截断随机变量的矩母函数的鞍点逼近
Saddlepoint approximation for moment generating functions of truncated random variables
摘要：
我们考虑的问题近似的矩母函数（MGF）的截断随机变量的MGF的基础（即，未截断的）随机变量。近似的MGF的目的是使鞍点近似的应用程序确定的截断随机变量的某些分布。两个重要的统计应用如下：某些多元累积分布函数的近似;和近似的通过时间分布的离子通道模型，其中包括时间间隔省略。我们推导出截断随机变量的MGF的两种类型的表示。这些表示之一是通过指数倾斜。第二种类型的表示有两个版本，称为指数卷积表示。每一种表示都激发了不同的近似。事实证明，这三个近似值中的每一个在它所适用的情况下都是非常精确的。此外，有一个简单的经验法则来决定在给定的情况下使用哪种近似，如果遵循这一规则，那么我们的数值和理论结果表明，得到的近似将是非常准确的。
英文摘要：
We consider the problem of approximating the moment generating function (MGF) of a truncated random variable in terms of the MGF of the underlying (i.e., untruncated) random variable. The purpose of approximating the MGF is to enable the application of saddlepoint approximations to certain distributions determined by truncated random variables. Two important statistical applications are the following: the approximation of certain multivariate cumulative distribution functions; and the approximation of passage time distributions in ion channel models which incorporate time interval omission. We derive two types of representation for the MGF of a truncated random variable. One of these representations is obtained by exponential tilting. The second type of representation, which has two versions, is referred to as an exponential convolution representation. Each representation motivates a different approximation. It turns out that each of the three approximations is extremely accurate in those cases ``to which it is suited.'' Moreover, there is a simple rule of thumb for deciding which approximation to use in a given case, and if this rule is followed, then our numerical and theoretical results indicate that the resulting approximation will be extremely accurate.
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