Relative errors for bootstrap approximations of the serial correlation coefficient

TL;DR

This paper develops saddle-point and bootstrap approximation methods for the serial correlation coefficient in AR(1) models with non-Gaussian errors, providing accurate tail probability estimates with known relative errors.

Contribution

It extends classical Gaussian-based results to non-Gaussian errors using saddle-point and conditional bootstrap techniques, with proven relative error bounds.

Findings

Bootstrap tail probabilities closely approximate true tail probabilities with known relative errors.

Conditional bootstrap maintains similar accuracy as the unconditional approach.

Saddle-point approximations effectively estimate tail probabilities in non-Gaussian AR(1) models.

Abstract

We consider the first serial correlation coefficient under an AR(1) model where errors are not assumed to be Gaussian. In this case it is necessary to consider bootstrap approximations for tests based on the statistic since the distribution of errors is unknown. We obtain saddle-point approximations for tail probabilities of the statistic and its bootstrap version and use these to show that the bootstrap tail probabilities approximate the true values with given relative errors, thus extending the classical results of Daniels [Biometrika 43 (1956) 169-185] for the Gaussian case. The methods require conditioning on the set of odd numbered observations and suggest a conditional bootstrap which we show has similar relative error properties.