Reduced bias nonparametric lifetime density and hazard estimation
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Reduced bias nonparametric lifetime density and hazard estimation Arthur Berg1
· Dimitris Politis2 · Kagba Suaray3 · Hui Zeng1
Received: 16 October 2018 / Accepted: 7 August 2019 © Sociedad de Estadística e Investigación Operativa 2019
Abstract Kernel-based nonparametric hazard rate estimation is considered with a special class of infinite-order kernels that achieves favorable bias and mean square error properties. A fully automatic and adaptive implementation of a density and hazard rate estimator is proposed for randomly right censored data. Careful selection of the bandwidth in the proposed estimators yields estimates that are more efficient in terms of overall mean square error performance, and in some cases, a nearly parametric convergence rate is achieved. Additionally, rapidly converging bandwidth estimates are presented for use in second-order kernels to supplement such kernel-based methods in hazard rate estimation. Simulations illustrate the improved accuracy of the proposed estimator against other nonparametric estimators of the density and hazard function. A real data application is also presented on survival data from 13,166 breast carcinoma patients. Keywords Bandwidth estimation · Density estimation · Fourier transform · Hazard function estimation · Infinite-order kernels · Nonparametric estimation · Survival analysis Mathematics Subject Classification 62G07 · 62G20 · 62N99
1 Introduction Hazard rate estimation has been extensively studied in the literature as it encompasses fundamental characteristics of time-to-event data with applications spanning medicine, engineering and economics. The first kernel-based nonparametric estimator of the hazard function with non-censored data appeared in Watson and Leadbetter (1964).
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Arthur Berg [email protected]
1
Division of Biostatistics and Bioinformatics, Penn State College of Medicine, Hershey, PA, USA
2
Department of Mathematics, University of California, La Jolla, CA, USA
3
Department of Mathematics and Statistics, California State University Long Beach, Long Beach, CA, USA
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A. Berg et al.
For censored data, density estimation approaches are described in Földes et al. (1981) and Padgett and McNichols (1984), and an empirical hazard approach is described in Tanner and Wong (1983). Kernel-based estimation of the hazard function under censoring was studied by Yandell (1983), Ramlau-Hansen (1983), Tanner and Wong (1984) and Müller and Wang (1994), among others. However, all of these kernel-based approaches capitalized on traditional theory of second-order kernels when constructing their kernel-based estimates. Through the use of infinite-order kernels, we demonstrate that considerable asymptotic improvements are attainable. The benefit of using infinite-order kernels, also called superkernels, in estimating the probability density function under iid data is well known; cf. Devroye (1992). More recently, Politis and others have investigated a class of infinite-order kernels that are constructed by taking the Fourier transform of flat-top function
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