Conditions for quantile process approximations

• Published in: Communications in statistics. Stochastic models Vol. 15; no. 3; pp. 485 - 502
• Main Authors: Drees, Holger, Haan, Laurens de
• Format: Journal Article
• Language: English
• Published: Monticello, NY Marcel Dekker, Inc 01.01.1999; Dekker
• Subjects: AMS Subject classification: Primary 60F15, 60F17; secondary 62G30; approximation; Chibisov-O'reilly weight function; Csörgó- Révész condition; Exact sciences and technology; extended regular variation; Limit theorems; Mathematics; Nonparametric inference; O-regular variation; Probability and statistics; Probability theory and stochastic processes; quantile process; Sciences and techniques of general use; Statistics; tail quantile process; Density estimation; Almost sure convergence; Tail probability; Probability theory; Stochastic approximation; Quantile; Limit theorem
• ISSN: 0882-0287
• DOI: 10.1080/15326349908807546

Csörgo[dacute]and Révész (1978) introduced a condition on the density of a distribution function that is sufficient to obtain weighted approximations for the pertaining normalized quantile process. We prove that this condition implies the extended regular variation of the density quantile function and that therefore it is substantially stronger than another sufficient condition due to Shorack, which is implied by -regular variation. The relationship between these conditions is clarified by introducing a new Csörgó- Révész type condition that is equivalent to -regular variation. Then we show that the Csörgo[dacute]- Révész condition is sufficient to establish stochastic and almost sure approximations of the tail quantile function, which were proven in previous papers under the stronger assumption that the density quantile function is regularly varying