We model an overdispersed count as a dependent measurement, by means of the Negative Binomial distribution. We consider a quantitative covariate that is fixed by design. The expectation of the dependent variable is assumed to be a known function of a linear combination involving the possibly multidimensional covariate and its coefficients. In the NB1-parametrization of the Negative Binomial distribution, the variance is a linear function of the expectation, inflated by the dispersion parameter, and the distribution not a generalized linear model. For the maximum likelihood estimator for all parameters we apply a general result of Bradley and Gart (Biometrika 49:205–214, 1962) to derive weak consistency and asymptotic normality and a technique in Fahrmeir and Kaufmann (Ann Stat 13:342–368, 1985) for strong consistency. To this end, we show (1) how to bound the logarithmic density by a function that is linear in the outcome of the dependent variable, independently of the parameter. Furthermore (2) the positive definiteness of the matrix related to the Fisher information is shown with the Cauchy–Schwarz inequality.

中文翻译：

---

具有固定协变量的负二项式回归的一致性

我们通过负二项分布将过度分散的计数建模为依赖测量。我们考虑一个由设计固定的定量协变量。因变量的期望被假定为一个已知的线性组合函数，该组合包括可能的多维协变量及其系数。在负二项分布的 NB1 参数化中，方差是期望的线性函数，由离散参数膨胀，分布不是广义线性模型。对于所有参数的最大似然估计量，我们应用 Bradley 和 Gart (Biometrika 49:205–214, 1962) 的一般结果来推导出弱一致性和渐近正态性以及 Fahrmeir 和 Kaufmann (Ann Stat 13:342–368, 1985）以获得强一致性。为此，我们展示了 (1) 如何通过与因变量的结果呈线性关系的函数来限制对数密度，而与参数无关。此外 (2) 与 Fisher 信息相关的矩阵的正定性用 Cauchy-Schwarz 不等式表示。