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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク 2-EPT probability density function ： ウィキペディア英語版 2-EPT probability density function In probability theory, a 2-EPT probability density function is a class of probability density functions on the real line. The class contains the density functions of all distributions that have characteristic functions that are strictly proper rational functions. ==Definition== A 2-EPT probability density function is a probability density function on $\backslash mathbb$ with a strictly proper rational characteristic function. On either $[0,\; \backslash infty)$ or $(-\backslash infty,\; 0)$ these probability density functions are exponential-polynomial-trigonometric (EPT) functions. Any EPT density function on $(-\backslash infty,\; 0)$ can be represented as :$f(x)=\backslash textbf\_Ne^\backslash textbf\_N\; ,$ where ''e'' represents a matrix exponential, $(\backslash textbf\_N,\backslash textbf\_P)$ are square matrices, $(\backslash textbf\_N,\backslash textbf\_P)$ are column vectors and $(\backslash textbf\_N,\backslash textbf\_P)$ are row vectors. Similarly the EPT density function on $[0,\; -\backslash infty)$ is expressed as :$f(x)=\backslash textbf\_Pe^\backslash textbf\_P.$ The parameterization $(\backslash textbf\_N,\backslash textbf\_N,\backslash textbf\_N,\backslash textbf\_P,\backslash textbf\_P,\backslash textbf\_P)$ is the minimal realization〔Kailath, T. (1980) ''Linear Systems'', Prentice Hall, 1980〕 of the 2-EPT function. The general class of probability measures on $\backslash mathbb$ with (proper) rational characteristic functions are densities corresponding to mixtures of the pointmass at zero ("delta distribution") and 2-EPT densities. Unlike phase-type and matrix geometric〔Neuts, M. "Probability Distributions of Phase Type", Liber Amicorum Prof. Emeritus H. Florin pages 173-206, Department of Mathematics, University of Louvain, Belgium 1975〕 distributions, the 2-EPT probability density functions are defined on the whole real line. It has been shown that the class of 2-EPT densities is closed under many operations and using minimal realizations these calculations have been illustrated for the two-sided framework in Sexton and Hanzon.〔Sexton, C. and Hanzon,B.,"State Space Calculations for two-sided EPT Densities with Financial Modelling Applications", ''www.2-ept.com''〕 The most involved operation is the convolution of 2-EPT densities using state space techniques. Much of the work centers on the ability to decompose the rational characteristic function into the sum of two rational functions with poles located in either the open left or open right half plane. The variance-gamma distribution density has been shown to be a 2-EPT density under a parameter restriction and the variance gamma process〔Madan, D., Carr, P., Chang, E. (1998) "The Variance Gamma Process and Option Pricing", ''European Finance Review'' 2: 79–105〕 can be implemented to demonstrate the benefits of adopting such an approach for financial modelling purposes. It can be shown using Parseval's theorem and an isometry that approximating the discrete time rational transform is equivalent to approximating the 2-EPT density itself in the L-2 Norm sense. The rational approximation software RARL2 is used to approximate the discrete time rational characteristic function of the density.〔Olivi, M. (2010) "Parametrization of Rational Lossless Matrices with Applications to Linear System Theory", HDR Thesis 〕 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「2-EPT probability density function」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.