Asymptotics for the linear kernel quantile estimator

Abstract

The method of linear kernel quantile estimator was proposed by Parzen (J Am Stat Assoc 74:105–121, 1979), which is a reasonable estimator for Value-at-risk (VaR). In this paper, we mainly investigate the asymptotic properties for linear kernel quantile estimator of VaR based on \(\varphi \)-mixing samples. At first, the Bahadur representation for sample quantiles under \(\varphi \)-mixing sequence is established. By using the Bahadur representation for sample quantiles, we further obtain the Bahadur representation for linear kernel quantile estimator of VaR in sense of almost surely convergence with the rate \(O\left( n^{-1/2}\log ^{-\alpha }n\right) \) for some \(\alpha >0\). In addition, the strong consistency for the linear kernel quantile estimator of VaR with the convergence rate \(O\left( n^{-1/2}(\log \log n)^{1/2}\right) \) is established, and the asymptotic normality for linear kernel quantile estimator of VaR based on \(\varphi \)-mixing samples is obtained. Finally, a simulation study and a real data analysis are undertaken to assess the finite sample performance of the results that we established.

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Acknowledgements

The authors are most grateful to the Editor and anonymous referees for carefully reading the manuscript and for valuable suggestions which helped in improving an earlier version of this paper.

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Correspondence to Xuejun Wang.

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Supported by the National Natural Science Foundation of China (11671012, 11871072, 11701004, 11701005), the Natural Science Foundation of Anhui Province (1508085J06), the Key Projects for Academic Talent of Anhui Province (gxbjZD2016005) and the Project on Reserve Candidates for Academic and Technical Leaders of Anhui Province (2017H123).

Appendix A

Appendix A

Lemma A.1

Assume that K(st) is a Kiefer process with covariance function \(\min (t,t')\varGamma (s,s')\). If \(p\in (0,1)\), then

$$\begin{aligned} K(p, n)=O\left( n^{1/2}(\log \log n)^{1/2}\right) ~~\mathrm{a.s.}, \end{aligned}$$

and there exists a \(\alpha >0\) such that

$$\begin{aligned} \sup _{|s-p|<\delta _n}|K(s, n)-K(p, n)|=O\left( n^{1/2}\log ^{-\alpha }n\right) ~~\mathrm{a.s.}, \end{aligned}$$

where \(\delta _n=n^{-1/4}(\log \log n)^{1/2}\).

This lemma follows from Lemmas 3.1 and 3.5 of Wei et al. (2010).

Lemma A.2

Suppose K(u) and g(x) are Borel measurable functions defined on \(\mathbb {R}\) and satisfy conditions: (1) K(u) is bounded; (2) \(\int _{-\infty }^\infty |K(u)|\mathrm{d}u<\infty \); (3) \(\lim _{|u|\rightarrow \infty }u|K(u)|=0\); (4) \(\int _{-\infty }^\infty |g(x)|\mathrm{d}x<\infty \). Denote \(g_n(x)=\frac{1}{h}\int _{-\infty }^\infty K\left( \frac{u-x}{h}\right) g(u)\mathrm{d}u\), where h is constant, and \(h\rightarrow 0\) as \(n\rightarrow \infty \). Then at every point x of continuity of g(x), we have

$$\begin{aligned} \lim _{n\rightarrow \infty }g_n(x)=g(x)\int _{-\infty }^\infty K(u) \mathrm{d}u. \end{aligned}$$

This lemma comes from Parzen (1962).

Lemma A.3

Let \(\{X_n,n\ge 1\}\) be a sequence of stationary \(\varphi \)-mixing random variables with common continuous distribution function F. Assume that F satisfies the Csörgö–Révész conditions, then there exists a Kiefer process K(sn) defined on the same probability space as \(\rho _n(x)\) with covariance function \(\min (n,n')\varGamma (s,s')\) and a constant \(\alpha >0\) such that

$$\begin{aligned} \sup _{\lambda _n\le s\le 1-\lambda _n}|\rho _n(s)-K(s,n)n^{-1/2}|=O\left( \log ^{-\alpha }n\right) ~~\mathrm{a.s.}, \end{aligned}$$

where \(\lambda _n=n^{-1/2}(\log n)^\alpha (\log \log n)\).

The proof of Lemma A.3 is similar to that of Lemma A.2 of Wei et al. (2010).

Lemma A.4

Let \(\{X_n,n\ge 1\}\) be a stationary \(\varphi \)-mixing sequence of random variables with \(EX_1=0\) and \(E|X_1|^r<\infty \) for some \(r>2\). Denote \(S_n=\sum _{t=1}^nX_t\). Further assume that the mixing coefficient satisfies \(\varphi (n)=O\left( n^{-\frac{r}{r-2}-\delta }\right) \) for some \(\delta >0\), and the series \(EX_1^2+2\sum _{i=2}^\infty EX_1X_i\) converges to a constant \(\sigma ^2\ge 0\). Then

$$\begin{aligned} \limsup _{n\rightarrow \infty }\frac{|S_n|}{\sqrt{2n\log \log n}}=1~~\mathrm{a.s.} \end{aligned}$$
(5.1)

The proof of the lemma is similar to that of Theorem 4.1 of Shao (1990).

Lemma A.5

Suppose that \(\{X_n, n\ge 1\}\) is a second-order stationary \(\varphi \)-mixing sequence with common marginal distribution function and \(EX_n=0\), \(|X_n|\le d<\infty \), \(n=1,2,\ldots \). Assume that \(\sum \nolimits _{n=1}^\infty \varphi ^{1/2}(n)<\infty \). If \(\mathrm{Var} (X_1)+2\sum \nolimits _{j=2}^\infty \mathrm{Cov}(X_1,X_j)=\sigma _0^2>0\), then

$$\begin{aligned} \sup \limits _{-\infty<t<\infty }\left| P\left( \frac{\sum \nolimits _{i=1}^n X_i}{\sqrt{n}\sigma _0}\le t\right) -\Phi (t)\right| \le C(\sigma _0^2)n^{-1/6}\cdot \log n\cdot \log \log n, \end{aligned}$$
(5.2)

where \(C(\sigma _0^2)\) is a positive constant depending only on \(\sigma _0^2\).

This lemma comes from Yang et al. (2014).

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Wang, X., Wu, Y., Yu, W. et al. Asymptotics for the linear kernel quantile estimator. TEST 28, 1144–1174 (2019). https://doi.org/10.1007/s11749-019-00627-9

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Keywords

  • Bahadur representation
  • Linear kernel quantile estimator
  • Value-at-risk
  • Strong consistency
  • Asymptotic normality

Mathematics Subject Classification

  • 62G30
  • 62G20
  • 62G05