Estimation and Test for Multidimensional Regression Models

TL;DR

This paper develops an asymptotically optimal estimator for multidimensional regression models by minimizing the log-determinant of the error covariance matrix, and introduces a simple test for model parameter selection, supported by numerical validation.

Contribution

It proposes a novel estimation method based on the log-determinant criterion and derives a straightforward asymptotic test for model complexity in multidimensional regression.

Findings

The estimator is asymptotically optimal under certain conditions.

The proposed test simplifies model selection in multidimensional regression.

Numerical experiments validate the theoretical properties.

Abstract

This work is concerned with the estimation of multidimensional regression and the asymptotic behaviour of the test involved in selecting models. The main problem with such models is that we need to know the covariance matrix of the noise to get an optimal estimator. We show in this paper that if we choose to minimise the logarithm of the determinant of the empirical error covariance matrix, then we get an asymptotically optimal estimator. Moreover, under suitable assumptions, we show that this cost function leads to a very simple asymptotic law for testing the number of parameters of an identifiable and regular regression model. Numerical experiments confirm the theoretical results.