标题： 数学统计中的符号微积分：综述 显示英文标题

标题： Symbolic Calculus in Mathematical Statistics: A Review

摘要： 在过去十年中，符号方法的应用大大扩展了统计学和概率论的理论和应用。 本综述回顾了由Rota和Taylor在$1994.$中引入的经典小标题微积分中出现的一种符号技术的发展。 这种符号技术的用处有两方面。 第一是展示如何通过新的代数恒等式来发现表面上彼此非常遥远且与概率和统计相关的主题之间的洞察力。 其中一个主要工具是对相同概率分布卷积的形式化推广，这使我们能够在仅有一些相互关联的主题中使用复合泊松随机变量。 有了不同的和更深入的视角，第二个目标是展示如何建立执行高效的代数计算的算法过程。 特别是，寻找这些符号过程的挑战将导致一种新方法，并提出涉及计算和概念问题的新问题。 将在统计推断、参数估计、Lévy过程以及更一般地涉及多元函数的问题中展示这种符号方法的效率证据。 Sheffer多项式序列的符号表示使我们能够建立经典、布尔和自由累积量的统一理论。 随机矩阵内的最近联系扩展了符号方法的应用。 显示英文摘要

摘要： In the last ten years, the employment of symbolic methods has substantially extended both the theory and the applications of statistics and probability. This survey reviews the development of a symbolic technique arising from classical umbral calculus, as introduced by Rota and Taylor in $1994.$ The usefulness of this symbolic technique is twofold. The first is to show how new algebraic identities drive in discovering insights among topics apparently very far from each other and related to probability and statistics. One of the main tools is a formal generalization of the convolution of identical probability distributions, which allows us to employ compound Poisson random variables in various topics that are only somewhat interrelated. Having got a different and deeper viewpoint, the second goal is to show how to set up algorithmic processes performing efficiently algebraic calculations. In particular, the challenge of finding these symbolic procedures should lead to a new method, and it poses new problems involving both computational and conceptual issues. Evidence of efficiency in applying this symbolic method will be shown within statistical inference, parameter estimation, L\'evy processes, and, more generally, problems involving multivariate functions. The symbolic representation of Sheffer polynomial sequences allows us to carry out a unifying theory of classical, Boolean and free cumulants. Recent connections within random matrices have extended the applications of the symbolic method.