Consistency for the negative binomial regression with fixed covariate
Consistency for the negative binomial regression with fixed covariate
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DOI：
10.1007/s00184-019-00750-5
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发表时间：
2019-10-24
期刊：
影响因子：
0.7
通讯作者：
Radloff, Lucas
中科院分区：
文献类型：
--
作者：
Weissbach, Rafael;Radloff, Lucas
我们通过负二项分布将过度分散的计数建模为相关测量。我们考虑一个定量协变量，是固定的设计。因变量的期望被假设为涉及可能的多维协变量及其系数的线性组合的已知函数。在负二项分布的NB 1-参数化中，方差是期望的线性函数，由离差参数膨胀，并且分布不是广义线性模型。对于所有参数的最大似然估计，我们应用布拉德利和加特（Biometrika 49：205-214，1962）的一般结果来推导弱一致性和渐近正态性，并应用Fahrmeir和Kaufmann（Ann Stat 13：342-368，1985）中的技术来推导强一致性。为此，我们展示了（1）如何通过一个函数来约束对数密度，该函数在因变量的结果中是线性的，与参数无关。（2）利用Cauchy-Schwarz不等式证明了与Fisher信息有关的矩阵的正定性。
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.