Hadeler K P
School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ 85287, USA.
J Math Biol. 2012 Mar;64(4):613-45. doi: 10.1007/s00285-011-0454-0. Epub 2011 Jul 8.
A multitype pair formation model for a one-sex population, without separation, with given type distribution of singles, produces a distribution of pairs with the given type distribution as a marginal distribution. The pair distribution can be seen as a nonnegative symmetric matrix. For this matrix representation formulas have been given years ago and have been widely used. The goal of the paper is to understand these formulas in probabilistic terms and give a meaning to their coefficients. Our approach connects the formulas to the problem of completing a substochastic matrix to a stochastic matrix. In this way the coefficients in the representation formula can be interpreted as preferences and insight can be gained into the set of distributions respecting given preferences. In order to put these questions into a wider perspective, the classical two-sex pair formation models are reviewed and embedded into the class of one-sex models, and dynamic models are designed that yield pair distributions as limit elements.
一个适用于单性别群体的多类型配对形成模型，无隔离，具有给定的单身类型分布，会产生一种配对分布，其边际分布为给定的类型分布。该配对分布可视为一个非负对称矩阵。多年前就已给出针对此矩阵表示的公式，且已被广泛使用。本文的目标是以概率术语理解这些公式，并赋予其系数意义。我们的方法将这些公式与把一个次随机矩阵完备为一个随机矩阵的问题联系起来。通过这种方式，表示公式中的系数可被解释为偏好，并且能够深入了解符合给定偏好的分布集。为了从更广泛的角度看待这些问题，对经典的两性配对形成模型进行了回顾，并将其嵌入到单性别模型类别中，还设计了动态模型，这些模型会产生作为极限元素的配对分布。
