Minimax Hypothesis Testing about the Density Support

Abstract

The paper is concerned with testing nonparametric hypotheses about the underlying support G of independent and identically distributed observations. It is assumed that G belongs to a class G of compact sets with smooth upper surface called boundary fragments. It is required to distinguish the simple null hypothesis speci®ed by a known set G 0 in G against nonparametric alternatives that G belongs to a class obtained by removing a certain neighbourhood of G 0 in G . Using the asymptotic minimax approach, the problem is to determine the order of the smallest distance between the null hypothesis H 0 and the alternatives for which one is able to test the null hypothesis against the alternatives with a given summarized error.

Testing Hypotheses About Contours in Images

“…Other procedures should be designed for this purpose. Some first results in this direction are given in Gayraud, 2001 for the problem of testing hypotheses about the support of a density. These results can be extended to our binary image model with proper modifications.…”

“…Meanwhile, for many purposes, for example in shape analysis, it is interesting not only to estimate but also to test hypotheses about the contours in images. First results on this subject are given by Gayraud, 2001 who considered testing hypotheses about the density support contours. Gayraud, 2001 suggests a test of a simple hypothesis against a nonparametric class of smooth alternatives.…”

“…First results on this subject are given by Gayraud, 2001 who considered testing hypotheses about the density support contours. Gayraud, 2001 suggests a test of a simple hypothesis against a nonparametric class of smooth alternatives. Here we study another problem: goodness-of-fit testing for contours in a binary image model.…”

A change-point problem and inference for segment signals

“…For the design (RD), N G is a random binomial variable, with parameters n and |G|. By conditioning to the design and using Markov's inequality, (5.8) becomes 10) where C = 1 − e − σ 2 8 . In both cases (5.9) and (5.10), the right side does not depend on G, and goes to zero as n → ∞.…”

“…Lower bound. We more or less reproduce the proof of Theorem 3.1 in [10]. Here, the noise is supposed to be zero-mean Gaussian, with variance σ 2 .…”