Adaptive estimator of the memory parameter and goodness-of-fit test using a multidimensional increment ratio statistic

TL;DR

This paper extends the increment ratio (IR) statistic for estimating the memory parameter of stationary processes, establishing a multidimensional CLT, and developing adaptive estimators and goodness-of-fit tests with proven oracle properties and robustness.

Contribution

It introduces a multidimensional CLT for IR statistics, constructs adaptive estimators and tests, and demonstrates their effectiveness and robustness in non-Gaussian scenarios.

Findings

Multidimensional CLT for IR statistics established

Adaptive estimator exhibits oracle property

Simulations show high accuracy and robustness

Abstract

The increment ratio (IR) statistic was first defined and studied in Surgailis {\it et al.} (2007) for estimating the memory parameter either of a stationary or an increment stationary Gaussian process. Here three extensions are proposed in the case of stationary processes. Firstly, a multidimensional central limit theorem is established for a vector composed by several IR statistics. Secondly, a goodness-of-fit $\chi^2$-type test can be deduced from this theorem. Finally, this theorem allows to construct adaptive versions of the estimator and test which are studied in a general semiparametric frame. The adaptive estimator of the long-memory parameter is proved to follow an oracle property. Simulations attest of the interesting accuracies and robustness of the estimator and test, even in the non Gaussian case.