# Testing One Hypothesis Multiple Times: The Multidimensional Case

## TL;DR

This paper extends the TOHM hypothesis testing method to multidimensional cases, providing a computationally efficient algorithm for inference involving complex random fields and high significance levels.

## Contribution

It introduces a novel algorithm for computing Euler Characteristics in multiple dimensions, enabling efficient hypothesis testing in complex, high-dimensional settings.

## Key findings

- Efficient algorithm for Euler Characteristic computation in multiple dimensions.
- Enables hypothesis testing with high significance levels in complex models.
- Provides a generalizable tool for non-standard regularity conditions.

## Abstract

The identification of new rare signals in data, the detection of a sudden change in a trend, and the selection of competing models, are among the most challenging problems in statistical practice. These challenges can be tackled using a test of hypothesis where a nuisance parameter is present only under the alternative, and a computationally efficient solution can be obtained by the "Testing One Hypothesis Multiple times" (TOHM) method. In the one-dimensional setting, a fine discretization of the space of the non-identifiable parameter is specified, and a global p-value is obtained by approximating the distribution of the supremum of the resulting stochastic process. In this paper, we propose a computationally efficient inferential tool to perform TOHM in the multidimensional setting. Here, the approximations of interest typically involve the expected Euler Characteristics (EC) of the excursion set of the underlying random field. We introduce a simple algorithm to compute the EC in multiple dimensions and for arbitrary large significance levels. This leads to an highly generalizable computational tool to perform inference under non-standard regularity conditions.