| Abstract: | 貝氏 Edgeworth expansion 及其應用 Edgeworth expansion 是一個關於機率分配的近似展開式.其主要之精神在於看出 一機率分配的動差(cumulants)可以用來描述與近似該機率分配.這樣一個展開式的 想法可以追溯到一百多年前(Chebyshev 1890 and Edgeworth 1896, 1905). 而 由於它能幫助探索當代統計方法,因此過去幾十年再度引起相當的興趣.關於這個主題 的文獻回顧 可以參考 Hall (The Bootstrap and the Edgeworth expansion 1992). 除了統計學,Edgeworth expansion 也被應用在其他領域; 比方天文物理 (Blinnikov and Moessner (1998) 比較Gram-Charlier, Gauss-Hermite 與 Edgeworth expansions 一天文物理問題之表現), 和財務數學(Filho and Rosenfeld (2004) 藉由Edgeworth expansion討論選擇權定價問題). 另一方面,Edgeworth expansion可以視為中央極限定理的精緻化. 中央極限定理在 貝氏統計中所對應的理論是關於後驗分配的常態性,也稱為 Bernstein-von Mises theorem. 而且不論是中央極限定理或是後驗分配的常態性,兩者都有若干高次漸近結 果. 然而貝氏 Edgeworth expansion 卻未曾被發展出來. 本研究計畫的目的如下: (1) 利用Stein`s Identity與Hermite多項式,在適當條件下推展出個別參數的後驗 分配的貝氏 Edgeworth expansion. (2) 應用此展開式來診斷後驗分配的蒙地卡羅模擬結果是否收歛. 主要的想法是: 若模 擬結果已經收歛到真實分配,則以此模擬所得之樣本所算出之後驗動差代入我們 的貝氏 Edgeworth expansion後,得到的機率密度函數應該要接近直接由該模 擬結果畫出之機率密度圖. (3) 除了個別參數,我們也要研究所有參數的非線性函數的貝氏 Edgeworth expansion.
The Edgeworth expansion, named in honor of F. Y. Edgeworth (1845-1926),is a series that approximates a probability distribution in terms of its cumulants. It is over a century old and in the past decades it has received a revival of interest due to its usefulness for exploring properties of contemporary statistical methods; see Hall (1992) for an excellent review of this subject. The Edgeworth expansion has been applied to other areas as well; for example, Blinnikov and Moessner (1998) compared Gram-Charlier, Gauss- Hermite and Edgeworth expansions in astrophysics problems, and Filho and Rosenfeld (2004) considered the option pricing problem using Edgeworth expansion, among others. The Bayesian counterpart of the central limit theorem is the Bernsteinvon Mises theorem, which states the asymptotic normality of the posterior. There have been extensive studies on refinements of the Bernstein-von Mises theorem. These refinements are Bayesian counterpart of the higher order asymptotic normality. However, an Edgeworth expansion for posterior distributions has not been developed yet. The goals of present project are the following: (1) Show that by Stein`s Identity and some properties of Hermite polynomials, one can obtain an expansion of the marginal posterior distribution that is valid under mild conditions and resembles the Edgeworth expansion. (2) Apply the expansions to validate convergence of simulation results. The idea is that if the posterior sample has converged to the true distribution, the density induced by the sample should agree with the one obtained by putting the empirical moments of the sample into the expansion. (3) In addition to marginal posterior distribution of the parameter, develop a Bayesian Edgeworth expansion for nonlinear functions of parameter. |
