On the rate of convergence of an empirical process constructed from a sample of random size. (English. Russian original) Zbl 0788.62042

Mirzakhmedov, M. A.; Tursunov, T. G.

Theory Probab. Math. Stat. 44, 89-93 (1992); translation from Teor. Veroyatn. Mat. Stat., Kiev 44, 92-96 (1991).

Summary: Suppose \(\{\xi_ n,\;n\geq 1\}\) is a sequence of independent, identically distributed random variables with continuous distribution function \(F(x)\), \(\{N_ n,\;n\geq 1\}\) is a sequence of nonnegative integer random variables. An estimate of the rate of convergence of the empirical process \(\zeta_{N_ n}(x)= \sqrt{N_ n}(F_{N_ n}(x)- F(x))\) to the corresponding Gaussian process is obtained, where \(F_{N_ n}(x)\) is the empirical distribution function of the sample \((\xi_ 1,\dots,\xi_{N_ n})\).

MSC:

62G30	Order statistics; empirical distribution functions
62G20	Asymptotic properties of nonparametric inference
62F12	Asymptotic properties of parametric estimators

Keywords:

rate of convergence; empirical process; empirical distribution; Gaussian process