Abstract
Quantiles, which are also known as values-at-risk in finance, frequently arise in practice as measures of risk. Confidence interval for quantiles are typically constructed via large sample theory or the sectioning. One of the ways for achieving the confidence interval for quantiles is direct use of a central limit theorem. In this approach, we require a good estimator of the quantile density function. In this paper, we consider the nonparametric estimator of the quantile density function from Soni et al. (2012) and we obtain confidence interval for quantiles. In the following, by using simulation, the coverage probability and mean square error of this confidence interval is calculated. Also, we compare our proposed approach with alternative approaches such as sectioning and jackknife.
Highlights
The quantile function Q F 1associated with a distribution function F defined asQ (u ) F 1(u ) inf{x ;F (x ) u}, for 0 u 1, is sometimes the object of more direct interest than the F itself
The approach consists of three steps: a point estimate of the cumulative distribution function, a confidence Interval (CI) for the point estimate, and converting it into CI for the quantile
As the sample size increases to 100 (Tables 6-10) or 200 (Tables 11-15), results of nonparametric method improve in center and tails, such that for u 0.05,0.95 in most cases the nonparametric method outperforms the two other methods
Summary
The quantile function Q F 1associated with a distribution function F defined as. Q (u ) F 1(u ) inf{x ;F (x ) u}, for 0 u 1, is sometimes the object of more direct interest than the F itself. We try to improve it by using nonparametric estimator of the quantile density function. The second attempt is to use sectioning or batching to calculate both a point estimate of the quantile and an. The last attempt is to apply a jackknife approach to reducing the bias of previously method (based on sectioning). We consider the first approach and by using nonparametric estimator of the quantile density function introduced by Soni et al (2012) construct a CI for quantile. The layout of the paper is as follows: In Section 2, we express three methods to achieve CI for quantile. Confidence Interval Attempts In the following we will review three attempts to achieve CI for quantiles
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