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arXiv 0909.1046math.STstat.MEstat.TH
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拟合Mat{é}rn模型的“吉布斯能量（GE）和平均方差”估计函数的渐近近效率-- II：通过“条件GE均值”解释测量误差

Asymptotic near-efficiency of the ''Gibbs-energy (GE) and empirical-variance'' estimating functions for fitting Mat{é}rn models -- II: Accounting for measurement errors via ''conditional GE mean''

Didier A. Girard

备注：
The previous version (version 2) considered both the case with measurement errors (also called ''nugget-effect'' or simply ''noise'') and the no-noise case. The no-noise case is now in Girard (2016) with more detailed proofs and two additional (wrt version 2) results: a consistency result is proved and the restriction $ ν\geq 1/2$ is eliminated. This version 3 is devoted to the case with measurement errors, and also gives the analogs of these two additional results
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摘要：

考虑一个连续时间高斯过程$Z$的实现，它属于具有已知"正则性“指数$ν>0$的Mat\' ern族。为了从精细网格上的$n$观测值估计自相关范围和$Z$的方差，我们在Girard（2016）中研究了GE-EV方法，该方法简单地保留了经验方差（EV）并将其等同于候选“吉布斯能量（GE）”，即~二次形式${\bf z}^T R^{-1} {\bf z}/n$，其中${\bf z}$是观测向量，$R$是与候选范围相关联的${\bf z}$的自相关矩阵。本研究考虑的情况下，观察是${\bf z}$加上高斯白噪声的方差是已知的。我们建议简单地偏置校正EV，并在给定观测值的情况下，用其条件均值代替GE。我们证明了距离参数的CGEM-EV估计的大$n$均方误差与其最大似然估计的比值，以及方差参数的模拟比值，与无噪声情况下具有相同的行为：当网格步长趋向于$0$时，它们都收敛于一个常数，仅为$ν$的函数，如果$ν$不是太大，则令人惊讶地接近于$1$。我们还得到，对于所有的$ν$，收敛到1的模拟比的微遍历参数。

英文摘要：

Consider one realization of a continuous-time Gaussian process $Z$ which belongs to the Mat\' ern family with known ``regularity'' index $ν>0$. For estimating the autocorrelation-range and the variance of $Z$ from $n$ observations on a fine grid, we studied in Girard (2016) the GE-EV method which simply retains the empirical variance (EV) and equates it to a candidate ``Gibbs energy (GE)'' i.e.~the quadratic form ${\bf z}^T R^{-1} {\bf z}/n$ where ${\bf z}$ is the vector of observations and $R$ is the autocorrelation matrix for ${\bf z}$ associated with a candidate range. The present study considers the case where the observation is ${\bf z}$ plus a Gaussian white noise whose variance is known. We propose to simply bias-correct EV and to replace GE by its conditional mean given the observation. We show that the ratio of the large-$n$ mean squared error of the resulting CGEM-EV estimate of the range-parameter to the one of its maximum likelihood estimate, and the analog ratio for the variance-parameter, have the same behavior than in the no-noise case: they both converge, when the grid-step tends to $0$, toward a constant, only function of $ν$, surprisingly close to $1$ provided $ν$ is not too large. We also obtain, for all $ν$, convergence to 1 of the analog ratio for the microergodic-parameter.

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