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Constructing confidence intervals for quantiles when using variance-reduction techniques

Nakayama, M; Chu, F (New Jersey Institute of Technology)
Monday 21 June 2010, 15:15-15:40

Seminar Room 1, Newton Institute


Consider a random variable $X$ having cumulative distribution function (CDF) $F$. For $0 < p < 1$, the $p$-quantile of $X$ is defined to be $\xi_p = F^{-1}(p) = \inf\{ x : F(x) \geq p \}$. Quantiles are often used to assess risk. For example, a project planner may be interested in the $0.95$-quantile $\xi_{0.95}$ of the time to complete a project, so there is a $95\%$ chance that the project will complete by time $\xi_{0.95}$. In the finance industry, a quantile is known as a value-at-risk (VaR), and VaRs are widely employed as measures of portfolio risk. We develop methods to construct an asymptotically valid confidence interval for $\xi_p$ when applying a variance-reduction technique (VRT). We establish our results within a general framework for VRTs, which encompasses antithetic variates (AV), control variates (CV), and a combination of importance sampling and stratified sampling (IS+SS).

The basic method to construct a point estimator for $\xi_p$ is as follows. Run a simulation applying a VRT, and use the output to construct an estimator $\tilde{F}_n$ of the CDF $F$, where $n$ denotes the computational budget (e.g., sample size). The VRT quantile estimator is then $\tilde{\xi}_{p,n} = \tilde{F}_n^{-1}(p)$. Our framework specifies a set of assumptions on $\tilde{F}_n$, which we show holds for AV, CV, and IS+SS.

To produce a confidence interval for $\xi_p$, we first prove that the VRT quantile estimator satisfies Ghosh's~\cite{Ghos:1971} weaker form of a Bahadur representation~\cite{Baha:1966}, which implies $\tilde{\xi}_{p,n}$ obeys a central limit theorem (CLT). The variance constant $\kappa_p^2$ in this CLT can be expressed as $\kappa_p = \psi_p \phi_p$, where $\psi_p$ depends on the VRT applied but $\phi_p = 1/f(\xi_p)$ is independent of the estimation technique. It turns out that $\psi_p^2$ is the variance constant appearing in the CLT for $\tilde{F}_n(\xi_p)$, and defining a consistent estimator of $\psi_p$ is straightforward. The main issue is providing a consistent estimator of $\phi_p$ when applying a VRT, and we derive such an estimator using the Bahadur representation.

  • [1] R.R. Bahadur. A note on quantiles in large samples. Annals of Mathematical Statistics, 37:577--580, 1966.
  • [2] J.K. Ghosh. A new proof of the Bahadur representation of quantiles and an application. Annals of Mathematical Statistics, 42:1957--1961, 1971.


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