Abstract We consider the estimation of the variance and spatial scale parameters of the covariance function of a one-dimensional Gaussian process with fixed smoothness parameter s . We study the fixed-domain asymptotic properties of composite likelihood estimators. As an improvement of previous references, we allow for any fixed number of neighbor observation points, both on the left and on the right sides, for the composite likelihood. First, we examine the case where only the variance parameter is unknown. We prove that for small values of s , the composite likelihood estimator converges at a sub-optimal rate and we provide its non-Gaussian asymptotic distribution. For large values of s , the estimator converges at the optimal rate. Second, we consider the case where the variance and the spatial scale are jointly estimated. We obtain the same conclusion as for the first case for the estimation of the microergodic parameter. The theoretical results are confirmed in numerical simulations.

中文翻译：

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高斯过程最大复合似然估计量的固定域渐近性质

摘要 我们考虑了具有固定平滑度参数 s 的一维高斯过程的协方差函数的方差和空间尺度参数的估计。我们研究了复合似然估计量的固定域渐近特性。作为对先前参考的改进，我们允许在左侧和右侧有任意固定数量的相邻观察点，用于复合似然。首先，我们检查只有方差参数未知的情况。我们证明对于较小的 s 值，复合似然估计量以次优速率收敛，并且我们提供其非高斯渐近分布。对于较大的 s 值，估计量以最佳速率收敛。其次，我们考虑方差和空间尺度联合估计的情况。对于微遍历参数的估计，我们得到了与第一种情况相同的结论。理论结果在数值模拟中得到证实。