The Annals of Statistics

On Some Global Measures of the Deviations of Density Function Estimates

P. J. Bickel and M. Rosenblatt

Full-text: Open access


We consider density estimates of the usual type generated by a weight function. Limt theorems are obtained for the maximum of the normalized deviation of the estimate from its expected value, and for quadratic norms of the same quantity. Using these results we study the behavior of tests of goodness-of-fit and confidence regions based on these statistics. In particular, we obtain a procedure which uniformly improves the chi-square goodness-of-fit test when the number of observations and cells is large and yet remains insensitive to the estimation of nuisance parameters. A new limit theorem for the maximum absolute value of a type of nonstationary Gaussian process is also proved.

Article information

Ann. Statist., Volume 1, Number 6 (1973), 1071-1095.

First available in Project Euclid: 12 April 2007

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Density function estimates weight function bandwidth global measure of deviation asymptotic distribution functionals of stochastic process Gaussian process


Bickel, P. J.; Rosenblatt, M. On Some Global Measures of the Deviations of Density Function Estimates. Ann. Statist. 1 (1973), no. 6, 1071--1095. doi:10.1214/aos/1176342558.

Export citation


  • See Correction: P. J. Bickel, M. Rosenblatt. Notes: Corrections to "On Some Global Measures of the Deviations of Density Function Estimates". Ann. Statist., Volume 3, Number 6 (1975), 1370--1370.