Optimal Sequential Kernel Detection for Dependent Processes

TL;DR

This paper develops optimal sequential kernel detection methods for dependent processes, providing theoretical guarantees and practical kernel selection strategies under weak dependence conditions.

Contribution

It introduces new theorems on the convergence of detection delay and explicit solutions for optimal kernels in dependent time series.

Findings

Convergence theorem for normed delay under dependence

Explicit representation of optimal kernels

Practical kernel selection procedure

Abstract

In many applications one is interested to detect certain (known) patterns in the mean of a process with smallest delay. Using an asymptotic framework which allows to capture that feature, we study a class of appropriate sequential nonparametric kernel procedures under local nonparametric alternatives. We prove a new theorem on the con- vergence of the normed delay of the associated sequential detection procedure which holds for dependent time series under a weak mixing condition. The result suggests a simple procedure to select a kernel from a finite set of candidate kernels, and therefore may also be of interest from a practical point of view. Further, we provide two new theorems about the existence and an explicit representation of optimal kernels minimizing the asymptotic normed delay. The results are illustrated by some examples.