Copula之估計法比較與模型診斷

Estimating method comparison and model diagnosis of Copula

陳冠璋(GUAN-JHANG CHEN)

指導教授 ： 白惠明

國立臺北大學/商學院/統計學系/碩士(2011年)

摘要

在使用copula建立描述資料的模型時，copula的參數估計通常是第一個重要的程序。基與此，估計方法對於模型的建立就變得相當具有影響力。Kim et al. (2007) 利用模擬的方式對於三種估計copula參數的方法進行比較。這三種估計方法為Maximum likelihood method (ML法)，Inference function for margins (IFM法) 與Canonical maximum likelihood method (CML法)，其中CML法的表現是較為穩定的。copula的另一種估計方法來自於Kendall’s tau與估計參數之函數關係。若將coupla限制於Archimedean copula，藉由Genest and Mackay (1986) 提出的Kendall’s tau與估計參數之函數關係計算式，copula之參數估計值可以更為容易地得到。我們的研究目的是想要比較這兩種估計方法的優劣。最後，將這兩種估計方法應用於建立欲描述的資料模型，並試著觀察是否有所差異。

關鍵字

Archimedean copula ； copula ； CML法 ； G&M法 ； Sklar 定理

並列摘要

Copula is a useful tool to build a joint distribution function. When we use a copula to build a joint distribution function, the estimation of the copula parameters is an essential procedure. Cherubini et al. (2004) systematically introduces some estimation methods in Chapter 5. Among those estimation methods, Kim et al. (2007) uses the simulation studies to compare Maximum likelihood method (ML) and Inference function for margins (IFM) method with Canonical maximum likelihood method (CML). The main conclusion is that the ML and IFM methods are nonrobust against misspecification of marginal distributions, and that the CML method performs better than the ML and IFM methods overall. The functional relationship between Kendall’s tau and copula parameter is another way to estimate copula. If we restrict our attention to Archimedean copula, then the parameter can be estimated more easily in terms of the functional relationship between Kendall’s tau and copula parameter proposed by Genest and Mackay (1986). This method is called G&M method. The purpose of this paper is to compare CML method with G&M method and employ copula in practical applications.

並列關鍵字

Archimedean copula ； copula ； CML method ； G&M method ； Sklar’s theorem

參考文獻

Allcroft, D.J., and Glasbey, C.A. (2003). A simulation-based method for model evaluation. Statistical Modelling:An International Journal. 3(1), 1–13.

Berg, D. (2007). Copula goodness-of-fit testing : An overview and power comparison. Working paper, University of Oslo & Norwegian computing center.

Genest, C., and MacKay, J. (1986). The joy of Copulas: bivariate distributions with uniform marginals. The American Statistician. 40(4), 280-283.

Genest, C., and Rivest, L.-P. (1993). Statistical inference procedures for bivariate archemedean copulas. Journal of American Statistical Association. 88, 1034–1043.

Genest, C., Quessy, J.-F., and Remillard, B. (2006). Goodness-of-fit procedures for copula models based on the probability integral transformation. Scandinavian Journal of Statistics. 33, 337–366.