This paper develops optimal statistical tests for testing certain class of non-linear time series models contiguous to a non-linear autoregressive processes with $\beta$-ARCH errors. The statistical tests are based on the Local Asymptotic Normality -LAN-(established via the quadratic mean differentiability) of the log-likelihood ratio of the studied Model when its parameters are estimated. Their local power are also computed. Some examples related to financial models are considered. Our approach uses mainly an efficient estimators of Bouzebda and Lounis (2019).

中文翻译：

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非线性自回归模型的渐近最优检验 $\beta$-ARCH 错误

本文开发了最佳统计测试，用于测试与非线性自回归过程相邻的某些类型的非线性时间序列模型 $\beta$-ARCH 错误。统计检验基于所研究模型在估计其参数时的对数似然比的局部渐近正态性 -LAN（通过二次平均可微性建立）。还计算了它们的局部功率。考虑了与财务模型相关的一些示例。我们的方法主要使用 Bouzebda 和 Lounis (2019) 的有效估计器。