Sparse M-estimators in semi-parametric copula models

In this paper, we consider the parsimonious estimation of copula models within the semi-parametric framework: margins are left unspecified and a parametric copula model is assumed. The sparse assumption is motivated by the model complexity that occurs in copula modelling, where the parameterization may require the estimation of a large number of parameters. For instance, the variance-covariance matrix of a $d$-dimensional Gaussian copula involves the estimation of $d(d-1)/2$ parameters, the components of an unknown correlation matrix; in single-index copula, the underlying conditional copula is parameterized through a link function that depends on a potentially large number of covariates and thus parameters, which may not be all relevant for describing these conditional distributions. Since the seminal work of [12], a significant amount of literature dedicated to sparsity-based M-estimators has been flourishing in a broad range of settings. In contrast, the sparse estimation of copula based M-estimators has benefited from a very limited attention so far. In [7], the authors considered a mixture of copulas with a joint estimation of the weight parameters and the copula parameters, while penalizing the former ones only. However, a strong limitation of their approach is the parametric assumption formulated for the marginals, which greatly simplifies the large sample inference. [37] specified a penalized estimating equation-based estimator for single-index copula models and derived the corresponding large sample properties but assuming known margins. A theory covering the sparse estimation for semi-parametric copulas is an important missing stone in the literature. One of the key difficulties is the treatment of the values close to the boundaries of $[0,1]^{d}$, where some loss functions potentially “explode”. The latter situation is not pathological. It often occurs for the standard Canonical Maximum Likelihood method or CML (see, e.g., [19, 33, 34]) and many usual copula log-densities, as pointed out in [32] in particular. Since the seminal works of [31, 30], the large sample properties of the empirical copula process $\widehat{{\mathbb{C}}}_{n}$ were established by, e.g., [16], and such properties were applied to the asymptotic analysis of copula-based maximum likelihood estimators with pseudo-observations. In that case, the empirical copula process is indexed by the likelihood function: see [34, 8], who considered some regularity conditions on the likelihood function to manage the values close to the boundaries of $[0,1]^{d}$. Similar conditions were stated likewise in [8, 39, 22], among others, where some bracketing number conditions on a suitable class of functions are assumed. These works share a similar spirit with [36, 11], who considered a general framework of empirical processes indexed by classes of functions under entropy conditions. Thanks to a general integration by parts formula, [28] established the conditions for the weak convergence of the empirical copula process $\int f\text{d}\widehat{{\mathbb{C}}}_{n}$ indexed by a class of functions $f\in\mathcal{F}$ of bounded variation, the so-called Hardy–Krause variation. Their results do not require explicit entropy conditions on the class of functions. In the same vein, [6] assumed similar regularity conditions on the indexing functions but restricted their analysis to the two-dimensional copula case. It is worth mentioning that the techniques for stating the large sample analysis of semi-parametric copulas amply differ from the fully parametric viewpoint, for which the classical M-estimation theory obviously applies.

The present paper is then motivated by the lack of links between sparsity and semi-parametric copulas. Our asymptotic analysis for sparse M-estimators in the context of semi-parametric copulas builds upon the theoretical frameworks of [6] and [28]. The contribution of our paper is fourfold: first, we provide the asymptotic theory (consistency, oracle property for variable selection and asymptotic normality in the same spirit as [12]) for general penalized semi-parametric copula models, where the margins are estimated by their empirical counterparts. In particular, our setting includes the Canonical Maximum Likelihood method. Second, these asymptotic results are extended for (a sequence of) copula models in large dimensions, a framework that corresponds to the diverging dimension case, as in, e.g., [13]. Third, we prove the asymptotic normality of multivariate-rank statistics for any arbitrary dimension $d\geq 2$, extending Theorem 3.3 of [6]. Fourth and finally, we prove the weak convergence of the empirical copula process indexed by functions of bounded variation, extending Theorem 5 of [28] to cover the prevailing situation of unbounded copula densities. We emphasize that our theory is not restricted to i.i.d. data and potentially covers the case of dependent observations, as in [28].

The rest of the paper is organized as follows. Section 2 details the framework and fix our notations. The large sample properties of our penalized estimator are provided in Section 3. The situation of conditional copulas is managed in Section 4. Section 5 discusses some examples and two simulated experiments to illustrate the relevance of our method. The theoretical results about multivariate rank statistics and empirical copula processes are stated in Appendix A. All the proofs, the theoretical results in the case of a diverging number of parameters and additional simulated experiments are provided in the Appendix.

This section details the sparse estimation framework for copula models when the marginal distributions are non-parametrically managed. We consider a sample of $n$ realizations of a random vector $\mbox{\boldmath$X$}\in{\mathbb{R}}^{d}$, $\mbox{\boldmath$X$}:=(X_{1},\ldots,X_{d})$. This sample is denoted as ${\mathcal{X}}_{n}=(\mbox{\boldmath$X$}_{1},\ldots,\mbox{\boldmath$X$}_{n})$. The observations may be dependent or not. As usual in the copula world, we are more interested in the “reduced” random variables $U_{k}=F_{k}(X_{k})$, $k\in\{1,\ldots,d\}$, where $F_{k}$ denotes the cumulative distribution function (c.d.f.) of $X_{k}$. Throughout this paper, we make the blanket assumption that all $X_{k}$ have continuous marginals. Then, the variables $U_{k}$ are uniformly distributed on $[0,1]$ and the joint law of $\mbox{\boldmath$U$}:=(U_{1},\ldots,U_{d})$ is the uniquely defined copula of $X$ denoted by $C$. To study the latter copula, it would be tempting to work with the sample ${\mathcal{U}}_{n}:=(\mbox{\boldmath$U$}_{1},\ldots,\mbox{\boldmath$U$}_{n})$ instead of ${\mathcal{X}}_{n}$. Unfortunately, since the marginal c.d.f.s’ $X_{k}$ are unknown in general, this is still the case of ${\mathcal{U}}_{n}$, and the marginal c.d.f.s’ have to be replaced by consistent estimates. Therefore, it is common to build a sample of pseudo-observations $\widehat{\mbox{\boldmath$U$}}_{i}=(\widehat{U}_{i,1},\ldots,\widehat{U}_{i,d})$, $i\in\{1,\ldots,n\}$, obtained from the initial sample ${\mathcal{X}}_{n}$. Here and as usual, set $\widehat{U}_{i,k}=F_{n,k}(X_{i,k})$ for every $i\in\{1,\ldots,n\}$ and every $k\in\{1,\ldots,d\}$, using the $k$-th re-scaled empirical c.d.f. $F_{n,k}(s):=(n+1)^{-1}\sum^{n}_{i=1}\mathbf{1}\{X_{i,k}\leq s\}.$ We will denote by $G_{n,k}$, $k\in\{1,\ldots,d\}$, the empirical c.d.f. of the (unobservable) random variable $U_{k}$, i.e. $G_{n,k}(u):=(n+1)^{-1}\sum^{n}_{i=1}\mathbf{1}\{F_{k}(X_{i,k})\leq u\}.$ The empirical c.d.f. of $U$ is $G_{n}$, i.e. $G_{n}(\mbox{\boldmath$u$}):=(n+1)^{-1}\sum^{n}_{i=1}\mathbf{1}\{\mbox{\boldmath$U$}_{i}\leq\mbox{\boldmath$u$}\}$ for any $\mbox{\boldmath$u$}\in[0,1]^{d}$. We denote by $\alpha_{n}$ the usual empirical process associated with the sample $(\mbox{\boldmath$U$}_{i})_{i=1,\ldots,n}$, i.e.

The natural estimator of the true underlying copula $C$, i.e. the c.d.f. of $U$, is the empirical copula map

and the associated empirical copula process is $\widehat{\mathbb{C}}_{n}:=\sqrt{n}(\widehat{C}_{n}-C)$.

Hereafter, we select a parametric family of copulas $C_{\theta}$, $\theta\in\Theta\subset{\mathbb{R}}^{p}$, and we assume it contains the true copula $C$: there exists $\theta_{0}$ (the “true value” of the parameter) s.t. $C=C_{\theta_{0}}$. We want to cover the usual case of semi-parametric dependence models, for which there is an orthogonality condition of the type ${\mathbb{E}}\big{[}\nabla_{\theta}\ell(\mbox{\boldmath$U$};\theta_{0})\big{]}=0,$ for some family of loss functions $\ell:(0,1)^{d}\times\Theta\rightarrow{\mathbb{R}}$. The dimension $p$ of the copula parameter $\theta$ will be fixed hereafter. In the Appendix, it will be allowed to tend to the infinity with the sample size $n$. The function $\ell$ is usually defined as a quadratic loss or minus a log-likelihood function. Note that $\ell$ has not to be defined on the boundaries of $[0,1]^{d}$ at this stage because the law of $U$ was assumed to be continuous. Moreover, an important contribution of the paper will be to deal with some maps $\ell$ that cannot be continuously extended on $[0,1]^{d}$.

For the sake of the estimation of $\theta_{0}$, let us specify a statistical criterion. Consider a global loss function ${\mathbb{L}}_{n}$ from $\Theta\times(0,1)^{dn}$ to ${\mathbb{R}}$. The value ${\mathbb{L}}_{n}(\theta;\mbox{\boldmath$u$}_{1},\ldots,\mbox{\boldmath$u$}_{n})$ evaluates the quality of the “fit” given $\mbox{\boldmath$U$}_{i}=\mbox{\boldmath$u$}_{i}$ for every $i\in\{1,\ldots,n\}$ and under ${\mathbb{P}}_{\theta}$. Hereafter, we assume there exists a continuous function $\ell:\Theta\times(0,1)^{d}\rightarrow{\mathbb{R}}$ such that

for every $\theta\in\Theta$ and every $(\mbox{\boldmath$u$}_{1},\ldots,\mbox{\boldmath$u$}_{n})$ in $(0,1)^{dn}$. As usual for the inference of semiparametric copula models, the empirical loss ${\mathbb{L}}_{n}(\theta;{\mathcal{U}}_{n})$ cannot be calculated since we do not observe the realizations of $U$ in practice. Therefore, invoking the “pseudo-sample” $\widehat{{\mathcal{U}}}_{n}:=(\widehat{\mbox{\boldmath$U$}}_{1},\ldots,\widehat{\mbox{\boldmath$U$}}_{n})$, the empirical loss ${\mathbb{L}}_{n}(\theta;{\mathcal{U}}_{n})$ will be approximated by ${\mathbb{L}}_{n}(\theta;\widehat{\mathcal{U}}_{n})$, a quantity called “pseudo-empirical” loss function.

Now, we assume that the unknown parameter is sparse and we introduce a penalization term. Our criterion becomes

when such a minimizer exists. Here, $\mbox{\boldmath$p$}(\lambda_{n},x)$, for $x\geq 0$, is a penalty function, where $\lambda_{n}\geq 0$ is a tuning parameter which depends on the sample size and enforces a particular type of sparse structure. Throughout the paper, we will implicitly work under the following assumption.

Note that only the uniqueness of $\theta_{0}$ is required but not that of $\widehat{\theta}$. In [12, 13], $\Theta$ is assumed to be an open subset. Nonetheless, it may happen that $\theta_{0}$ belongs to the frontier of $\Theta$. This may be the case for the estimation of weights in mixture models (see Example 3 below), e.g. Actually, our results will apply even if $\Theta$ does not contain an open neighborhood of $\theta_{0}$. Indeed, it is sufficient that there exists a convex open neighborhood of $\theta_{0}$ in $\Theta$: for some open ball $B_{\delta}(\theta_{0})\subset{\mathbb{R}}^{p}$ centered at $\theta_{0}$ and with a radius $\delta>0$ and for every parameter $\theta_{1}\in\Theta\cap B_{\delta}(\theta_{0})$, every element of the segment $t\theta_{0}+(1-t)\theta_{1}$, $t\in[0,1]$ belongs to $\Theta$ and $\theta_{1}$ is the limit of a sequence of elements in $B_{\delta}(\theta_{0})\cap\text{Int}(\Theta)$, where $\text{Int}(\Theta)$ is the interior of $\Theta$. Therefore, we will define the partial derivatives of any map $h:\Theta\mapsto{\mathbb{R}}$ at some point on the frontier of $\Theta$ (in particular $\theta_{0}$) by continuity. For example, when the derivative of $\theta\mapsto\ell(\theta;\mbox{\boldmath$u$})$ exists in the interior of $\Theta$, for some $u$, it will be defined at $\theta_{0}$ by $\nabla_{\theta}\ell(\theta_{0};\mbox{\boldmath$u$}):=\lim_{\theta\rightarrow\theta_{0},\theta\in\Theta}\nabla_{\theta}\ell(\theta;\mbox{\boldmath$u$})$, assuming the latter limit exists. To lighten the presentation, we will keep Assumption  ‣ 2 as above. By slightly strengthening some regularity assumptions, the case of $\theta_{0}$ on the boundary of $\Theta$ will be straightforwardly obtained (essentially by imposing the continuity of some derivatives around $\theta_{0}$).

Some well-known penalty functions are the LASSO, where $\mbox{\boldmath$p$}(\lambda,{\color[rgb]{0,0,0}x})=\lambda{\color[rgb]{0,0,0}x}$ for every $x\geq 0$, and the non-convex SCAD and MCP. The SCAD penalty of [12] is defined as: for every $x\geq 0$,

where $a_{\text{scad}}>2$. The MCP due to [38] is defined for $b_{\text{mcp}}>0$ as follows: for every $x\geq 0$,

Note that when $a_{\text{scad}}\rightarrow\infty$ (resp. $b_{\text{mcp}}\rightarrow\infty$), the SCAD (resp. MCP) penalty behaves as the LASSO penalty since all coefficients of $\theta$ are equally penalized. The idea of sparsity for copulas naturally applies to a broad range of situations. In such cases, the parameter value zero usually plays a particular role, possibly after reparameterisation. This is in line with the usual previously cited penalties.

Dealing with conditional copulas ([17], e.g.), our framework will be slightly modified. Now, the law of a random vector $\mbox{\boldmath$X$}\in{\mathbb{R}}^{d}$ knowing the vector of covariates $\mbox{\boldmath$Z$}=\mbox{\boldmath$z$}\in{\mathbb{R}}^{m}$ is given by a parametric conditional copula whose parameter depends on a known map of $z$ and is denoted $\theta(\mbox{\boldmath$z$};\beta)$, $\beta\in{\mathbb{R}}^{q}$. Beside, the law of the margins $X_{k}$, $k\in\{1,\ldots,d\}$, given $\mbox{\boldmath$Z$}=\mbox{\boldmath$z$}$ are unknown and we assume they do not depend on $z$. In other words, the conditional law of $X$ given $Z$ is assumed to be

Therefore, as for the CML method and in the i.i.d. case, an estimator of $\beta$ would be

Surprisingly and to the best of our knowledge, the asymptotic theory of such estimators has apparently not been explicitly stated in the literature until now. This will be the topic of Section 4.

To prove the asymptotic results, we consider two sets of assumptions: one is related to the loss function; the other one concerns the penalty function. First, define the support of the true parameter as ${\mathcal{A}}:=\big{\{}k:\theta_{0,k}\neq 0,k=1,\ldots,p\big{\}}$. We will implicitly assume a sparsity assumption, i.e. the cardinal of ${\mathcal{A}}$ is ”significantly” smaller than $p$.

When $u$ does not belong to $(0,1)^{d}$ (i.e. when one of its components is zero or one), we have defined $\dot{C}_{j}(\mbox{\boldmath$u$}):=\lim\sup_{t\rightarrow 0}\big{\{}C(\mbox{\boldmath$u$}+t\mbox{\boldmath$e$}_{j})-C(\mbox{\boldmath$u$})\big{\}}/t$. It has been pointed out in [32] that Condition 2 is satisfied for bivariate Gaussian and bivariate extreme-value copulas in particular. We formally state this property for $d$-dimensional Gaussian copulas in Section E of the Appendix.

We will denote by $\partial_{2}\mbox{\boldmath$p$}(\lambda,x)$ (resp. $\partial^{2}_{2,2}\mbox{\boldmath$p$}(\lambda,x)$) the first order (resp. second order) derivative of $x\mapsto\mbox{\boldmath$p$}(\lambda,x)$, for any $\lambda$.

Assumption 1 is a standard regularity condition for asymptotically normal M-estimators. Assumptions 2 is a smoothness condition on the copula $C$ and is similar to Condition 4.1 of [32] and Condition 2.1 of [6]. It ensures that the second-order derivatives with respect to $u$ do not explode “too rapidly” when $u$ approaches the boundaries of $[0,1]^{d}$. Assumption 3 is related to the indexing function of the copula process, here by the first, second and third order derivatives of the loss function $\ell(\theta;.)$. The $g_{\omega}$-regularity of these functions ensures that they are of bounded Hardy-Krause variation - similar to assumption F of [28] - together with some integrability conditions. Here, $\omega$ is some fixed number in $(0,1/2)$ entering in the definition of a weight function $\min_{k}\min(u_{k},1-u_{k})^{\omega}$, as specified in Definition A, point (ii). Such a weight function is related to the theory of weighted empirical processes applied in [6]. Assumption 4 is dedicated to the regularity of the penalty function, and includes conditions in the same vein as [13], Assumption 3.1.1. Note that the LASSO, the SCAD and the MCP penalties fulfill Assumption 4. Our first result establishes the existence of a consistent penalized M-estimator.

The proof is postponed in Section D of the Appendix. The factor $\ln(\ln n)$ could be replaced by any sequence that tends to the infinity with $n$, as in Corollary A.2. It has been arbitrarily chosen for convenience. We will apply this rule throughout the article.

We now show that the penalized estimator $\widehat{\theta}$ satisfies the oracle property in the sense of [12]: the true support can be recovered asymptotically and the non-zero estimated coefficients are asymptotically normal. Denote by $\widehat{\mathcal{A}}:=\big{\{}k:\widehat{\theta}_{k}\neq 0,k=1,\ldots,p\big{\}}$ the (random) support of our estimator. The following theorem states the main results of this section. It uses some standard notations for concatenation of sub-vectors, as recalled in Appendix A.

The proof is relegated to Section D of the Appendix. The existence and the meaning of the integrals defining $W_{j}$ comes from Assumption 13. The proofs of the two latter theorems rely on a third-order limited expansion of our empirical loss w.r.t. its arguments. The negligible terms are managed thanks to a ULLN (Corollary A.2). An integration by parts formula (Theorem A.1) allows to write the main terms as sums of integrals of the empirical copula process w.r.t. some “sufficiently regular” functions. The latter ones are deduced from the derivatives of the loss, justifying the concept of $g_{\omega}$ regularity and Assumption 3. A weak convergence result of the weighted empirical copula process concludes the asymptotic normality proof.

Note that the previous results apply with dependent observations $(\mbox{\boldmath$X$}_{i})$. Indeed, in Theorem 3.2 (ii), we only require the weak convergence of the process $(\alpha_{n})$ to $\alpha_{C}$. In the i.i.d. case, $\alpha_{C}$ is a Brownian Bridge and $W$ is a Gaussian random vector. The latter assertion is still true when $(\mbox{\boldmath$X$}_{i})$ is a strongly stationary and geometrically alpha-mixing sequence, due to Proposition 4.4 in [6]. The existence and the meaning of the random variable $\int_{(0,1)^{|I|}}{\mathbb{C}}(\mbox{\boldmath$u$}_{I}:{\mathbf{1}}_{-I})\,\partial_{\theta_{j}}\ell(\theta_{0};d\mbox{\boldmath$u$}_{I};{\mathbf{1}}_{-I})$ come from Assumption 13. Our conditions on $(a_{n},\lambda_{n})$ allow to use the SCAD or the MCP penalty, but not the LASSO because $a_{n}=\lambda_{n}$ in the latter case. In other words, Theorem 3.1 may apply with LASSO but not Theorem 3.2. The fact that LASSO does not yield the oracle property has already been noted in the literature: see [40] and the references therein. Actually, consider any penalty such that $\mbox{\boldmath$p$}(\lambda,t)$ does not depend on $t>0$, when $\lambda$ is sufficiently small, as for the SCAD and MCP cases. Then,

when $n$ is sufficiently large because $\lambda_{n}\rightarrow 0$, implying $a_{n}=b_{n}=0$. Thus, Assumption 4 is satisfied and $\mathbf{A}_{n}(\theta_{0})=\mathbf{B}_{n}(\theta_{0})=\mathbf{0}$, for $n$ large enough. Therefore, for such penalty functions, the asymptotic law of $\widehat{\theta}$ becomes a lot simpler.

Theorem 3.2 establishes that the non-convex penalization procedure for semi-parametric copula models is an “oracle”: for a consistent estimator $\widehat{\theta}$, the true sparse support is correctly identified in probability, and the non-zero estimated parameters have the same limiting law as if the support of $\theta_{0}$ were known. The limit $W$ is in the same vein as the one in Theorem 5 of [28]. However, in contrast to the latter result which assumes bounded indexing functions on $[0,1]^{d}$, our framework covers the case of unbounded ones. In particular, our indexing functions $\mbox{\boldmath$u$}\mapsto\partial_{\theta_{j}}\ell(\theta_{0};\mbox{\boldmath$u$})$ are not constrained to be bounded on $(0,1)^{d}$.

In the particular case of CML, the asymptotic law of our estimator has been stated under a set of regularity assumptions (particularly Assumption 3) that significantly differs from those that have been proposed in the literature: see the seminal papers [34] (Assumption (A.1)) and [8] (Assumptions (A.2)-(A.4)). These competing sets of assumption are apparently not nested, due to two very different techniques of proofs: a general integration by parts formula in our case, and some old results from [31, 30] in the latter cases. Our set of assumptions should most often be weaker. Indeed, we typically do not require that some expectations as ${\mathbb{E}}\big{[}\{U_{1}(1-U_{1})U_{2}(1-U_{2})\}^{-a}\big{]}$ are finite for some $a>0$ (Assumption (A.3) in [8]). Moreover and importantly, our results may directly be applied to time series, contrary to the cited papers that are strongly restricted to i.i.d. observations.

Our proofs rely on the weak convergence of multivariate rank-order statistics: see Theorem A.1 in the Appendix, that is an extension of Theorem 3.3 of [6] in an arbitrary dimension. Note that some papers have already stated similar results, but under restrictive conditions: Theorem 6 in [15] assumed the existence of continuous partial derivatives of the true copula on the whole hypercube $[0,1]^{d}$ (contrary to our Assumption 2). Theorem 2.4 in [20] relies on numerous very technical assumptions and it is unclear whether the latter result is weaker/stronger than ours. Nonetheless, it “only” states weak convergence in the space of cadlag functions equipped with the Skorohod topology.

Applying Theorem 3.2 (ii) or its corollary, it is possible to approximate by plug-in the limiting law of $\sqrt{n}(\widehat{\theta}-\theta_{0})_{I}$, for every fixed subset $I\subset\widehat{\mathcal{A}}$ and assuming $\widehat{\mathcal{A}}={\mathcal{A}}$ (an event whose probability tends to one): just replace the unknown quantities with their empirical counterparts. This is obvious concerning the matrices $\mathbf{A}_{n}(\theta_{0})$, $\mathbf{B}_{n}(\theta_{0})$ and ${\mathbb{H}}$. Concerning the Gaussian random vector $W$, an estimator of its variance is proposed in Section B of the Appendix.

In several situations of interest, multivariate models are defined through some conditioning laws. In other words, there exist a random vector of covariates $\mbox{\boldmath$Z$}\in{\mathcal{Z}}$, for some borelian subset ${\mathcal{Z}}$ in ${\mathbb{R}}^{m}$, and we focus on the laws of $X$ given any value of $Z$. The natural framework is given by conditional copulas and the associated Sklar’s theorem ([17], e.g.). Generally speaking, conditional copula models have to manage conditional marginal distributions on one side, and conditional copulas on the other side. In our semiparametric approach, we do not specify the former ones. Indeed, this would be a source of complexities due to the need of kernel smoothing or other non parametric techniques ([17, 2], among others). Here, we make the following simplifying assumption.

In other words, the conditional margins and the unconditional ones are identical. Even if the latter assumption may be considered as relatively strong, it is not implausible. For instance, in typical copula-GARCH models ([9]), the marginal dynamics are filtered in a first stage, and a parametric copula is then postulated between the residuals. It is well-known that systemic risk measures strongly depend on the economic cycle. Thus, some macroeconomic explanatory variables may have a significant impact on the latter copula. But, concerning the marginal conditional distributions, this effect could be hidden due to the first-order phenomenon of “volatility clustering”.

Under Assumption 5, our dependence model of interest will be related to the laws of $U$ given $\mbox{\boldmath$Z$}=\mbox{\boldmath$z$}$, $\mbox{\boldmath$z$}\in{\mathcal{Z}}$, by keeping the same definition of $U$ as previously. Sparsity would then be related to the number of components of $z$ that are relevant to specify any copula of $X$ (or $U$, equivalently) given $\mbox{\boldmath$Z$}=\mbox{\boldmath$z$}$. Typically, the latter copulas belong to a given parametric $d$-dimensional family $\{C_{\theta};\theta\in\Theta\subset{\mathbb{R}}^{p}\}$ and the parameter $\theta$ depends on the underlying covariate: given $\mbox{\boldmath$Z$}=\mbox{\boldmath$z$}$, the law of $U$ is $C_{\theta(\mbox{\boldmath$z$};\beta_{0})}$, for some known map $\theta:{\mathcal{Z}}\times{\mathcal{B}}\rightarrow\Theta$, ${\mathcal{B}}\subset{\mathbb{R}}^{q}$. The problem is now to evaluate the true “new parameter” $\beta_{0}$, based on a sample $(\mbox{\boldmath$X$}_{i},\mbox{\boldmath$Z$}_{i})_{i=1,\ldots,n}$. Compared to the previous sections, the focus has switched from $(\theta_{0},\Theta,p)$ to $(\beta_{0},{\mathcal{B}},q)$. In particular, we will assume that the new parameter set ${\mathcal{B}}$ satisfies Assumption  ‣ 2 instead of $\Theta$.

Under Assumption 5, we define the same pseudo-observations as before, and keep the notation $\widehat{\mathcal{U}}_{n}$. In addition, set ${\mathcal{Z}}_{n}:=(\mbox{\boldmath$Z$}_{1},\ldots,\mbox{\boldmath$Z$}_{n})$. For example, the parameter $\beta_{0}$ may naturally be estimated without any penalty by CML as

Under sparsity and with penalties, the results of Sections 2 and 3 can be adapted to tackle this new problem and even more general ones. First of all, we need to distinguish the cases of known and/or unknown covariate distributions.