( Var P ( M ( b , b ) ) ) ( − 1 / 2 ) ℙ n M ( b , b ) ≤ q α N ( 0 , 1 ) / n ≃ 0 , 2533 / 7.0710 = 0.03582 }

Results of this optimisation can be found in Table 3 and Figure 1.

Therefore, we can conclude that H₀ is verified.

We are in dimension 2(=d), and we use the χ² divergence to perform our optimisations.

Let us consider a sample of 50(=n) values of a random variable X with a density law f defined by f ( x ) = Gumbel ( x 1 ) . Exponential ( x 2 )where the Gumbel distribution parameters are −1 and 1 and the exponential density parameter is 2.

Let g be an instrumental product of univariate Gaussian densities with diag(V ar(X₁), …, V ar(X_d)) as covariance matrix and with the same mean as f.

We theoretically obtain k = 2 and (a₁, a₂) = ((1, 0), (0, 1)). To get this result, we perform the following test: ( H 0 ) : ( a 1 , a 2 ) = ( ( 1 , 0 ) , ( 0 , 1 ) ) versus ( H 1 ) : ( a 1 , a 2 ) ≠ ( ( 1 , 0 ) , ( 0 , 1 ) )

Then, theorem 5.2 enables us to verify (H₀) by the following 0.9(=α) level confidence ellipsoid ℰ 2 = { b ∈ ℝ 2 ; ( Var P ( M ( b , b ) ) ) ( − 1 / 2 ) ℙ n M ( b , b ) ≤ q α N ( 0 , 1 ) / n ≃ 0.03582203 }

Results of this optimisation can be found in Table 4 and Figure 2.

Therefore, we can conclude that f = Π i = 1 d f i.

(On the choice of a ϕ-divergence). In this paragraph, we perform our algorithm several times. We first use several ϕ-divergences (see Appendix B for their definitions and their notations). We then perform a sensitivity analysis by varying the number n of simulated variables. Finally we introduce outliers.

At present, we consider a sample of n values of a random variable X with a density f defined by f(x) = Laplace(x₁).Gumbel(x₀),

where the Gumbel distribution parameters are (1, 2) and where the Laplace distribution parameters are 4 and 3. In theory, we get a₁ = (0, 1) and a₂ = (1, 0). Then, following the procedure of the first simulation, we get

Let us for instance study the moves in the stock prices of Renault and Peugeot from January 4, 2010 to July 25, 2010. We thus gather 140(=n) data from these stock prices, see Table 7 and Table 8 below.

Let us also consider X₁ (resp. X₂) the random variable defining the stock price of Renault (resp. Peugeot). We will assume—as it is commonly done in mathematical finance—that the stock market abides by the classical hypotheses of the Black-Scholes model—see Black and Scholes [34].

Consequently, X₁ and X₂ each present a log-normal distribution as probability distribution.

Let f be the density of vector (ln(X₁), ln(X₂)), let us now apply our algorithm to f with the Kullback-Leibler divergence as ϕ-divergence. Let us generate then a Gaussian random variable Y with a density—that we will name as g—presenting the same mean and variance as f.

We first assume that there exists a vector a such that D ϕ ( g f a g a , f ) = 0.

In order to verify this hypothesis, our reasoning will be the same as in Simulation 6.1. Indeed, we assume that this vector is a co-factor of f. Consequently, corollary 4.2 enables us to estimate a by the following 0.9(=α) level confidence ellipsoid ℰ 1 = { b ∈ ℝ 2 ; ( Var P ( M ( b , b ) ) ) ( − 1 / 2 ) ℙ n M ( b , b ) ≤ q α N ( 0 , 1 ) / n ≃ 0 , 2533 / 140 = 0.02140776 } .

Numerical results of the first projection are summarized in Table 5.

Therefore, our first hypothesis is confirmed.

However, our goal is to study the copula of (ln(X₁), ln(X₂)). Then, as explained in Section 5.4, we formulate another hypothesis assuming that there exists a vector a such that D ϕ ( g ( 1 ) f a g a ( 1 ) , f ) = 0.

In order to verify this hypothesis, we use the same reasoning as above. Indeed, we assume that this vector is a co-factor of f. Consequently, corollary 4.2 enables us to estimate a by the following 0.9(=α) level confidence ellipsoid ℰ 2 = { b ∈ ℝ 2 ; ( Var P ( M ( b , b ) ) ) ( − 1 / 2 ) ℙ n M ( b , b ) ≤ q α N ( 0 , 1 ) / n ≃ 0 , 2533 / 140 = 0.02140776 }. Numerical results of the second projection are summarized in Table 6.

Therefore, our second hypothesis is confirmed.

In conclusion, as explained in corollary 5.1, the copula of f is equal to 1 in the {a₁, a₂} basis.

This result has been illustrated at Figures 3, 4 and 5.

In the case where f is unknown, we will never be sure to have reached the minimum of the ϕ-divergence: the simulated annealing method has been used to solve our optimisation problem, and therefore it is only when the number of random jumps tends in theory towards infinity that the probability to get the minimum tends to 1. We also note that no theory on the optimal number of jumps to implement does exist, as this number depends on the specificities of each particular problem.

Moreover, we choose the 50 − 4 4 + d for the AMISE of the two simulations. This choice leads us to simulate 50 random variables—see Scott [23] page 151, none of which have been discarded to obtain the truncated sample.

This has also been the case in our application to real datasets.

Finally, the shape of the copula in the case of real datasets in the {a₁, a₂} basis is also noteworthy.

Figure 4 shows that the curve reaches a quite wide plateau around 1, whereas Figure 5 shows that this plateau prevails on almost the entire [0, 1]² set. We can therefore conclude that the theoretical analysis is indeed confirmed by the above simulation.

Projection pursuit is useful in evidencing characteristic structures as well as one-dimensional projections and their associated distribution in multivariate data. This article clearly demonstrates the efficiency of the φ-projection pursuit methodology for goodness-of-fit tests for copulas. Indeed, the robustness as well as the convergence results that we achieved convincingly fulfilled our expectations regarding the methodology used.