For spatial statistics, two asymptotic approaches are usually considered: increasing domain asymptotics and fixed-domain asymptotics (or infill asymptotics).For increasing domain asymptotics, sampled data increase with the increasing spatial domain, while under infill asymptotics, data are observed on a fixed region with the distance betweenneighboring observations tending to zero. The consistency and asymptotic results under these two asymptotic frameworks can be quite different. For example, not all parameters are consistently estimated under infill asymptotics while consistency holds for those parameters under increasing asymptotics (Zhang 2004).For a stationary Gaussian random field on $mathbb{R}^d$ with the spectral density $f(ulambda)$ that satisfies $f(ulambda) ~sim~ c, |ulambda|^{-theta} $ as $|ulambda| ~rightarrow~ infty$, the parameters $c$ and $theta$ control the tail behavior of the spectral density where $theta$ is related to the smoothness of random field and $c$ can be used to determine the orthogonality of probability measures for a fixed $theta$. Specially, c corresponds to the microergodic parameter mentioned in Du et al. (2009) when Mat'{e}rn covariance is assumed. Additionally, under infill asymptotics, the tail behavior of the spectral density dominates the performance of the prediction, and the equivalence of theprobability measures. Based on those reasons, it is significant in statistics to estimate $c$ and $theta$.When the explicit form of $f$ is known, its corresponding covariance structure can be computed through the Fourier transformation. Therefore, spatial domain methodologies like Maximum Likelihood Estimator (MLE) or Tapering MLE can be used for the estimation of $c$ and $theta$. Unfortunately, the exact form of $f$ should be unknown in practice. Under this situation, spatial domain methods will not be applied without the covariance information. In my work, for data observed on grid points, two methods which utilize tail frequency information are proposed to estimate $c$ and $theta$.One of them can be viewed as a weighted local Whittletype estimator. Under proposed approaches, the explicit form of $f$ and the restriction of the dimension are not necessary.The asymptotic properties of the proposed estimators under infill asymptotics (or fixed-domain asymptotics) are investigated in this dissertation together with simulation studies.
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