Electronic Journal of Statistics

Nonparametric estimation of a maximum of quantiles

Georg C. Enss, Benedict Götz, Michael Kohler, Adam Krzyżak, and Roland Platz

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Abstract

A simulation model of a complex system is considered, for which the outcome is described by $m(p,X)$, where $p$ is a parameter of the system, $X$ is a random input of the system and $m$ is a real-valued function. The maximum (with respect to $p$) of the quantiles of $m(p,X)$ is estimated. The quantiles of $m(p,X)$ of a given level are estimated for various values of $p$ from an order statistic of values $m(p_{i},X_{i})$ where $X,X_{1},X_{2},\dots$ are independent and identically distributed and where $p_{i}$ is close to $p$, and the maximal quantile is estimated by the maximum of these quantile estimates. Under assumptions on the smoothness of the function describing the dependency of the values of the quantiles on the parameter $p$ the rate of convergence of this estimate is analyzed. The finite sample size behavior of the estimate is illustrated by simulated data and by applying it in a simulation model of a real mechanical system.

Article information

Source
Electron. J. Statist., Volume 8, Number 2 (2014), 3176-3192.

Dates
First available in Project Euclid: 26 January 2015

Permanent link to this document
https://projecteuclid.org/euclid.ejs/1422281611

Digital Object Identifier
doi:10.1214/14-EJS970

Mathematical Reviews number (MathSciNet)
MR3303681

Zentralblatt MATH identifier
1308.62061

Subjects
Primary: 62G05: Estimation 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 62G30: Order statistics; empirical distribution functions

Keywords
Conditional quantile estimation maximal quantile rate of convergence supremum norm error

Citation

Enss, Georg C.; Götz, Benedict; Kohler, Michael; Krzyżak, Adam; Platz, Roland. Nonparametric estimation of a maximum of quantiles. Electron. J. Statist. 8 (2014), no. 2, 3176--3192. doi:10.1214/14-EJS970. https://projecteuclid.org/euclid.ejs/1422281611


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