Asymptotically Optimal Tests when Parameters are Estimated

TL;DR

This paper develops an asymptotically optimal Neyman-Pearson-type test for models with estimated parameters, demonstrating its effectiveness in AR(1) and ARCH models through derived asymptotic power functions.

Contribution

It introduces a new test statistic that remains optimal when parameters are estimated, with specific applications to AR(1) and ARCH models.

Findings

Derived asymptotic power functions for AR(1) and ARCH models

Proposed estimators enable asymptotic optimality of the test

Validated the test's effectiveness through theoretical analysis

Abstract

The main purpose of this paper is to provide an asymptotically optimal test. The proposed statistic is of Neyman-Pearson-type when the parameters are estimated with a particular kind of estimators. It is shown that the proposed estimators enable us to achieve this end. Two particular cases, AR(1) and ARCH models were studied and the asymptotic power function was derived.