On the approximation of continuous time threshold ARMA processes
On the approximation of continuous time threshold ARMA processes
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连续时间阈值ARMA过程的逼近
DOI：
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发表时间：
1995
期刊：
影响因子：
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通讯作者：
O. Stramer
中科院分区：
文献类型：
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作者：
P. Brockwell;O. Stramer
Brockwellet 等人最近的几篇论文讨论了具有连续时间参数的阈值自回归 (AR) 和自回归移动平均 (ARMA) 过程。 (1991, Statist. Sinica,1, 401–410)、Tong 和 Yeung (1991, Statist. Sinica,1, 411–430)、Brockwell 和 Hyndman (1992,International Journal Forecasting,8, 157–173) 和 Brockwell (1994,J. Statist. Plann. Inference,39, 291–304)。边界宽度 2δ>0 的阈值 ARMA 过程很容易根据系数为分段线性和 Lipschitz 的随机微分方程的唯一强解来定义。正边界宽度是一种方便的数学手段，可以平滑边界处的系数变化，从而确保导出该过程的随机微分方程的强解的存在性和唯一性。在本文中，我们在仅自回归系数随过程水平变化的重要情况下给出了 δ=0 的阈值 ARMA 过程的直接定义。 （这当然包括具有恒定尺度参数的所有阈值 AR 过程。）这个想法是用某个随机微分方程的弱解来表达过程的分布。结果表明，δ=0 时该解的联合分布是 δ>0 时解的分布的弱极限，δ ↓ 0 。还研究了 Brockwell 和 Hyndman (1992, International Journal Forecasting,8, 157–173) 使用的近似过程序列收敛到该弱解的意义。一些数值示例说明了后一种近似值与从 Cameron-Martin-Girsanov 公式获得的过程的更直接表示的比较。它特别用于拟合太阳黑子和加拿大山猫系列的连续时间阈值模型。
Threshold autoregressive (AR) and autoregressive moving average (ARMA) processes with continuous time parameter have been discussed in several recent papers by Brockwellet al. (1991,Statist. Sinica,1, 401–410), Tong and Yeung (1991,Statist. Sinica,1, 411–430), Brockwell and Hyndman (1992,International Journal Forecasting,8, 157–173) and Brockwell (1994,J. Statist. Plann. Inference,39, 291–304). A threshold ARMA process with boundary width 2δ>0 is easy to define in terms of the unique strong solution of a stochastic differential equation whose coefficients are piecewise linear and Lipschitz. The positive boundary-width is a convenient mathematical device to smooth out the coefficient changes at the boundary and hence to ensure the existence and uniqueness of the strong solution of the stochastic differential equation from which the process is derived. In this paper we give a direct definition of a threshold ARMA processes with δ=0 in the important case when only the autoregressive coefficients change with the level of the process. (This of course includes all threshold AR processes with constant scale parameter.) The idea is to express the distributions of the process in terms of the weak solution of a certain stochastic differential equation. It is shown that the joint distributions of this solution with δ=0 are the weak limits as δ ↓ 0 of the distributions of the solution with δ>0. The sense in which the approximating sequence of processes used by Brockwell and Hyndman (1992,International Journal Forecasting,8, 157–173) converges to this weak solution is also investigated. Some numerical examples illustrate the value of the latter approximation in comparison with the more direct representation of the process obtained from the Cameron-Martin-Girsanov formula. It is used in particular to fit continuous-time threshold models to the sunspot and Canadian lynx series.