Minimum density power divergence estimator for covariance matrix based on skew $t$ distribution

Abstract

In this paper, we study the problem of estimating the covariance matrix of stationary multivariate time series based on the minimum density power divergence method that uses a multivariate skew $t$ distribution family. It is shown that under regularity conditions, the proposed estimator is strongly consistent and asymptotically normal. A simulation study is provided for illustration.

Appendix

We provide the proofs of Theorems 3.1 and 3.2 for the case of $\lambda >0$. Since the proofs are similar to those presented in Kim and Lee (2011, 2013) except for Lemma 6.1, we only provide the proof for Lemma 6.1. The proofs of Lemmas 6.2–6.5 and Theorem 3.1 are detailed in the supplementary material.

The following two lemmas are needed to prove Theorem 3.1.

Lemma 6.1

For all $\theta \in \Theta$ and $\varvec{x}\in \mathbb {R}^d$, there exist constants $l_{\lambda ,0},\ldots ,l_{\lambda ,3}$ such that $|V_\lambda (\theta ;\varvec{x})|\le l_{\lambda ,0}$ and for $1\le i_1,\ldots ,i_r\le \rho ~(r=1,2,3)$,

$$\begin{aligned} \left| \frac{\partial ^rV_\lambda (\theta ;\varvec{x})}{\partial \theta _{i_1}\cdots \partial \theta _{i_r}}\right| \le l_{\lambda ,r}. \end{aligned}$$

Proof

Note that

$$\begin{aligned} |V_\lambda (\theta ;\varvec{x})|&= \left| \int f_\theta ^{1+\lambda }(\varvec{z})d\varvec{z}-(1+\frac{1}{\lambda })f_\theta ^\lambda (\varvec{x})\right| \\&\le \int f_\theta ^{1+\lambda }(\varvec{z})d\varvec{z}+(1+\frac{1}{\lambda })f_\theta ^\lambda (\varvec{x})\\ {}&:= W_{\lambda ,0}(\theta )+Q_{\lambda ,0}(\theta ;\varvec{x}). \end{aligned}$$

Let $w_{\lambda ,0}:=\sup _{\theta \in \Theta }W_{\lambda ,0}(\theta )$ and $q_{\lambda ,0}(\varvec{x}):=\sup _{\theta \in \Theta }Q_{\lambda ,0}(\theta ;\varvec{x})$. Since $\Theta$ is compact and $W_{\lambda ,0}(\theta )$ and $Q_{\lambda ,0}(\theta ;\varvec{x})$ are continuous in $\theta \in \Theta , W_{\lambda ,0}(\theta )\le w_{\lambda ,0}<\infty ~\text{ and }~Q_{\lambda ,0}(\theta ;\varvec{x})\le q_{\lambda ,0}(\varvec{x})<\infty$ for all $\varvec{x}\in \mathbb {R}^d$. Further, $q_{\lambda ,0}(\varvec{x})=O\left( (\varvec{x}'\varvec{x})^{-\lambda (\nu +d)/2}\right)$, so that $q_{\lambda ,0}(\varvec{x})<q'_{\lambda ,0}$ for some constant $q'_{\lambda ,0}$. Let $l_{\lambda ,0}:=w_{\lambda ,0}+q'_{\lambda ,0}$. Then, $|V_{\lambda }(\theta ;\varvec{x})|\le l_{\lambda ,0}$.

Now, we consider the case of $r=1$. Note that

$$\begin{aligned} \left| \frac{\partial V_\lambda (\theta ;\varvec{x})}{\partial \alpha _i}\right|&= \left| \int \frac{\partial f_\theta ^{1+\lambda }(\varvec{z})}{\partial \alpha _i}d\varvec{z}-\left( 1+\frac{1}{\lambda }\right) \frac{\partial f_\theta ^{\lambda }(\varvec{x})}{\partial \alpha _i}\right| \\ {}&\le (1+\lambda )\left\{ \left| \int f_\theta ^\lambda (\varvec{z})\frac{\partial f_\theta (\varvec{z})}{\partial \alpha _i}d\varvec{z}\right| +\left| f_\theta ^\lambda (\varvec{x})\left( \frac{\partial f_\theta (\varvec{x})}{\partial \alpha _i}\right) /f_\theta (\varvec{x}) \right| \right\} \\ {}&:= (1+\lambda )\left\{ W_{\lambda ,1,1}(\theta )+Q_{\lambda ,1,1}(\theta ;\varvec{x}) \right\} ~\text{ for }~1\le i \le d. \end{aligned}$$

In the same way, we have

$$\begin{aligned} \left| \frac{\partial V_\lambda (\theta ;\varvec{x})}{\partial \omega _{ij}}\right|&\le (1+\lambda )\left\{ W_{\lambda ,1,2}(\theta )+Q_{\lambda ,1,2}(\theta ;\varvec{x}) \right\} ~\text{ for }~2\le i \le d,~1\le j\le i-1,\\\left| \frac{\partial V_\lambda (\theta ;\varvec{x})}{\partial \omega _{ii}}\right|&\le (1+\lambda )\left\{ W_{\lambda ,1,3}(\theta )+Q_{\lambda ,1,3}(\theta ;\varvec{x}) \right\} ~\text{ for }~1\le i \le d,\\\left| \frac{\partial V_\lambda (\theta ;\varvec{x})}{\partial \nu }\right|&\le (1+\lambda )\left\{ W_{\lambda ,1,4}(\theta )+Q_{\lambda ,1,4}(\theta ;\varvec{x}) \right\} . \end{aligned}$$

Since the exact form of $\partial f_\theta (\varvec{x})/\partial \theta _i$ is complicated, we do not provide the details here. However, it can be easily obtained. Let $w_{\lambda ,1,k}=\sup _{\theta \in \Theta }W_{\lambda ,1,k}(\theta )~\text{ and }~q_{\lambda ,1,k}(\varvec{x})=\sup _{\theta \in \Theta }Q_{\lambda ,1,k}(\theta ;\varvec{x}),~k=1,2,3,4.$ Since $\Theta$ is compact and $W_{\lambda ,1,k}(\theta )$ and $Q_{\lambda ,1,k}(\theta ;\varvec{x})$ are continuous in $\theta \in \Theta , W_{\lambda ,1,k}(\theta )\le w_{\lambda ,1,k}<\infty ~\text{ and }~Q_{\lambda ,1,k}(\theta ;\varvec{x})\le q_{\lambda ,1,k}(\varvec{x})<\infty$ for all $\varvec{x}\in \mathbb {R}^d, k=1,2,3,4$. Further, $q_{\lambda ,1,1}(\varvec{x})=q_{\lambda ,1,2}(\varvec{x})=q_{\lambda ,1,3}(\varvec{x})=O\left( (\varvec{x}'\varvec{x})^{-\lambda (\nu +d)/2}\right)$ and $q_{\lambda ,1,4}(\varvec{x})=O\left( (\varvec{x}'\varvec{x})^{-\lambda (\nu +d)/2}\ln (\varvec{x}'\varvec{x})\right)$, so that $q_{\lambda ,1,k}(\varvec{x})\le q'_{\lambda ,1,k}$ for some constants $q'_{\lambda ,1,k}, k=1,2,3,4$. Let $l_{\lambda ,1}=(1+\lambda )\max \{w_{\lambda ,1,k}+q'_{\lambda ,1,k},~k=1,2,3,4\}$. Then, $|{\partial V_{\lambda }(\theta ;\varvec{x})}/{\partial \theta _i}|\le l_{\lambda ,1}~\text{ for }~1\le i\le \rho$. The cases of $r=2,3$ can be derived similarly. This establishes the lemma. $\square$

The proofs of the following lemma and Theorem 3.1 are presented in the supplementary material.

Lemma 6.2

Let $\varvec{X}_1,\varvec{X}_2,\ldots$ be strictly stationary and ergodic. If

• (a) $\Theta$ is compact;

• (b) $A(\varvec{x},\theta )$ is continuous in $\theta$ for all $\varvec{x}$;

• (c) There exists $B(\varvec{x})$ such that $EB(\varvec{X})<\infty$ and $|A(\varvec{x},\theta )|\le B(\varvec{x})$ for all $\varvec{x}$ and $\theta$,

then

$$\begin{aligned} P\left\{ \lim _{n\rightarrow \infty }\sup _{\theta \in \Theta }\left| \frac{1}{n}\sum _{t=1}^{n}A(\varvec{X}_t,\theta )-a(\theta ) \right| =0 \right\} =1, \end{aligned}$$

(6.1)

where $a(\theta )=EA(\varvec{X},\theta )$.

In addition, if there exists $\theta _0={{\mathrm{argmin}}}_{\theta \in \Theta }a(\theta )$ and it is unique, then

$$\begin{aligned} P\{\hat{\theta }_n\rightarrow \theta _0,~n\rightarrow \infty \}=1, \end{aligned}$$

(6.2)

where $\hat{\theta }_n={{\mathrm{argmin}}}_{\theta \in \Theta }\frac{1}{n}\sum _{t=1}^{n}A(\varvec{X}_t,\theta )$.

Theorem 3.2 is a direct result of the next three lemmas whose proofs are detailed in the supplementary material.

Lemma 6.3

Under the conditions in Theorem 3.2,

$$\begin{aligned} \sqrt{n}U_{\lambda ,n}(\theta _\lambda )\mathop {\rightarrow }\limits ^{d}N(\mathbf{0},K_\lambda )~\text{ as }~n\rightarrow \infty . \end{aligned}$$

Lemma 6.4

Under the conditions in Theorem 3.2, $R_{\lambda ,n}$ converges to $J_\lambda$ a.s..

Lemma 6.5

Under the conditions in Theorem 3.2, $\sqrt{n}\Vert \Delta _{\lambda ,n}\Vert =o_p(1)$.