Presmoothed estimation of the density function with truncated and censored data

Abstract

A new kernel-type density estimator is presented in the context of left truncated and right censored data. This estimator is obtained by the convolution of a kernel with an estimator of the distribution function based on presmoothing ideas. Asymptotic properties, including an i.i.d. representation, consistency, asymptotic normality and mean integrated squared error expressions, are given. The practical performance of this estimator is illustrated in a simulation study and used to analyse the lifetime density in a real data example.

Keywords:

• almost sure representation
• kernel-type density estimator
• LTRC data
• presmoothing
• survival analysis

2000 Mathematics Subject Classifications :

• 62G07
• 62N01

Acknowledgements

We are very grateful to Professor Per Kragh Andersen (Department of Biostatistics, University of Copenhagen) for providing the Fyn diabetes data, and collected by Dr. Anders Green. We also acknowledge the economic support of the Grant MTM2005-00429 of the Spanish Ministerio de Educación y Ciencia, Grant MTM2008-00166 of the Spanish Ministerio de Ciencia e Innovación” and XUGA Grant PGIDT03PXIC10505PN for the first author, and Grant MTM2005-01274 of the Spanish Ministerio de Educación y Ciencia, Grant MTM2008-03129 of the Spanish Ministerio de Ciencia e Innovación and XUGA Grant PGIDIT07PXIB300191PR for the second author.