Testing and Estimation of Change Point in ARMA Model With Heavy‐Tailed G‐GARCH Noises
针对重尾G-GARCH噪声下的ARMA模型，提出基于自加权最小绝对偏差估计的两种得分型检验统计量用于变点检测，并证明估计量的渐近性质，适用于无限方差数据。
ABSTRACT We develop a procedure for testing and estimating the change point in the autoregressive moving average (ARMA) model with heavy‐tailed general generalized autoregressive conditional heteroskedasticity (G‐GARCH) noises. Based on the self‐weighted least absolute deviation estimator (SLADE), we propose two score‐type test statistics for change point detection. Under the null hypothesis, we show that one of them converges weakly to the maxima of a Brownian bridge, and the other one converges weakly to an extreme distribution. Furthermore, we prove that the SLADE of the change point converges weakly to the location of the maxima of a double‐sided random walk, and the SLADE of other parameters is asymptotically normal. Our two score‐type tests and SLADE are applicable for the data with an infinite variance, while not specifying the form of G‐GARCH noises. Finally, we use simulations and two real examples to demonstrate the usefulness of the proposed tests and estimator in handling the heavy‐tailed data with an infinite variance.