空間統計模型選取之大樣本理論

Abstract

在傳統迴歸模型中，模型選取的大樣本理論已被廣泛建立。然而在空間統計迴歸模型中，使用傳統模型選取準則的選模結果並未被完善的討論及研究，尤其當假設資料觀測空間為一固定區域而不隨著樣本增加而放大時，其大樣本理論可以預期會與傳統的理論結果有所差異。論文中，我們在一些常規假設下，建立了傳統模型選取準則的大樣本理論。而後在一維空間的一些例子下，我們發現這些常規假設的成立與否不僅與樣本空間放大的速度有關，也與所選取變數在空間中的平滑程度有緊密關係。當空間互變異函數參數未知時，我們同樣發現，參數估計及傳統模型選取準則的大樣本理論，也與樣本空間放大的速度和所選取變數在空間中的平滑程度有關。最後我們執行有限樣本的模擬實驗，並得到與大樣本理論一致的結果。

Information criteria, such as Akaike's information criterion (AIC), Bayesian information criterion (BIC), and conditional AIC (CAIC) are often applied in model selection. However, their asymptotic behaviors under geostatistical regression models have not been well studied particularly under the fixed domain asymptotic framework with more and more data observed in a bounded fixed region. In this thesis, we investigate two classes of criteria for geostatistical model selection: generalized information criterion (GIC) and conditional GIC (CGIC), which include AIC, BIC, and CAIC as special cases, under both the increasing domain asymptotic and fixed domain asymptotic frameworks. We establish conditions under which GIC and CGIC are selection consistent and asymptotically efficient even without assuming spatial covariance structure to be known. These conditions are further examined for GIC and CGIC in selecting one-dimensional geostatistical regression models with the exponential covariance function class under various settings. For example, under the fixed domain asymptotic framework, where some covariance parameters are not consistently estimable, we show that selection consistency not only depends on the tuning parameter of GIC, but also depends on smoothness of the explanatory variables in space. In addition, under the increasing domain framework, we show that asymptotic properties of GIC depend on the growing rates for the size of the domain. Moreover, some numerical experiments are provided to demonstrate the finite sample behavior of various criteria.