## The Annals of Statistics

### Sequential confidence bands for densities

#### Abstract

This paper proposes a fully sequential procedure for constructing a fixed-width confidence band for an unknown density on a finite interval and shows the procedure has the desired coverage probability asymptotically as the width of the band approaches zero. The procedure is based on a result of Bickel and Rosenblatt. Its implementation in the sequential setting cannot be obtained using Anscombe's theorem, because the normalized maximal deviations between the kernel estimate and the true density are not uniformly continuous in probability (u.c.i.p.). Instead, we obtain a slightly weaker version of the u.c.i.p. property and a correspondingly stronger convergence property of the stopping rule. These together yield the desired results.

#### Article information

Source
Ann. Statist., Volume 23, Number 6 (1995), 2218-2240.

Dates
First available in Project Euclid: 15 October 2002

https://projecteuclid.org/euclid.aos/1034713654

Digital Object Identifier
doi:10.1214/aos/1034713654

Mathematical Reviews number (MathSciNet)
MR1389872

Zentralblatt MATH identifier
0897.62093

Subjects
Primary: 62L12: Sequential estimation
Secondary: 62G07: Density estimation 62G20: Asymptotic properties

#### Citation

Xu, Yi; Martinsek, Adam T. Sequential confidence bands for densities. Ann. Statist. 23 (1995), no. 6, 2218--2240. doi:10.1214/aos/1034713654. https://projecteuclid.org/euclid.aos/1034713654

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