We consider estimation in a particular semiparametric regression model forthe mean of a counting process with ``panel count'' data. The basic modelassumption is that the conditional mean function of the counting process is ofthe form $E\{\mathbb{N}(t)|Z\}=\exp(\beta_0^TZ)\Lambda_0(t)$ where $Z$ is avector of covariates and $\Lambda_0$ is the baseline mean function. The ``panelcount'' observation scheme involves observation of the counting process$\mathbb{N}$ for an individual at a random number $K$ of random time points;both the number and the locations of these time points may differ acrossindividuals. We study semiparametric maximum pseudo-likelihood and maximumlikelihood estimators of the unknown parameters $(\beta_0,\Lambda_0)$ derivedon the basis of a nonhomogeneous Poisson process assumption. Thepseudo-likelihood estimator is fairly easy to compute, while the maximumlikelihood estimator poses more challenges from the computational perspective.We study asymptotic properties of both estimators assuming that theproportional mean model holds, but dropping the Poisson process assumption usedto derive the estimators. In particular we establish asymptotic normality forthe estimators of the regression parameter $\beta_0$ under appropriatehypotheses. The results show that our estimation procedures are robust in thesense that the estimators converge to the truth regardless of the underlyingcounting process.
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机译：我们考虑在特定的半参数回归模型中对具有``面板计数''数据的计数过程的平均值进行估计。基本模型假设是计数过程的条件均值函数形式为$ E \ {\ mathbb {N}（t）| Z \} = \ exp（\ beta_0 ^ TZ）\ Lambda_0（t）$，其中$ Z $是协变量的向量，$ \ Lambda_0 $是基线均值函数。 ``面板计数''观察方案涉及对随机时间点的随机数$ K $的个体的计数过程$ \ mathbb {N} $的观察;这些时间点的数量和位置可能因人而异。我们研究基于非齐次Poisson过程假设得出的未知参数$（\ beta_0，\ Lambda_0）$的半参数最大拟似然性和最大似然估计。伪似然估计量很容易计算，而最大似然估计量则从计算角度提出了更大的挑战。我们假设比例均值模型成立，但研究了两个估计量的渐近性质，但舍弃了用于推导估计量的泊松过程假设。特别地，我们在适当的假设下为回归参数$ \ beta_0 $的估计量建立渐近正态性。结果表明，我们的估计程序具有鲁棒性，因为无论基础计算过程如何，估计器都可以收敛到真相。
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