Estimation problems for some perturbations of the independence copula.

We call copula in general a bivariate copula, which is defined as a bivariate joint cumulative distibution function with uniform marginals on $[0,1]$. An equivalent definition can be found in Nelsen (2006). A stationary Markov chain can be represented by a copula and its stationary or marginal distribution. The copula of a stationary Markov chain is that of any of its consecutive variables. We say that a stationary sequence $(Y_{1},\cdots,Y_{n})$ with finite mean $\mu$ and variance $\sigma^{2}$ satisfies the central limit theorem if $\sqrt{n}(\bar{Y}-\mu)\to N(0,\sigma_{0}^{2})$, for some positive number $\sigma^{2}_{0}$. Kipnis and Varadhan (1986) showed that when the variance of partial sums of functions of the Markov chain satisfies $\sigma_{f}^{2}=\lim_{n\to\infty}n^{-1}var(S_{n})$, where $S_{n}=\sum_{k=1}^{n}f(X_{i})$, $Y=f(X)$, $\mu=\mathbb{E}(f(X))$, the central limit theorem holds. This theorem is combined with the Cramér-Wold device to prove a multivariate central limit theorem for parameter estimators in this paper. In doing so, we use the $n$-fold product of copulas to compute the limiting variance. The $n$-fold product of the copula uses partial derivatives to find a new copula. The notation $C_{,i}(u,v)$ is for the derivative of $C(u,v)$ with respect to the $i^{th}$ variable. The $n$-fold product of a copula is defined in Nelsen (2006) (or by Darsow et al. (1992)) by $C^{1}(u,v)=C(u,v),$

Given that we are dealing with Markov chains, we recall that the fold product for $n=2$ is the joint cumulative distribution of $(X_{1},X_{3})$ if $(X_{1},X_{2},X_{3})$ is a stationary Markov chain with copula $C(u,v)$ and the uniform marginal distribution (see Darsow et al. (1992)).

In general, the copula $C^{n}(u,v)$ is the joint distribution of $(X_{0},X_{n})$ when $X_{0},\cdots,X_{n}$ is a stationary Markov chain generated by $C(u,v)$ and the uniform distribution. These definitions can be found in different forms in various papers, but are equivalent when the study is concerned with absolutely continuous copulas and continuous marginal distributions. This is because Sklar’s theorem guarantees uniqueness of the representation of the joint distribution of a vector through its copula and marginal distributions (see Sklar(1959)). Copulas simplify the study of mixing properties of Markov chains due to the fact that for continuous random variables, the marginal distributions are scaled out in the computation; the formulas reduce to equivalent forms in terms of the uniform marginal distributions. This is one of the advantages of dealing with copulas while studying mixing.

Mixing coefficients, that we refer to, are defined in Bradley (2007), and were used in Longla (2023) to establish properties of Markov chains for the copulas that we are studying in this work. This work is not concerned by mixing in general, but rather uses the that fact, that mixing was established in an earlier work. This justifies the limitation of the background to the provided references. The mixing rate allows convergence of the variance of partial sums of functions of the Markov chains, and implies the needed ergodicity in the Theorem of Kipnis and Varadhan (1986). This theorem on its turn is used to establish central limit theorems for copula parameter estimators. In the simulation part of this work, we use a result of Longla and Peligrad (2021) to derive some estimators of copula parameter. These estimators are compared to the MLE, and a second class of estimators, that is derived based on the fact, that parameters are eigen-values of the operator induced by the copula of the Markov chain on $L^{2}(0,1)$.

We propose several estimators of copula parameters for the copulas created in Longla (2023). Large sample study of these estimators is provided in general via derivation of a multivariate central limit theorem. These results are applied to 3 concrete families of copulas. We have conducted a comparative simulation study in which 4 different estimators are used. For each of the examples, we perform maximum likelihood estimation and show that the $R$ codes for our proposed estimators take less time to run than the MLE and can be preferable in large samples.

The rest of the paper is organized as follows. In section 2 we provide the copula families of interest in this work along with some graphs of copula densities and level curves. In section 3 we present parameter estimation procedures for Markov chains generated by the copula families proposed by Longla (2023). This section includes our proposed estimators, central limit theorems for estimators and $\chi^{2}$ tests of independence of observations. A multivariate central limit theorem is provided here as well. These estimators and central limit theorems are results of this papers. They are applied to each of the 3 examples that are proposed in Section 2. In section 4 we provide a simulation study of parameter estimators for the considered copula families. For each of the proposed copulas, we prove existence of the MLE, and derive its asymptotic distribution. A comparative study is provided at the end of this section to show advantages of our proposed estimators. In Section 5 we provide comments and conclusions of the work. An appendix of proofs concludes the paper.

For the classes of copulas given above, the Spearman’s $\rho(C)$ is estimated by $\hat{\lambda}_{1}$.

1. 1.

   Sine-cosine copulas. Theorem 3.1 implies that for the copulas based on the sine-cosine basis,

   $$\displaystyle\sigma^{2}_{k}=1+\frac{\lambda_{2k}}{2}+\frac{\lambda_{k}^{2}\lambda_{2k}}{1-\lambda_{2k}},\quad\text{for}\quad\hat{\lambda}_{k}=\frac{1}{n}\sum_{i=1}^{n}2\cos 2\pi kU_{i}\cos 2\pi kU_{i-1}$$

   $$\text{and}\quad\sigma^{2}_{k}=1+\frac{\lambda_{2k}}{2}+\frac{\mu_{k}^{2}\lambda_{2k}}{1-\lambda_{2k}},\quad\text{for}\quad\hat{\mu}_{k}=\frac{1}{n}\sum_{i=1}^{n}2\sin 2\pi kU_{i}\sin 2\pi kU_{i-1}.$$

   It is also clear that for sine-cosine copulas, with $i\neq j$, $\Sigma_{i,j}$ is given by

   $\varphi_{k_{i}}/\varphi_{k_{j}}$	$\varphi_{1}(x)=\sqrt{2}\cos 2\pi k_{j}x$	$\varphi_{2}(x)=\sqrt{2}\sin 2\pi k_{j}x$
   $\varphi_{1}(x)$	$\frac{1}{2}(\lambda_{k_{i}+k_{j}}+\lambda_{\lvert{}k_{i}-k_{j}\lvert{}})-\lambda_{k_{i}}\lambda_{k_{j}}$	$\frac{1}{2}(\mu_{k_{i}+k_{j}}+\mu_{\lvert{}k_{i}-k_{j}\lvert{}})-\lambda_{k_{i}}\mu_{k_{j}}$
   $\varphi_{2}(x)$	$\frac{1}{2}(\mu_{k_{i}+k_{j}}+\mu_{\lvert{}k_{i}-k_{j}\lvert{}})-\mu_{k_{i}}\lambda_{k_{j}}$	$\frac{1}{2}(\lambda_{k_{i}+k_{j}}+\lambda_{\lvert{}k_{i}-k_{j}\lvert{}})-\mu_{k_{i}}\mu_{k_{j}}$

   Therefore, the multivariate central limit theorem holds with $\Sigma_{ij}$ as entries of the covariance matrix $\Sigma$. A test of independence can be formed based on this limit theorem. Under the hypothesis of independence of observations agains that of a Markov chain with copula $C(u,v)$, all coefficients are equal to $0$. The estimators are asymptotically independent with $\sigma^{2}=1$. We can formulate this in the following proposition.

   Proposition 3.4.

   For any square integrable copula with representation in sine-cosine series, under the assumption that all parameters are equal to zero, any vector of estimators with finitely many components satisfies the $\sqrt{n}$-central limit theorem with variance-covariance matrix $\Sigma=\mathbb{I}_{s}$, where $s$ is the length of the vector and $\mathbb{I}_{s}$ is the identity matrix of dimension $s$. Moreover, if a sequence $W_{n_{i}}$ takes on values of $\hat{\lambda}_{k}$ or $\hat{\mu}_{k}$ for the Markov chain $U_{0},\cdots,U_{n}$, then

   $$n\sum_{i=1}^{s}W_{n_{i}}^{2}\to\chi^{2}(s)\quad\text{as}\quad n\to\infty\quad\text{and}\quad\frac{1}{\sqrt{2s}}(n\sum_{i=1}^{s}W_{n_{i}}^{2}-s)\to N(0,1)$$

   $$\text{as}\quad n\to\infty,\quad\text{then}\quad s\to\infty.$$

2. 2.

   Sine copulas. Considering sine copulas, we have $\displaystyle\sigma^{2}_{k}=1+\frac{\lambda_{2k}}{2}+\frac{\lambda_{k}^{2}\lambda_{2k}}{1-\lambda_{2k}}.$ Estimators are

   $$\hat{\lambda}_{k}=\frac{1}{n}\sum_{i=1}^{n}2\cos\pi kU_{i}\cos\pi kU_{i-1}.$$

   Therefore, Proposition 3.4 holds exactly with the same conditions.

We consider 3 types of copulas with densities of the form (1). A sine copula, a sine-cosine copula and a Legendre copula. The first two present several tops and bottoms, while the second is an extension of the FGM copula family. This third copula can be used to test departure from the FGM copula family in modelling. They are defined as follows.

1. 1.

   Sine copulas
   

   $$C(u,v)=uv+\dfrac{2}{\pi^{2}}\lambda_{1}\sin(\pi u)\sin(\pi v)+\dfrac{1}{2\pi^{2}}\lambda_{2}\sin(2\pi u)\sin(2\pi v),$$

   (9)

   for $\lvert\lambda_{1}\lvert+\lvert\lambda_{2}\lvert\leq 0.5$ and density

   $$c(u,v)=1+2\lambda_{1}\cos(\pi u)\cos(\pi v)+2\lambda_{2}\cos(2\pi u)\cos(2\pi v).$$

2. 2.

   Sine-cosine copulas
   

   	$\displaystyle C(u,v)$	$\displaystyle=$	$\displaystyle uv+\dfrac{1}{2\pi^{2}}(\lambda_{1}\sin(2\pi u)\sin(2\pi v)+\mu_{1}(1-\cos(2\pi u))(1-\cos(2\pi v))$
   			$\displaystyle+\dfrac{1}{8\pi^{2}}(\lambda_{2}\sin(4\pi u)\sin(4\pi v)+\mu_{2}(1-\cos(4\pi u))(1-\cos(4\pi v)),$

   for $\lvert\lambda_{1}\lvert+\lvert\lambda_{2}\lvert+\lvert\mu_{1}\lvert+\lvert\mu_{2}\lvert\leq 0.5$ and density

   	$\displaystyle c(u,v)$	$\displaystyle=$	$\displaystyle 1+2\lambda_{1}\cos(2\pi u)\cos(2\pi v)+2\mu_{1}\sin(2\pi u)\sin(2\pi v)$		(10)
   			$\displaystyle+2\lambda_{2}\cos(4\pi u)\cos(4\pi v)+2\mu_{2}\sin(4\pi u)\sin(4\pi v)).$

3. 3.

   Legendre copulas
   If $\varphi_{k}\in\{\sqrt{3}(2x-1),~{}\sqrt{5}(6x^{2}-6x+1)\}$ and $3|\lambda_{1}|+5|\lambda_{2}|\leq 1$, we obtain the following Legendre copula:

   $$C(u,v)=1+3\lambda_{1}(u^{2}-v)(v^{2}-v)+5\lambda_{2}(2u^{3}-3u^{2}+u)(2v^{3}-3v^{2}+v),$$

   (11)

   with density

   $$c(u,v)=1+3\lambda_{1}(2u-1)(2v-1)+5\lambda_{2}(6u^{2}-6u+1)(6v^{2}-6v+1).$$

   (12)

Each of these copulas is a perturbation of the independence copula and can be used to seek some level of dependence in the data. We generate Markov chains using copulas from these families for known values of parameters. Then, we use them to build confidence intervals, and compute coverage probabilities based on the asymptotic distrubtions of estimators that are proposed.

As shown in Darsow et al. (1992), the transition kernel for a copula-based Markov chain is given by the derivative with respect to the first variable of the copula by $P(x,[0,y])=C_{,1}(x,y)$. Therefore, to generate a realization of a Markov chain $(U_{0},\cdots,U_{n})$ using each of the three considered copulas and the uniform marginal distribution, we use the following algorithm.

1. 1.

   generate $U_{0}\sim Unif(0,1)$;

2. 2.

   $U[1]=U_{0}$;

3. 3.

   For $2\leq i\leq n+1$:

   a). generate $w\sim Unif(0,1)$;

   b). $U[i]$ is the unique root on $[0,1]$ of the equation $C_{,1}(U[i-1],U[i])-w=0$.

For each of the used copulas, the last step uses the $R$ function $Uniroot$. For n=999, we have obtained the following graphs, displaying time series copula 1, copula 2 and copula 3 respectively for sine copula, sine-cosine copula and Legendre copula.

Assume a Markov chain $(U_{0},\cdots,U_{n})$ is generated using a copula of the form (1). The estimator of $\lambda_{k}$ is given by

and $\hat{\Lambda}=(\hat{\lambda}_{1},...,\hat{\lambda}_{s})$, satisfies $\sqrt{n}(\hat{\Lambda}-\Lambda)\longrightarrow{N}\left({\bf 0},\Sigma\right),$ where $\Sigma$ is the variance-covariance matrix defined in the Proposition 4.1 below.

Proposition 4.1 shows that the estimators are asymptotically independent if $\lambda_{1}=0$ or $\lambda_{2}=0.5$ for the sine copula. This is true for the Legendre copula if $\lambda_{1}=0$. These matrices allow us to construct confidence intervals for single copula parameters. Any test of hypotheses on these parameters can use the obtained multivariate central limit theorem.

For the Markov chain $(U_{0},\cdots,U_{n})$ generated by each of the considered copulas and the uniform marginal distribution, the log-likelihood function is:

1. 1.

   For the sine copula:

   $$\ell(\Lambda)=\sum\limits_{i=1}^{n}\ln(1+2\lambda_{1}\cos(\pi U_{i})\cos(\pi U_{i-1})+2\lambda_{2}\cos(2\pi U_{i})\cos(2\pi U_{i-1})),$$

   and the MLE of $\Lambda=(\lambda_{1},\lambda_{2})$ is given by:

   $$\hat{\Lambda}^{ML}=\underset{\lvert\lambda_{1}\lvert+\lvert\lambda_{2}\lvert\leq.5}{Argmax}\ell(\Lambda).$$

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2. 2.

   For the sine-cosine copula:

   	$\displaystyle\ell(\Lambda)=\sum\limits_{i=1}^{n}\ln(1+2\lambda_{1}\cos(2\pi U_{i})\cos(2\pi U_{i-1})+2\mu_{1}\sin(2\pi U_{i})\sin(2\pi U_{i-1})$
   	$\displaystyle+2\lambda_{2}\cos(4\pi U_{i})\cos(4\pi U_{i-1})+2\mu_{2}\sin(4\pi U_{i})\sin(4\pi U_{i-1})),$

   and the MLE of $\Lambda=(\lambda_{1},\lambda_{2},\mu_{1},\mu_{2})$ is given by:

   $$\hat{\Lambda}^{ML}=\underset{\lvert\lambda_{1}\lvert+\lvert\lambda_{2}\lvert+\lvert\mu_{1}\lvert+\lvert\mu_{2}\lvert\leq.5}{Argmax}\ell(\Lambda).$$

   (19)

3. 3.

   For the Legendre copula:

   $$\ell=\sum\limits_{i=1}^{n}\ln(1+3\lambda_{1}(2U_{i}-1)(2U_{i-1}-1)+5\lambda_{2}(6U_{i}^{2}-6U_{i}+1)(6U_{i-1}^{2}-6U_{i-1}+1)),$$

   and the MLE of $\Lambda$ is given by:

   $$\hat{\Lambda}^{ML}=\underset{3\lvert\lambda_{1}\lvert+5\lvert\lambda_{2}\lvert\leq 1}{Argmax}\ell(\Lambda).$$

   (20)

Billingsley (1961) proved asymptotic likelihood theory for the MLE of parameters of a Markov process ($\sqrt{n}(\hat{\Lambda}^{ML}-\Lambda)\to N({\bf 0},\Sigma^{-1})$) under the following regularity conditions.

1. 1.

   The Markov chain is ergodic and not necessarity stationary.

2. 2.

   For $u\in[0,1]$, the set of $v$ for which $c(u,v)>0$ does not depend on $\Lambda$.

3. 3.

   For $1\leq i,j,k\leq s$, where $s$ is the length of the vector $\Lambda$,
   

   $$\frac{\partial}{\partial\lambda_{i}}\ln c(u,v),\frac{\partial^{2}}{\partial\lambda_{i}\lambda_{j}}\ln c(u,v)\text{ and }\frac{\partial^{3}}{\partial\lambda_{i}\lambda_{j}\lambda_{k}}\ln c(u,v)$$

   exist and are continuous on the set of all considered $\Lambda$.

4. 4.

   For $1\leq i,j,k\leq s$, $A^{\prime}$ an open subset of the support of the vector $\Lambda$:
   $\mathbb{E}_{\Lambda}\left[\left(\frac{\partial}{\partial\lambda_{i}}\ln c(u,v)\right)^{2}\right]<\infty$ and $\mathbb{E}_{\Lambda}\left[\underset{\Lambda\in A^{\prime}}{\sup}\lvert\frac{\partial^{3}}{\partial\lambda_{i}\lambda_{j}\lambda_{k}}\ln c(u,v)\lvert\right]<\infty$.

5. 5.

   $$\sigma_{i,j}=\mathbb{E}_{\Lambda}\left[\frac{\partial}{\partial\lambda_{i}}\ln c(u,v)\frac{\partial}{\partial\lambda_{j}}\ln c(u,v)\right]$$

   (21)

   form a non-singular matrix $\Sigma$.