Estimation and prediction of Gaussian processes using generalized Cauchy covariance model under fixed domain asymptotics
Estimation and prediction of Gaussian processes using generalized Cauchy covariance model under fixed domain asymptotics
复制标题
定域渐近下广义柯西协方差模型的高斯过程估计与预测
DOI：
10.1214/19-ejs1597
复制
发表时间：
2017
影响因子：
1.1
通讯作者：
Tarik Faouzi
中科院分区：
文献类型：
--
作者：
M. Bevilacqua;Tarik Faouzi
研究了具有广义Cauchy（GC）族协方差模型的高斯过程在固定区域渐近下的估计和预报问题。具有这种协方差函数的高斯过程提供了分形维数和长程相关性的单独表征，这在许多物理，生物或地质系统中是一个吸引人的特征。本文的研究结果分为三个部分。在第一部分中，我们用GC协方差函数刻画了两个高斯测度的等价性。然后给出了两个具有Mat{e}rn（MT）和GC协方差函数的高斯测度与两个具有广义Wendland（GW）和GC协方差函数的高斯测度等价的充分条件.在第二部分中，我们建立了GC协方差模型中微遍历参数的极大似然估计在固定区域渐近下的强相合性和渐近分布。最后一部分讨论了GC模型的最优预测问题，具体地，在固定区域渐近下，给出了错误指定的GC、MT或GW模型的渐近有效预测和均方误差的渐近正确估计的条件。我们的研究结果说明通过模拟研究：第一个比较有限样本的行为的GC模型的微遍历参数的最大似然估计与给定的渐近分布。然后，我们比较了有限样本的预测行为及其相关的均方误差时，真正的模型是GC和预测是使用真正的模型和错误指定的GW模型。
We study estimation and prediction of Gaussian processes with covariance model belonging to the generalized Cauchy (GC) family, under fixed domain asymptotics. Gaussian processes with this kind of covariance function provide separate characterization of fractal dimension and long range dependence, an appealing feature in many physical, biological or geological systems. The results of the paper are classified into three parts. In the first part, we characterize the equivalence of two Gaussian measures with GC covariance function. Then we provide sufficient conditions for the equivalence of two Gaussian measures with Mat{e}rn (MT) and GC covariance functions and two Gaussian measures with Generalized Wendland (GW) and GC covariance functions. In the second part, we establish strong consistency and asymptotic distribution of the maximum likelihood estimator of the microergodic parameter associated to GC covariance model, under fixed domain asymptotics. The last part focuses on optimal prediction with GC model and specifically, we give conditions for asymptotic efficiency prediction and asymptotically correct estimation of mean square error using a misspecified GC, MT or GW model, under fixed domain asymptotics. Our findings are illustrated through a simulation study: the first compares the finite sample behavior of the maximum likelihood estimation of the microergodic parameter of the GC model with the given asymptotic distribution. We then compare the finite-sample behavior of the prediction and its associated mean square error when the true model is GC and the prediction is performed using the true model and a misspecified GW model.