重尾非线性自回归模型自加权M-估计的渐近分布

Abstract

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

CLC Number:

O212.1

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.

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Figures/Tables 3

References 17

1	Tong H. Non-linear Time Series:A Dynamical Systems Approach. Oxford:Oxford University Press, 1990
2	Fan J, Yao Q. Nonlinear Time Series:Nonparametric and Parametric Methods. Berlin:Springer, 2003
3	金阳, 安鸿志. 带有重尾扰动项的非线性自回归模型. 中国科学, 2005, 35 (1): 39- 45
	Jin Y , An H . Nonlinear autoregressive models with heavy-tailed innovation. Sci China Ser A Math, 2005, 35 (1): 39- 45
4	金阳, 安鸿志. 非线性自回归序列的矩的存在性. 应用数学学报, 2006, 29 (1): 1- 8
	Jin Y , An H . Existence of moment for the nonlinear autoregressive series. Acta Math Appl Sinica, 2006, 29 (1): 1- 8
5	张韧, 张绍义. 非线性自回归序列的平稳解及其矩的存在性. 数学物理学报, 2013, 33 (2): 260- 266 doi: 10.3969/j.issn.1003-3998.2013.02.009
	Zhang R , Zhang S . Stochastic stability for the nonlinear autoregressive series. Acta Math Sci, 2013, 33A (2): 260- 266 doi: 10.3969/j.issn.1003-3998.2013.02.009
6	周杰, 刘三阳, 张正策. 非线性自回归序列L₁估计的渐近分布. 高校应用数学学报, 2007, 22 (4): 427- 432 doi: 10.3969/j.issn.1000-4424.2007.04.008
	Zhou J , Liu S , Zhang Z . Asymptotical distribution of L₁-estimator for nonlinear autoregression. Appl Math J Chinese Univ Ser A, 2007, 22 (4): 427- 432 doi: 10.3969/j.issn.1000-4424.2007.04.008
7	周杰, 刘三阳, 张正策. 方差无穷非线性自回归序列的自加权L₁估计. 高校应用数学学报, 2008, 23 (2): 193- 199 doi: 10.3969/j.issn.1000-4424.2008.02.010
	Zhou J , Liu S , Zhang Z . Self-weighted L₁-estimator for infinite variance nonlinear autoregressive model. Appl Math J Chinese Univ Ser A, 2008, 23 (2): 193- 199 doi: 10.3969/j.issn.1000-4424.2008.02.010
8	Cheng F , Sun S . A goodness-of-fit test of the errors in nonlinear autoregressive time series models. Stat Prob Lett, 2008, 78 (1): 50- 59 doi: 10.1016/j.spl.2007.05.003
9	Fu K , Yang X . Asymptotics of kernel error density estimators in nonlinear autoregressive models. J Math Chem, 2008, 44 (3): 831- 838 doi: 10.1007/s10910-008-9379-2
10	Cheng F . Global property of error density estimation in nonlinear autoregressive time series models. Stat Infer Stoch Proc, 2010, 13 (1): 43- 53 doi: 10.1007/s11203-009-9036-9
11	Li J . Asymptotics of the L_p-norms of density estimators in the nonlinear autoregressive models. Commun Stat-Theo Meth, 2014, 43 (22): 4845- 4855 doi: 10.1080/03610926.2012.724503
12	Cheng F . Variance estimation in nonlinear autoregressive time series models. J Stat Plan Infer, 2011, 141 (4): 1588- 1592 doi: 10.1016/j.jspi.2010.11.010
13	Koenker R , Bassett G . Regression quantiles. Econometrica, 1978, 46, 33- 49 doi: 10.2307/1913643
14	Ling S . Self-weighted least absolute deviation estimation for infinite variance autoregressive models. J R Stat Soc B, 2005, 67 (3): 381- 393 doi: 10.1111/j.1467-9868.2005.00507.x
15	Huber P . Robust regression:asymptotics, conjectures and monte carlo. Ann Stat, 1973, 1 (5): 799- 821 doi: 10.1214/aos/1176342503
16	Yohai V . High breakdown-point and high efficiency robust estimates for regression. Ann Stat, 1987, 15 (2): 642- 656 doi: 10.1214/aos/1176350366
17	Davis R A , Knight K , Liu J . M-estimaton for autoregressions with infinite variance. Stoch Proc Appl, 1992, 40, 145- 180 doi: 10.1016/0304-4149(92)90142-D

$\theta_1$	$\theta_2$		$\hat{\theta}_{2_{LS}}$	$\hat{\theta}_{2_{{L_1}}}$	$\hat{\theta}_{2_{SM_1}}$	$\hat{\theta}_{2_{SM_2}}$
$n = 200$
-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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Asymptotics for the Self-Weighted M-Estimation of Nonlinear Autoregressive Models with Heavy-Tailed Errors

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