We propose a new joint mean and correlation regression model for correlated multivariate discrete responses, that simultaneously regresses the mean of each response against a set of covariates, and the correlations between responses against a set of similarity/distance measures. A set of joint estimating equations are formulated to construct an estimator of both the mean regression coefficients and the correlation regression parameters. Under a general setting where the number of responses can tend to infinity, the joint estimator is demonstrated to be consistent and asymptotically normally distributed, with differing rates of convergence due to the mean regression coefficients being heterogeneous across responses. An iterative estimation procedure is developed to obtain parameter estimates in the required, constrained parameter space. We apply the proposed model to a multivariate abundance dataset comprising overdispersed counts of 38 Carabidae ground beetle species sampled throughout Scotland, along with information about the environmental conditions of each site and the traits of each species. Results show in particular that the relationships between the mean abundances of various beetle species and environmental covariates are different and that beetle total length has statistically important effect in driving the correlations between the species. Simulations demonstrate the strong finite sample performance of the proposed estimator in terms of point estimation and inference.

翻译：我们提出一种针对相关多元离散响应的新型联合均值与相关回归模型，该模型同时将各响应的均值对协变量集进行回归，并将响应间的相关性对相似性/距离度量集进行回归。通过构建一组联合估计方程，我们构造了均值回归系数与相关回归参数的联合估计量。在响应数量可趋于无穷的一般设定下，该联合估计量被证明具有相合性与渐近正态性，且由于均值回归系数在不同响应间存在异质性，其收敛速度存在差异。我们开发了一种迭代估计程序，以在所需的约束参数空间中获取参数估计值。将该模型应用于苏格兰全境采集的38种步甲科地面甲虫物种的过离散计数多变量丰度数据集，并包含各地点环境条件与各物种性状信息。结果表明：不同甲虫物种的平均丰度与环境协变量之间的关系存在差异，且甲虫体长对驱动物种间相关性具有统计显著影响。模拟实验展示了所提估计量在点估计与推断方面的强有限样本性能。