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Two likelihood-based semiparametric estimation methods for panel count data with covariates. (English) Zbl 1126.62084

Summary: We consider estimation in a particular semiparametric regression model for the mean of a counting process with “panel count” data. The basic model assumption is that the conditional mean function of the counting process is of the form \(E\{\mathbb N (t)|Z\}=\exp(\beta _{0}^TZ)\Lambda _{0}(t)\) where \(Z\) is a vector of covariates and \(\Lambda _{0}\) is the baseline mean function. The “panel count” 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 across individuals.
We study semiparametric maximum pseudo-likelihood and maximum likelihood estimators of the unknown parameters \((\beta _{0}, \Lambda _{0})\) derived on the basis of a nonhomogeneous Poisson process assumption. The pseudo-likelihood estimator is fairly easy to compute, while the maximum likelihood estimator poses more challenges from the computational perspective. We study asymptotic properties of both estimators assuming that the proportional mean model holds, but dropping the Poisson process assumption used to derive the estimators. In particular we establish asymptotic normality for the estimators of the regression parameter \(\beta _{0}\) under appropriate hypotheses. The results show that our estimation procedures are robust in the sense that the estimators converge to the truth regardless of the underlying counting process.

MSC:

62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
62G08 Nonparametric regression and quantile regression
62M09 Non-Markovian processes: estimation
60F05 Central limit and other weak theorems
60F17 Functional limit theorems; invariance principles
62F12 Asymptotic properties of parametric estimators

Software:

Mathematica
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References:

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