Inequalty Constrained Minimum Density Power Divergence Estimation in Panel Count Data

Abstract

Analysis of panel count data has drawn considerable amount of attention in the literature. Panel count data is common when the study subjects are exposed to recurrent events, observed only at discrete time points, instead of being observed continuously in time. They might occur when continuous monitoring of the subjects under study is either too expensive or unfeasible. In practice, we see frequent application of panel count data in medical research, industrial applications, social sciences. In the literature, it is common practice to characterize the occurrences of recurrent events by leveraging counting processes. Further, due to incomplete nature of the panel count data, it is more convenient to model the mean function of the counting processes. Moreover, each subject under study might get exposed to multiple recurrent events where the correlation among the events reflects individual-specific heterogeneity. In this work, we develop inferential analysis of panel count data where each subject under study is exposed to multiple recurrent events characterized by non-homogeneous Poisson processes correlated through a frailty variable.

Conventionally, in the literature, most of the model estimation works under panel count data, are based on maximum likelihood estimation methods. However, with small deviations from the assumed model conditions, those methods tend to lead biased and inefficient estimates arising the need of robust estimation methods. In this work, our objective is to perform the statistical inference based on multivariate panel count data applying the robust density power divergence estimation methods. The minimum density power divergence estimation (MDPDE) was first proposed by Basu et al. [1998] and thereafter used in plenty of works. Additionally, upon reviewing the literature, it is found that none of the existing studies on panel count data did address robust methods for parameter estimation.

Further, in real life scenario, it is often evident that model parameter space is imposed with some constraints. There have been limited studies on restricted minimum density power divergence estimators, with all existing works focusing exclusively on equality constraints. For instance, Basu et al. [2018], Felipe et al. [2023], proposed equality constraints within the parameter space and subsequently obtained the restricted minimum density power divergence estimators. A significant contribution of our study is the novel incorporation of inequality constraints into the parameter space. The principal novelty in this work lies in deriving the asymptotic distribution of the restricted minimum density power divergence estimators while incorporating this inequality restriction.

Moreover, the MDPDE is characterized by a tuning parameter $\gamma$, which controls the trade-off between robustness and efficiency. In the literature, Warwick and Jones [2005] was the first to propose a data-driven algorithm for determining the optimal value of the tuning parameter, which has since been studied and refined by various authors. Recently, Sugasawa and Yonekura [2021] and Yonekura and Sugasawa [2023] delved into this problem by exploiting Hyvärinen score-matching method. But this approach is restricted to continuous-valued data, where the densities have to be differentiable. In the literature, Lyu [2012] demonstrated a generalized score-matching approach For discrete data, by incorporating a marginalization operator. This study was further extended by Jiazhen et al. [2022] for discrete INID multivariate data, where a general linear operator was used, in place of the gradient operator. In our study, we leverage this generalized score-matching method to find the optimal value of the tuning parameter $\gamma .$ Further a comparative study was executed to access the performances of some existing methods and generalized score-matching method.

• Department of Mathematics and Computing, IIT(Indian School of Mines) Dhanbad, Jharkhand, India [shuvasri29@iitism.ac.in]

• Department of Mathematics and Computing, IIT(Indian School of Mines) Dhanbad, Jharkhand, India [23dr0196@iitism.ac.in]

Keywords: Asymptotic properties, – Inequality constraints, – Minimum density power divergence, – Optimal tuning parameter, – Panel count data, – Restricted Estimation,