Efficient Importance Sampling for Copula Models with Applications

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徐慈陽(Tzu-Yang Hsu) 查詢紙本館藏

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統計研究所

論文名稱

(Efficient Importance Sampling for Copula Models with Applications)

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摘要(中)

本篇研究是針對在copula 模型之下，提出一種有效率的重點抽樣演算法，使得罕見事件的模擬可以得到改善。其最佳機率測度的尋找方式為：固定在一個參數化的指數傾斜性家族，接而最小化蒙地卡羅估計的變異數而得之。由於copula 模型是由一個copula 函數和單維度的邊際累積分配函數構成，具有較為複雜的形式，其動差生成函數計算上較為困難，因此，我們先行應用轉換概似函數 (TLR) 方法使其得到另一個指數傾斜性家族，接著在這個新的指數傾斜性家族找最佳機率測度的解。本研究方法具有普遍性，可應用在許多copula 模型。為了描述本研究方法的廣大應用性，本篇研究有幾個應用：第一，使用拔靴法估計信賴區間的應用；第二，信用風險管理上，其優先損失的計算。由模擬結果可得，本研究方法相對於一般蒙地卡羅方法，變異數減少的量很大，具有顯著的效率性。

摘要(英)

In this thesis, we propose an efficient importance sampling algorithm for rare event simulation under copula models. The derived optimal probability is based on the criterion of minimizing the variance of the Monte Carlo estimator within a parametric exponential tilting family. Since copula model is defined on a copula function for one-dimensional marginal cumulative distribution functions of a random vector, and its moment generating function is not easy to get, we apply the transform likelihood ratio (TLR) method to have an alternative exponential tilting family first. And then obtain a simple and explicit expression of the optimal alternative distribution under this transformed exponential tilting family. The importance sampling framework we propose is quite general and can be implemented for many classes of copula models from which sampling is feasible. To illustrate the broad applicability of our method, we study bootstrap confidence intervals for multivariate distributions based on copula models, to which substantial variance reduction was obtained in comparison to standard Monte Carlo estimators.

關鍵字(中)

★ Copula
★ 重點抽樣
★ 模擬
★ 拔靴法

關鍵字(英)

★ Copula
★ importance sampling
★ simulation
★ bootstrap

論文目次

Contents
1 Introduction 1
1.1 Motivation and Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2 Preliminaries for Copulas 4
2.1 Sklar’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2 Simulation methods for copulas . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2.1 Conditional Inverse Method . . . . . . . . . . . . . . . . . . . . . . . 5
2.2.2 A Simulation Method for Elliptical Copulas . . . . . . . . . . . . . . 6
2.2.3 Marshal-Olkin Method . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.3 Parameter Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3.1 Full Maximum Likelihood Estimation [FML] . . . . . . . . . . . . . . 8
2.3.2 Inference Function for Margins [IFM] . . . . . . . . . . . . . . . . . . 8
2.3.3 Canonical Maximum Likelihood [CML] . . . . . . . . . . . . . . . . . 9
3 Methodology 10
3.1 Importance Sampling Method . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.2 The Transform Likelihood Ratio Method . . . . . . . . . . . . . . . . . . . . 13
3.3 Importance Sampling with Transform Likelihood Ratio Method . . . . . . . 15
3.4 Importance Sampling for Copula Models . . . . . . . . . . . . . . . . . . . . 18
3.4.1 Importance Sampling for General Copula . . . . . . . . . . . . . . . . 19
3.4.2 Importance Sampling for Elliptical Copula . . . . . . . . . . . . . . . 20
3.4.3 Importance Sampling for Archimedean Copula . . . . . . . . . . . . . 24
4 Simulation Study 26
Applications 35
5.1 Importance Resampling for Bootstrap Confidence Intervals of the Dependence
Parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
5.1.1 Bootstrap Confidence Intervals for the Dependence Parameter in Copula 37
5.1.2 A Real Data Example . . . . . . . . . . . . . . . . . . . . . . . . . . 38
6 Conclusion and Furture Work 41
Appendix 1 41
Appendix 2 44
Bibliography 48

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Forthcoming.

指導教授

傅承德、鄧惠文
(Cheng-der Fuh、Huei-wen Teng)

審核日期

2014-6-16

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