樣本半偏差極限分佈的探討

A Note of the Asymptotic Distribution of the Sample Semivarogram

半偏差被廣泛應用於地球統計資料的分析，而樣本半偏差便是分析者最常使用的估計量。因此很多有關於樣本半偏差的統計推論諸如半偏差模型適合度的檢定，與方向相關的對稱性質檢定等等，都須了解樣本半偏差的抽樣分佈方可進行。至今學術上曾證明得到在任一m相依的高斯隨機場上，某一固定位移量的樣本半偏差之邊際極限分佈。然而由於其定理條件太嚴格且由於空間性相關的因素，此結果缺乏實用性。而且只知樣本半偏差的邊際分佈仍無法進行很多統計推論。雖然有些文章曾利用模擬的方法探討樣本半偏差的聯合極限分佈，但仍欠缺理論證明。因此，在本篇文章中，我們提出一個比m相依更為寬鬆且為多數半偏差模型滿足的混合條件，證明在此條件下，樣本半偏差的聯合極限分佈為多變量常態。文中我們先導出在任意隨機場上，樣本半偏差的極限分佈，然後探討在高斯隨機場上，此極限分佈所須之條件及結果都大為簡化且易於應用。

The semivariogram is widely used in geostatistical data analysis. Most often, it is estimated by the sample semivariogram. Knowledge about the sampling distribution of the sample semivariogram is necessary in many statistical inferences, such as assessing the goodness of fit of any proposed semivariogram model, and testing for directional symmetry properties. Some theoretical works have been done on the marginal distribution of the sample semivariogram for a Gaussian m-dependent process, but it is not of immediate practical use due to the restrictive conditions and the presence of spatial inter-correlations. Although the joint distrbution of the sample semivariogram has been discussed through simulation studies, theoretical support is needed. In this paper, we prove the joint asymptotic normality of the sample semivariogram under a mixing condition which is less restrictive than m-dependence and is satisfied by most geostatistical models. The distributional result is established for random fields without Gaussianity assumption and then specialized to Gaussian random fields.