## The Annals of Statistics

### New goodness-of-fit tests and their application to nonparametric confidence sets

Lutz Dümbgen

#### Abstract

Suppose one observes a process V on the unit interval, where $dV = f_o + dW$ with an unknown parameter $f_o \epsilon L_1[0, 1]$ and standard Brownian motion W. We propose a particular test of one-point hypotheses about $f_o$ which is based on suitably standardized increments of V. This test is shown to have desirable consistency properties if, for instance, $f_o$ is restricted to various Hölder classes of functions. The test is mimicked in the context of nonparametric density estimation, nonparametric regression and interval-censored data. Under shape restrictions on the parameter, such as monotonicity or convexity, we obtain confidence sets for $f_o$ adapting to its unknown smoothness.

#### Article information

Source
Ann. Statist., Volume 26, Number 1 (1998), 288-314.

Dates
First available in Project Euclid: 28 August 2002

Permanent link to this document
https://projecteuclid.org/euclid.aos/1030563987

Digital Object Identifier
doi:10.1214/aos/1030563987

Mathematical Reviews number (MathSciNet)
MR1611768

Zentralblatt MATH identifier
0930.62034

Subjects
Primary: 62G07: Density estimation 62G15: Tolerance and confidence regions

#### Citation

Dümbgen, Lutz. New goodness-of-fit tests and their application to nonparametric confidence sets. Ann. Statist. 26 (1998), no. 1, 288--314. doi:10.1214/aos/1030563987. https://projecteuclid.org/euclid.aos/1030563987

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