A Wilks' theorem for grouped data

TL;DR

This paper develops a new Wilks' theorem tailored for grouped data, enabling more robust hypothesis testing when observations are concentrated in short time intervals, overcoming issues caused by arbitrary time origin choices.

Contribution

It introduces a Wilks' theorem for grouped data that improves hypothesis testing robustness by addressing the arbitrary choice of time origin, enhancing test power.

Findings

The new procedure overcomes the arbitrary time origin problem.

It is more powerful than classical Wilks' theorem-based methods.

The method is applicable to large sample sizes with grouped observations.

Abstract

Consider $n$ independent measurements, with the additional information of the times at which measurements are performed. This paper deals with testing statistical hypotheses when $n$ is large and only a small amount of observations concentrated in short time intervals are relevant to the study. We define a testing procedure in terms of multiple likelihood ratio (LR) statistics obtained by splitting the observations into groups, and in accordance with the following principles: P1) each LR statistic is formed by gathering the data included in $G$ consecutive vectors of observations, where $G$ is a suitable time window defined a priori with respect to an arbitrary choice of the `origin of time'; P2) the null statistical hypothesis is rejected only if at least $k$ LR statistics are sufficiently small, for a suitable choice of $k$. We show that the application of the classical Wilks' theorem may be affected by the arbitrary choice of the "origin of time", in connection with P1). We then introduce a Wilks' theorem for grouped data which leads to a testing procedure that overcomes the problem of the arbitrary choice of the `origin of time', while fulfilling P1) and P2). Such a procedure is more powerful than the corresponding procedure based on Wilks' theorem.