Finite-Sample Confidence Envelopes for Shape-Restricted Densities

Abstract

A conservative finite-sample simultaneous confidence envelope for a density can be found by solving a finite set of finite-dimensional linear programming problems if the density is known to be monotonic or to have at most $k$ modes relative to a positive weight function. The dimension of the problems is at most $(n/\log n)^{1/3}$, where $n$ is the number of observations. The linear programs find densities attaining the largest and smallest values at a point among cumulative distribution functions in a confidence set defined using the assumed shape restriction and differences between the empirical cumulative distribution function evaluated at a subset of the observed points. Bounds at any finite set of points can be extrapolated conservatively using the shape restriction. The optima are attained by densities piecewise proportional to the weight function with discontinuities at a subset of the observations and at most five other points. If the weight function is constant and the density satisfies a local Lipschitz condition with exponent $\varrho$, the width of the bounds converges to zero at the optimal rate $(\log n/n)^{\varrho/(1+2\varrho)}$ outside every neighborhood of the set of modes, if a "bandwidth" parameter is chosen correctly. The integrated width of the bounds converges at the same rate on intervals where the density satisfies a Lipschitz condition if the intervals are strictly within the support of the density. The approach also gives algorithms to compute confidence intervals for the support of monotonic densities and for the mode of unimodal densities, lower confidence intervals on the number of modes of a distribution and conservative tests of the hypothesis of $k$-modality. We use the method to compute confidence bounds for the probability density of aftershocks of the 1984 Morgan Hill, CA, earthquake, assuming aftershock times are an inhomogeneous Poisson point process with decreasing intensity.