Refinements of the Kiefer-Wolfowitz Theorem and a Test of Concavity

06/25/2019
by   Zheng Fang, et al.
0

This paper studies estimation of and inference on a distribution function F that is concave on the nonnegative half line and admits a density function f with potentially unbounded support. When F is strictly concave, we show that the supremum distance between the Grenander distribution estimator and the empirical distribution is of order O(n^-2/3( n)^2/3) almost surely, which reduces to an existing result of Kiefer and Wolfowitz when f has bounded support. We further refine this result by allowing F to be not strictly concave or even non-concave and instead requiring it be "asymptotically" strictly concave. Building on these results, we then develop a test of concavity of F or equivalently monotonicity of f, which is shown to have asymptotically pointwise level control under the entire null as well as consistency under any fixed alternative. In fact, we show that our test has local size control and nontrivial local power against any local alternatives that do not approach the null too fast, which may be of interest given the irregularity of the problem. Extensions to settings involving testing concavity/convexity/monotonicity are discussed.

READ FULL TEXT

page 1

page 2

page 3

page 4

research
01/09/2023

A Sequential Test for Log-Concavity

On observing a sequence of i.i.d. data with distribution P on ℝ^d, we as...
research
11/22/2020

Least favorability of the uniform distribution for tests of the concavity of a distribution function

A test of the concavity of a distribution function with support containe...
research
02/18/2021

Convolution of a symmetric log-concave distribution and a symmetric bimodal distribution can have any number of modes

In this note, we show that the convolution of a discrete symmetric log-c...
research
02/14/2020

Local continuity of log-concave projection, with applications to estimation under model misspecification

The log-concave projection is an operator that maps a d-dimensional dist...
research
02/01/2021

How can one test if a binary sequence is exchangeable? Fork-convex hulls, supermartingales, and Snell envelopes

Suppose we observe an infinite series of coin flips X_1,X_2,…, and wish ...
research
12/15/2020

Signaling Games for Arbitrary Distributions: Number of Bins and Properties of Equilibria

We investigate the equilibrium behavior for the decentralized quadratic ...
research
05/11/2023

The Cardinality Bound on the Information Bottleneck Representations is Tight

The information bottleneck (IB) method aims to find compressed represent...

Please sign up or login with your details

Forgot password? Click here to reset