重尾非线性自回归模型自加权M-估计的渐近分布

数学物理学报 ›› 2020, Vol. 40 ›› Issue (2): 475-483.

重尾非线性自回归模型自加权M-估计的渐近分布

傅可昂(),丁丽,李君巧

^{浙江工商大学统计与数学学院杭州 310018}

收稿日期:2018-09-27 出版日期:2020-04-26 发布日期:2020-05-21
作者简介:傅可昂, E-mail:fukeang@hotmail.com
基金资助:
国家自然科学基(11971432);浙江省自然科学基金(LY17A010004);浙江省一流学科A类(浙江工商大学统计学)

Asymptotics for the Self-Weighted M-Estimation of Nonlinear Autoregressive Models with Heavy-Tailed Errors

Keang Fu(),Li Ding,Junqiao Li

^{School of Statistics and Mathematics, Zhejiang Gongshang University, Hangzhou 310018}

Received:2018-09-27 Online:2020-04-26 Published:2020-05-21
Supported by:
the NSFC(11971432);the NSF of Zhejiang Province(LY17A010004);the First Class Discipline of Zhejiang-A(浙江工商大学统计学)

摘要/Abstract

摘要：

考虑非线性自回归模型x_t=f（x_t-1，…，x_t-p，θ）+ε_t，其中θ为q维未知参数，ε_t为随机误差.在允许误差方差无穷的重尾条件下，构造θ的自加权M-估计，并证明了该估计的渐近正态性.最后通过数值模拟，在随机误差服从某些重尾分布的条件下，说明自加权M-估计比最小二乘和L₁估计更有效.

关键词: 非线性自回归, 自加权M-估计, 重尾, 渐近正态

Abstract:

Consider the nonlinear autoregressive model x_t=f(x_t-1, …, x_t-p, θ)+ε_t, where θ is the q-dimensional unknown parameter and ε_t's are random errors with possibly infinite variance. In this paper, the self-weighted M-estimator of θ is constructed, and the asymptotic normality of the proposed estimator is also established. Some simulation studies are also given to show that the self-weighted M-estimators have good performances with some heavy-tailed random errors.

Key words: Nonlinear autoregression, Self-weighted M-estimator, Heavy tail, Asymptotic normality

中图分类号:

O212.1

傅可昂,丁丽,李君巧. 重尾非线性自回归模型自加权M-估计的渐近分布[J]. 数学物理学报, 2020, 40(2): 475-483.

Keang Fu,Li Ding,Junqiao Li. Asymptotics for the Self-Weighted M-Estimation of Nonlinear Autoregressive Models with Heavy-Tailed Errors[J]. Acta mathematica scientia,Series A, 2020, 40(2): 475-483.

图/表 3

表 1

表 2

表 3

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$\theta_1$	$\theta_2$		$\hat{\theta}_{2_{LS}}$	$\hat{\theta}_{2_{{L_1}}}$	$\hat{\theta}_{2_{SM_1}}$	$\hat{\theta}_{2_{SM_2}}$
$n = 200$
-0.9	2	Mean	2.0802	2.0592	2.0279	2.0215
		SD	0.9983	0.7977	0.6060	0.6419
0.9	-2	Mean	-1.8743	-1.9307	-2.0247	-2.0507
		SD	1.0449	0.8511	0.6223	0.6847
$n = 400$
-0.9	2	Mean	2.0136	1.9881	1.9997	2.0135
		SD	0.6956	0.5466	0.4115	0.4540
0.9	-2	Mean	-1.9341	-1.9648	-2.0044	-2.0023
		SD	0.7366	0.5947	0.4401	0.4714

$\theta_1$	$\theta_2$		$\hat{\theta}_{2_{LS}}$	$\hat{\theta}_{2_{{L_1}}}$	$\hat{\theta}_{2_{SM_1}}$	$\hat{\theta}_{2_{SM_2}}$
$n = 200$
-0.9	2	Mean	1.9244	2.0396	2.0043	2.0182
		SD	2.1240	0.9085	0.7293	0.7503
0.9	-2	Mean	-1.9520	-2.0495	-2.0119	-2.0155
		SD	2.3319	0.9362	0.7937	0.8031
$n = 400$
-0.9	2	Mean	1.9621	2.0164	2.0065	2.0172
		SD	1.6561	0.6101	0.5121	0.3659
0.9	-2	Mean	-1.9583	-1.9832	-2.0018	-2.0112
		SD	1.6165	0.6406	0.5269	0.3847

$\theta_1$	$\theta_2$		$\hat{\theta}_{2_{LS}}$	$\hat{\theta}_{2_{{L_1}}}$	$\hat{\theta}_{2_{SM_1}}$	$\hat{\theta}_{2_{SM_2}}$
$n = 200$
-0.9	2	Mean	14.4757	2.6779	2.0648	2.0772
		SD	813.7240	28.9289	2.3877	2.2688
0.9	-2	Mean	-23.0939	-1.8470	-2.0652	-2.0545
		SD	627.0616	295.3715	3.6210	2.5467
$n = 400$
-0.9	2	Mean	3.7147	2.0554	2.0279	2.0320
		SD	233.3033	1.2508	1.2325	1.1376
0.9	-2	Mean	-7.3892	-2.1109	-2.0242	-2.0132
		SD	326.2041	1.2015	1.1631	1.1284

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