Nonparametric estimation of expected shortfall for α-mixing financial losses

Abstract

In this paper, we investigate the Bahadur type representation of a nonparametric expected shortfall estimator for $\alpha$-mixing financial losses. Based on the Bahadur type representation, we further establish the Berry–Esseen bound of the nonparametric expected shortfall estimator. It is shown that the optimal rate can achieve nearly $O(n^{-1/8})$ under some appropriate conditions. We also carry out some numerical simulations and a real data example to support our theoretical results based on finite samples.

Appendix A

To prove our main results, the following lemmas are indispensable. The first one can be found in Roussas and Ioannides (1987).

Lemma A.1

Suppose that $\xi$ and $\eta$ are $\mathcal {F}_{1}^{k}$-measurable and $\mathcal {F}_{k+n}^{\infty }$-measurable random variables, respectively. If $E|\xi |^{p}<\infty$, $E|\eta |^{q}<\infty$ for some $p,q,t>1$ with $p^{-1}+q^{-1}+t^{-1}=1$. Then

$$\begin{aligned} |E\xi \eta -E\xi E\eta |\le 10\alpha ^{1/t}(n)\Vert \xi \Vert _{p}\Vert \eta \Vert _{q}. \end{aligned}$$

The next lemma is Theorem 2.1 of Yang (2007).

Lemma A.2

Let $r>2,\eta >0$, and $\{e_{i},i\ge 1\}$ be a sequence of $\alpha$-mixing random variables with $Ee_{i}=0$ and $E|e_{i}|^{r+\eta }<\infty$ for each $i\ge 1$. Suppose that $\alpha (n)=O(n^{-\lambda })$ for some $\lambda >r(r+\eta )/(2\eta )$. Then for any $\epsilon >0$, there exists a positive constant C depending only on $\varepsilon ,r,\eta$, and $\lambda$ such that for each $n\ge 1$,

$$\begin{aligned} E\left( \max _{1\le m\le n}\left| \sum _{i=1}^{m}e_{i}\right| ^{r}\right) \le C\left\{ n^{\epsilon }\sum _{i=1}^{n}E|e_{i}|^{r}+\left( \sum _{i=1}^{n}\Vert e_{i}\Vert _{r+\eta }^{2}\right) ^{r/2}\right\} . \end{aligned}$$

The following lemma comes from Yang (2000).

Lemma A.3

Let $\{a_{i},i\ge 1\}$ be a real sequence and $\{Y_{i},i\ge 1\}$ be a sequence of zero mean $\alpha$-mixing random variables with $E|Y_{i}|^{2+\delta }<\infty$ for some $\delta >0$. Then for all $n\ge 1$,

$$\begin{aligned} E\left( \sum _{i=1}^{n}a_{i}Y_{i}\right) ^{2}\le \left( 1+20\sum _{m=1}^{n}\alpha ^{\delta /(2+\delta )}(m)\right) \sum _{i=1}^{n}a_{i}^{2}\Vert Y_{i}\Vert _{2+\delta }^{2}. \end{aligned}$$

The following lemma can be seen in Yang and Li (2006).

Lemma A.4

Let p and q be positive integers and $\{e_{i},i\ge 1\}$ be a sequence of $\alpha$-mixing random variables. Set $\eta _{l}=:\sum _{j=(l-1)(p+q)+1}^{(l-1)(p+q)+p}e_{j}, 1\le l\le k$. Assume that $r>0$ and $s>0$ such that $1/r+1/s=1$. Then there exists some positive constant C such that for any $t\in \mathbb {R}$,

$$\begin{aligned} \left| E\exp \left\{ it\sum _{l=1}^{k}\eta _{l}\right\} -\prod _{l=1}^{k}E\exp \{it\eta _{l}\}\right| \le C|t|\alpha ^{1/s}(q)\sum _{l=1}^{k}\Vert \eta _{l}\Vert _{r}. \end{aligned}$$

The last one can be found in Yang (2003) for instance.

Lemma A.5

Suppose that $\{\zeta _{n},n\ge 1\}$ and $\{\eta _{n},n\ge 1\}$ are two sequences of random variables, $\{\gamma _{n},n\ge 1\}$ is a positive constant sequence with $\gamma _{n}\rightarrow 0$ as $n\rightarrow \infty$. If

$$\begin{aligned} \sup _{-\infty<u<\infty }|F_{\zeta _{n}}(u)-\Phi (u)|\le C\gamma _{n}, \end{aligned}$$

the for any $\varepsilon >0$,

$$\begin{aligned} \sup _{-\infty<u<\infty }|F_{\zeta _{n}+\eta _{n}}(u)-\Phi (u)|\le C[\gamma _{n}+\varepsilon +P(|\eta _{n}|>\varepsilon )]. \end{aligned}$$