Some questions of nonparametric statistics of weakly dependent observations. (English. Russian original) Zbl 0748.62023

Bondarev, B. V.

Theory Probab. Math. Stat. 43, 21-27 (1991); translation from Teor. Veroyatn. Mat. Stat., Kiev 43, 19-26 (1990).

Summary: The author establishes exponential estimates of the probability of exiting from a level by the variables \(w^ 2_ n\) and \[ T_ n=\int^{+\infty}_{-\infty}| f_ n(x)-f(x)|^ 2a_ 0(x)dx \] (\(\omega^ 2_ n\) are statistics of the \(\omega^ 2\)-test, \(f_ n(x)\) is a nonparametric estimate of the unknown density of the stationary distribution of a sequence of observations) under the assumption that the sequence of observations \(\{x_ 1,x_ 2,\dots,x_ n\}\) forms a stationary random process in the strict sense, satisfying a strong mixing condition (s.m.c.) with coefficient \(\alpha(C_ m)\).

MSC:

62G07	Density estimation
62M09	Non-Markovian processes: estimation
60G10	Stationary stochastic processes
62G30	Order statistics; empirical distribution functions

Keywords:

exit probability; weakly dependent observations; Rosenblatt-Parzen statistic; exponential estimates; stationary distribution; stationary random process; strong mixing condition