AISM 52, 459-470

## Nonparametric estimation of a conditional quantile

for *alpha*-mixing processes

### Toshio Honda

Institute of Social Sciences, University of Tsukuba, Tsukuba, Ibaraki 305-8571, Japan

(Received August 5, 1997; revised March 23, 1999)

Abstract.
Let (*X*_{i}', Y_{i})' be a set of observations form
a stationary *alpha*-mixing process and *theta*(*x*) be the conditional *alpha*-th quantile of *Y* given *X*=*x*. Several authors considered nonparametric estimation of *theta*(*x*) in the i.i.d. setting. Assuming the smoothness of *theta*(*x*), we estimate it by local polynomial fitting and prove the asymptotic normality and the uniform convergence.

Key words and phrases:
Conditional quantile, local polynomial fitting, *alpha*-mixing process, exponential inequality, Bahadur representation, uniform convergence.

**Source**
( TeX , DVI )