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CENTRAL LIMIT THEORMS IN C[0,1] FOR A CLASS OF ESTIMATORS OF A DISTRIBUTION FUNCTION
Author(s) -
Nixdorf R.
Publication year - 1985
Publication title -
statistica neerlandica
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.52
H-Index - 39
eISSN - 1467-9574
pISSN - 0039-0402
DOI - 10.1111/j.1467-9574.1985.tb01143.x
Subject(s) - mathematics , combinatorics , sequence (biology) , limit (mathematics) , zero (linguistics) , distribution (mathematics) , brownian bridge , estimator , function (biology) , distribution function , weak convergence , mathematical analysis , brownian motion , physics , statistics , thermodynamics , linguistics , philosophy , genetics , computer security , evolutionary biology , computer science , asset (computer security) , biology
As non–parametric estimates of an unknown distribution function (d.f.) F based on i.i.d. observations X 1 X n with this d.f. are used, where H n is a sequence of d.f.'s converging weakly to the unit mass at zero. Under regularity conditions on F and the sequence ( H n ) it is shown that √ n ( F n – F ) and √ n ( R n – F ) in C [0,1] converge in distribution to a process G with G( t ) = W° ( F ( t )), where W ° is a Brownian bridge in C [0,1]. Further the a.s. uniform convergence of R., is considered and some examples are given.