학술논문
Nonparametric estimation of a maximum of quantiles.
Document Type
Journal
Author
Enss, Georg C. (D-DARM-SZM) AMS Author Profile; Götz, Benedict (D-DARM-SZM) AMS Author Profile; Kohler, Michael (D-DARM) AMS Author Profile; Krzyżak, Adam (3-CONC-CSW) AMS Author Profile; Platz, Roland (D-FISDS) AMS Author Profile
Source
Subject
60 Probability theory and stochastic processes -- 60K Special processes
60K35Interacting random processes; statistical mechanics type models; percolation theory
62Statistics -- 62G Nonparametric inference
62G30Order statistics; empirical distribution functions
60K35
62
62G30
Language
English
Abstract
Summary: ``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.''