Abstract
A plug-in type bandwidth selector is presented for density estimation with truncated and censored data. It is based on a representation of the MISE function obtained in the paper. Rate of convergence and limit distribution are derived for this selector. A bootstrap method is introduced to estimate the MISE whose minimizer is an alternative bandwidth selector. A simulation study was carried out to assess the behavior with small samples. This methodology is applied to a real-data problem consisting of reporting delay of AIDS cases. The almost sure representation of the product-limit estimator is a key tool in our proofs.
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REFERENCES
Arcones, M. A. and Giné, E. (1995). On the law of iterated logarithm for canonical U-statistics and processes, Stochastic Process. Appl., 58, 217-245.
Cao, R. (1993). Bootstrapping the mean integrated squared error, J. Multivariate Anal., 45, 137-160.
Cao, R., Cuevas, A. and González-Manteiga, W. (1994). A comparative study of several smoothing methods in density estimation, Comput. Statist. Data Anal., 17, 153-176.
Dvoretzki, A., Kiefer, J. and Wolfowitz, J. (1956). Asymptotic minimax character of the sample distribution function and of the classical multinomial estimator, Ann. Math. Statist., 27, 642-669.
Gijbels, I. and Wang, J. L. (1993). Strong representations of the survival function estimator for truncated and censored data with applications, J. Multivariate Anal., 47, 210-229.
Hall, P. (1984). Central limit theorem for integrated square error of multivariate nonparametric density estimators, J. Multivariate Anal., 14, 1-16.
Hall, P. and Marron, J. S. (1987). Extent to which least-squares cross-validation minimises integrated square error in nonparametric density estimation, Probab. Theory Related Fields, 74, 567-581.
Jones, M. C., Marron, J. S. and Sheather, S. J. (1996). A brief survey of bandwidth selection for density estimation, J. Amer. Statist. Assoc., 91, 401-407.
Lo, S. H., Mack, Y. P. and Wang, J. L. (1989). Density and hazard rate estimation for censored data via strong representation of the Kaplan-Meier estimator, Probab. Theory Related Fields, 80, 461-473.
Patil, P. N. (1993). Bandwidth choice for nonparametric hazard rate estimation, J. Statist. Plann. Inference, 35, 15-30.
Sánchez Sellero, C., Vázquez, E., González, W., Otero, X. L., Hervada, X., Fernández, E. and Taboada, X. A. (1995). Reporting delay: A review with a simulation study and application to Spanish AIDS data, Statistics in Medicine, 15, 305-321.
Serfling, R. (1980). Approximation Theorems of Mathematical Statistics, Wiley, New York.
Sheather, S. J. and Jones, M. C. (1991). A reliable data-based bandwidth selection method for kernel density estimation, J. Roy. Statist. Soc. Ser. B, 53, 683-690.
Struthers, C. A. and Farewell, V. T. (1989). A mixture model for time to AIDS data with left truncation and an uncertain origin, Biometrika, 76, 814-817.
Stute, W. (1993). Almost sure representations of the product-limit estimator for truncated data, Ann. Statist., 21, 146-156.
Tsai, W. Y., Jewell, N. P. and Wang, M. C. (1987). A note on the product limit estimator under right censoring and left truncation, Biometrika, 74, 883-886.
Uzunogullari, U. and Wang, J. L. (1992). A comparison of hazard rate estimators for left truncated and right censored data, Biometrika, 79, 297-310.
Woodroofe, M. (1985). Estimating a distribution function with truncated data, Ann. Statist., 13, 163-177.
Zhou, Y. (1996) A note on the TJW product-limit estimator for truncated and censored data, Statist. Probab. Lett., 26, 381-387.
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Sánchez-Sellero, C., González-Manteiga, W. & Cao, R. Bandwidth Selection in Density Estimation with Truncated and Censored Data. Annals of the Institute of Statistical Mathematics 51, 51–70 (1999). https://doi.org/10.1023/A:1003879001416
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DOI: https://doi.org/10.1023/A:1003879001416