Limit Theorems and Inequalities for the Uniform Empirical Process Indexed by Intervals

The uniform empirical process U,, is considered as a process indexed by intervals. Powerful new exponential bounds are established for the process indexed by both "points" and intervals. These bounds trivialize the proof of the Chibisov-O'Reilly theorem concerning the convergence of the process with respect to 11 ./q 11-metrics and are used to prove an interval analogue of the Chibisov-O'Reilly theorem. A strong limit theorem related to the well-known Holder condition for Brownian bridge U is also proved. Connections with related work of Csdki, Eicker, Jaeschke, and Stute are mentioned. As an application we introduce a new statistic for testing uniformity which is the natural interval analogue of the classical Anderson-Darling statistic.

[1]  H. McKean,et al.  Diffusion processes and their sample paths , 1996 .

[2]  Winfried Stute,et al.  THE OSCILLATION BEHAVIOR OF EMPIRICAL PROCESSES , 1982 .

[3]  G. Shorack.  SOME LAW OF THE ITERATED LOGARITHM TYPE RESULTS FOR THE EMPIRICAL PROCESS , 1980 .

[4]  Pál Révész,et al.  How Big are the Increments of a Wiener Process , 1979 .

[5]  G. Shorack.  Weak convergence of empirical and quantile processes in sup-norm metrics via kmt-constructions , 1979 .

[6]  J. Wellner,et al.  Linear Bounds on the Empirical Distribution Function , 1978 .

[7]  J. Wellner.  Limit theorems for the ratio of the empirical distribution function to the true distribution function , 1978 .

[8]  W. Philipp.  A Functional Law of the Iterated Logarithm for Empirical Distribution Functions of Weakly Dependent Random Variables , 1977 .

[9]  E. Csáki.  The law of the iterated logarithm for normalized empirical distribution function , 1977 .

[10]  Barry R. James.  A Functional Law of the Iterated Logarithm for Weighted Empirical Distributions , 1975 .

[11]  N. O'Reilly.  On the Weak Convergence of Empirical Processes in Sup-norm Metrics , 1974 .

[12]  M. J. Wichura,et al.  Some Strassen-Type Laws of the Iterated Logarithm for Multiparameter Stochastic Processes with Independent Increments , 1973 .

[13]  W. Hoeffding.  Probability inequalities for sum of bounded random variables , 1963 .

[14]  G. Bennett.  Probability Inequalities for the Sum of Independent Random Variables , 1962 .

[15]  G. S. Watson,et al.  Goodness-of-fit tests on a circle. II , 1961 .

[16]  Paul Erdös,et al.  On the Lipschitz's condition for Brownian motion , 1959 .

[17]  J. Cassels.  An extension of the law of the iterated logarithm , 1951, Mathematical Proceedings of the Cambridge Philosophical Society.

[18]  F. Eicker.  The Asymptotic Distribution of the Suprema of the Standardized Empirical Processes , 1979 .

[19]  D. Jaeschke.  The Asymptotic Distribution of the Supremum of the Standardized Empirical Distribution Function on Subintervals , 1979 .

[20]  J. Durbin.  Distribution theory for tests based on the sample distribution function , 1973 .

[21]  Geoffrey S. Watson,et al.  Distribution Theory for Tests Based on the Sample Distribution Function , 1973 .

[22]  A. Skorokhod.  Limit Theorems for Stochastic Processes , 1956 .