关于低频采样下$α$ -稳定CARMA过程估计的注记
A note on estimation of $α$-stable CARMA processes sampled at low frequencies
摘要：
本文研究了等距采样对称$α$稳定CARMA过程的估计.模拟研究表明，Whittle估计量和Garc 'ıa等人（2011）提出的估计量是稳定CARMA过程参数的一致估计量。对于具有有限二阶矩的CARMA过程，Whittle估计是相合的且渐近正态分布。因此，在轻尾设置的CARMA过程的Whittle估计的性质是类似的ARMA过程的Whittle估计。然而，在本文中，我们证明，在一般情况下，对称$α$稳定的CARMA过程在低频率采样的惠特尔估计是不一致的，并强调为什么模拟研究表明别的东西。因此，与轻尾设置相比，重尾ARMA过程的Whittle估计的性质不能转移到重尾CARMA过程。我们也阐述了Garc 'ıa et al.（2011）中提出的估计量面临同样的问题。然而，稳定CAR（1）过程的Whittle估计是一致的。
英文摘要：
In this paper, we investigate estimators for symmetric $α$-stable CARMA processes sampled equidistantly. Simulation studies suggest that the Whittle estimator and the estimator presented in Garc\'ıa et al. (2011) are consistent estimators for the parameters of stable CARMA processes. For CARMA processes with finite second moments it is well-known that the Whittle estimator is consistent and asymptotically normally distributed. Therefore, in the light-tailed setting the properties of the Whittle estimator for CARMA processes are similar to those of the Whittle estimator for ARMA processes. However, in the present paper we prove that, in general, the Whittle estimator for symmetric $α$-stable CARMA processes sampled at low frequencies is not consistent and highlight why simulation studies suggest something else. Thus, in contrast to the light-tailed setting the properties of the Whittle estimator for heavy-tailed ARMA processes can not be transferred to heavy-tailed CARMA processes. We elaborate as well that the estimator presented in Garc\'ıa et al. (2011) faces the same problems. However, the Whittle estimator for stable CAR(1) processes is consistent.
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