Published

2015-07-01

Global Polynomial Kernel Hazard Estimation

Ajuste polinomial global para la estimación kernel de la función de riesgo

DOI:

https://doi.org/10.15446/rce.v38n2.51668

Keywords:

Kernel Estimation, Hazard Function, Local Linear Estimation, Boundary Kernels, Polynomial Correction (en)
Estimación kernel, Funciones de riesgo, Estimación local lineal, Kernels de frontera, Corrección polinomial. (es)

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Authors

  • Munir Hiabu Cass Business School, City University London, United Kingdom
  • María Dolores Martínez-Miranda University of Granada, Spain
  • Jens Perch Nielsen Cass Business School, City University London, United Kingdom
  • Jaap Spreeuw Cass Business School, City University London, United Kingdom
  • Carsten Tanggaard CREATES, Aarhus University, Denmark
  • Andrés M. Villegas Cass Business School, City University London, United Kingdom

This paper introduces a new bias reducing method for kernel hazard estimation. The method is called global polynomial adjustment (GPA). It is a global correction which is applicable to any kernel hazard estimator. The estimator works well from a theoretical point of view as it asymptotically reduces bias with unchanged variance. A simulation study investigates the finite-sample properties of GPA. The method is tested on local constant and local linear estimators. From the simulation experiment we conclude that the global estimator improves the goodness-of-fit. An especially encouraging result is that the bias-correction works well for small samples, where traditional bias reduction methods have a tendency to fail.

En este artículo se introduce un nuevo método de correción del sesgo para la estimación núcleo de la función de riesgo. El método, denominado ajuste polinomial global (APG), consiste en una corrección global que es aplicable a cualquier tipo de estimador núcleo de la función de riesgo. Se comprueba que APG posee buenas propiedades asintóticas y que consigue reducir el sesgo sin incrementar la varianza. Se realizan estudios de simulación para evaluar las propiedades del APG en muestras finitas. Dichos estudios muestran un buen comportamiento en la práctica del APG. Esto es especialmente alentador dado que para muestras finitas los métodos tradicionales de reducción del sesgo tienden a tener un comportamiento bastante pobre.

https://doi.org/10.15446/rce.v38n2.51668

Global Polynomial Kernel Hazard Estimation

Ajuste polinomial global para la estimación kernel de la función de riesgo

MUNIR HIABU1, MARÍA DOLORES MARTÍNEZ-MIRANDA2, JENS PERCH NIELSEN3, JAAP SPREEUW4, CARSTEN TANGGAARD5, ANDRÉS M. VILLEGAS6

1City University London, Cass Business School, Faculty of Actuarial Science and Insurance, United Kingdom. Ph.D. Student. Email: Munir.Hiabu.1@cass.city.ac.uk
2City University London, Cass Business School, Faculty of Actuarial Science and Insurance, United Kingdom. University of Granada, Faculty of Sciences, Department of Statistics and O.R., Spain. Associate Professor. Email: mmiranda@ugr.es
3City University London, Cass Business School, Faculty of Actuarial Science and Insurance, United Kingdom. Professor. Email: Jens.Nielsen.1@city.ac.uk
4City University London, Cass Business School, Faculty of Actuarial Science and Insurance, United Kingdom. Senior Lecturer. Email: J.Spreeuw@city.ac.uk
5Aarhus University, CREATES, Denmark. Professor. Email: ctanggaard@creates.au.dk
6City University London, Cass Business School, Faculty of Actuarial Science and Insurance, United Kingdom. Ph.D. Student. Email: Andres.Villegas.1@cass.city.ac.uk

Abstract

This paper introduces a new bias reducing method for kernel hazard estimation. The method is called global polynomial adjustment (GPA). It is a global correction which is applicable to any kernel hazard estimator. The estimator works well from a theoretical point of view as it asymptotically reduces bias with unchanged variance. A simulation study investigates the finite-sample properties of GPA. The method is tested on local constant and local linear estimators. From the simulation experiment we conclude that the global estimator improves the goodness-of-fit. An especially encouraging result is that the bias-correction works well for small samples, where traditional bias reduction methods have a tendency to fail.

Key words: Kernel Estimation, HazardFunction, Local Linear Estimation, Boundary Kernels, Polynomial Correction.

Resumen

En este artículo se introduce un nuevo método de correción del sesgo para la estimación núcleo de la función de riesgo. El método, denominado ajuste polinomial global (APG), consiste en una corrección global que es aplicable a cualquier tipo de estimador núcleo de la función de riesgo. Se comprueba que APG posee buenas propiedades asintóticas y que consigue reducir el sesgo sin incrementar la varianza. Se realizan estudios de simulación para evaluar las propiedades del APG en muestras finitas. Dichos estudios muestran un buen comportamiento en la práctica del APG. Esto es especialmente alentador dado que para muestras finitas los métodos tradicionales de reducción del sesgo tienden a tener un comportamiento bastante pobre.

Palabras clave: estimación kernel, funciones de riesgo, estimación local lineal, kernels de frontera, corrección polinomial.

Texto completo disponible en PDF

References

1. Andersen, P., Borgan, O., Gill, R. & Keiding, N. (1993), Statistical Models Based on Counting Processes, Springer, New York.

2. Copas, J. B. (1995), 'Local likelihood based on kernel censoring', Journal of the Royal Statistical Society, Series B 57(1), 221-235.

3. Efron, B. & Tibshirani, R. (1996), 'Using specially designed exponential families for density estimation', Annals of Statistics 24(6), 2431-2461.

4. Eguchi, S. & Copas, J. (1998), 'A class of local likelihood methods and near-parametric asymptotics', Journal of the Royal Statistical Society, Series B 60(4), 709-724.

5. Gámiz Pérez, M. L., Janys, L., Martínez Miranda, M. D. & Nielsen, J. P. (2013), 'Bandwidth selection in marker dependent kernel hazard estimation', Computational Statistics & Data Analysis 68, 155-169.

6. Gámiz Pérez, M. L., Martínez Miranda, M. D. & Nielsen, J. P. (2013), 'Smoothing survival densities in practice', Computational Statistics & Data Analysis 58, 368-382.

7. Hjort, N. L. & Glad, I. K. (1995), 'Nonparametric density estimation with a parametric start', Annals of Statistics 23(3), 882-904.

8. Hjort, N. L. & Jones, M. C. (1996), 'Locally parametric nonparametric density estimation', Annals of Statistics 24(4), 1619-1647.

9. Jones, M. C., Linton, O. B. & Nielsen, J. P. (1995), 'A simple bias reduction method for density estimation', Biometrika 82(2), 327-338.

10. Jones, M. C. & Signorini, D. F. (1999), 'A comparison of higher-order bias kernel density estimators', Journal of the American Statistical Association 439, 1063-1073.

11. Jones, M. C., Signorini, D. F. & Hjort, N. L. (1999), 'On multiplicative bias correction in kernel density estimation', Sankhya 61(1), 422-430.

12. Koul, H. L. & Song, W. (2013), 'Large sample results for varying kernel regression estimates', Journal of Nonparametric Statistics 25(4), 829-853.

13. Lemonte, A., Martínez-Florez, G. & Moreno-Arenas, G. (2014), 'Multivariate Birnbaum-Saunders distribution: Properties and associated inference', Journal of Statistical Computation and Simulation 85(2), 374-392.

14. Loader, C. R. (1996), 'Local likelihood density estimation', Annals of Statistics 24(4), 1602-1618.

15. Martínez-Flórez, G., Moreno-Arenas, G. & Vergara-Cardozo, S. (2013), 'Properties and inference for proportional hazard models', Revista Colombiana de Estadística 36(1), 95-114.

16. Nielsen, J. P. (1998), 'Multiplicative bias correction in kernel hazard estimation', Scandinavian Journal of Statistics 25(3), 541-553.

17. Nielsen, J. P. (2003), 'Variable bandwidth kernel hazard estimators', Journal of Nonparametric Statistics 15(3), 355-376.

18. Nielsen, J. P. & Linton, O. (1995), 'Kernel estimation in a nonparametric marker dependent hazard model', Annals of Statistics 23(5), 1735-1748.

19. Nielsen, J. P. & Tanggaard, C. (2001), 'Boundary and bias correction in kernel hazard estimation', Scandinavian Journal of Statistics 28(4), 675-698.

20. Nielsen, J. P., Tanggaard, C. & Jones, M. C. (2009), 'Local linear density estimation for filtered survival data', Statistics 43(2), 167-186.

21. Otneim, H., Karlsen, H. A. & Tjøstheim, D. (2013), 'Bias and bandwidth for local likelihood density estimation', Statistics and Probability Letters 83(4), 1382-1387.

22. Ramlau-Hansen, H. (1983), 'Smoothing counting process intensities by means of kernel functions', Annals of Statistics 11(2), 453-466.

23. Salinas, V., Pérez, P., González, E. & Vaquera, H. (2012), 'Goodness of fit tests for the Gumbel distribution with type II right censored data', Revista Colombiana de Estadística 35(3), 407-424.

24. Spreeuw, J., Nielsen, J. P. & Jarner, S. F. (2013), 'A nonparametric visual test of mixed hazard models', SORT 37(2), 153-174.

[Recibido en diciembre de 2013. Aceptado en enero de 2015]

Este artículo se puede citar en LaTeX utilizando la siguiente referencia bibliográfica de BibTeX:

@ARTICLE{RCEv38n2a06,
    AUTHOR  = {Hiabu, Munir and Martínez-Miranda, María Dolores and Nielsen, Jens Perch and Spreeuw, Jaap and Tanggaard, Carsten and Villegas, Andrés M.},
    TITLE   = {{Global Polynomial Kernel Hazard Estimation}},
    JOURNAL = {Revista Colombiana de Estadística},
    YEAR    = {2015},
    volume  = {38},
    number  = {2},
    pages   = {399-411}
}

References

Andersen, P., Borgan, O., Gill, R. & Keiding, N. (1993), Statistical Models Based on Counting Processes, Springer, New York.

Copas, J. B. (1995), ‘Local likelihood based on kernel censoring’, Journal of the Royal Statistical Society, Series B 57(1), 221–235.

Efron, B. & Tibshirani, R. (1996), ‘Using specially designed exponential families for density estimation’, Annals of Statistics 24(6), 2431–2461.

Eguchi, S. & Copas, J. (1998), ‘A class of local likelihood methods and near-parametric asymptotics’, Journal of the Royal Statistical Society, Series B 60(4), 709–724.

Gámiz Pérez, M. L., Janys, L., Martínez Miranda, M. D. & Nielsen, J. P. (2013), ‘Bandwidth selection in marker dependent kernel hazard estimation’, Computational Statistics & Data Analysis 68, 155–169.

Gámiz Pérez, M. L., Martínez Miranda, M. D. & Nielsen, J. P. (2013), ‘Smoothing survival densities in practice’, Computational Statistics & Data Analysis 58, 368–382.

Hjort, N. L. & Glad, I. K. (1995), ‘Nonparametric density estimation with a parametric start’, Annals of Statistics 23(3), 882–904.

Hjort, N. L. & Jones, M. C. (1996), ‘Locally parametric nonparametric density estimation’, Annals of Statistics 24(4), 1619–1647.

Jones, M. C., Linton, O. B. & Nielsen, J. P. (1995), ‘A simple bias reduction method for density estimation’, Biometrika 82(2), 327–338.

Jones, M. C. & Signorini, D. F. (1999), ‘A comparison of higher-order bias kernel density estimators’, Journal of the American Statistical Association 439, 1063–1073.

Jones, M. C., Signorini, D. F. & Hjort, N. L. (1999), ‘On multiplicative bias correction in kernel density estimation’, Sankhya 61(1), 422–430.

Koul, H. L. & Song, W. (2013), ‘Large sample results for varying kernel regression estimates’, Journal of Nonparametric Statistics 25(4), 829–853.

Lemonte, A., Martínez-Florez, G. & Moreno-Arenas, G. (2014), ‘Multivariate Birnbaum-Saunders distribution: Properties and associated inference’, Journal of Statistical Computation and Simulation 85(2), 374–392.

Loader, C. R. (1996), ‘Local likelihood density estimation’, Annals of Statistics 24(4), 1602–1618.

Martínez-Flórez, G., Moreno-Arenas, G. & Vergara-Cardozo, S. (2013), ‘Properties and inference for proportional hazard models’, Revista Colombiana de Estadística 36(1), 95–114.

Nielsen, J. P. (1998), ‘Multiplicative bias correction in kernel hazard estimation’, Scandinavian Journal of Statistics 25(3), 541–553.

Nielsen, J. P. (2003), ‘Variable bandwidth kernel hazard estimators’, Journal of Nonparametric Statistics 15(3), 355–376.

Nielsen, J. P. & Linton, O. (1995), ‘Kernel estimation in a nonparametric marker dependent hazard model’, Annals of Statistics 23(5), 1735–1748.

Nielsen, J. P. & Tanggaard, C. (2001), ‘Boundary and bias correction in kernel hazard estimation’, Scandinavian Journal of Statistics 28(4), 675–698.

Nielsen, J. P., Tanggaard, C. & Jones, M. C. (2009), ‘Local linear density estimation for filtered survival data’, Statistics 43(2), 167–186.

Otneim, H., Karlsen, H. A. & Tjøstheim, D. (2013), ‘Bias and bandwidth for local likelihood density estimation’, Statistics and Probability Letters 83(4), 1382– 1387.

Ramlau-Hansen, H. (1983), ‘Smoothing counting process intensities by means of kernel functions’, Annals of Statistics 11(2), 453–466.

Salinas, V., Pérez, P., González, E. & Vaquera, H. (2012), ‘Goodness of fit tests for the gumbel distribution with type II right censored data’, Revista Colombiana de Estadística 35(3), 407–424.

Spreeuw, J., Nielsen, J. P. & Jarner, S. F. (2013), ‘A nonparametric visual test of mixed hazard models’, SORT 37(2), 153–174.

How to Cite

APA

Hiabu, M., Martínez-Miranda, M. D., Nielsen, J. P., Spreeuw, J., Tanggaard, C. and Villegas, A. M. (2015). Global Polynomial Kernel Hazard Estimation. Revista Colombiana de Estadística, 38(2), 399–411. https://doi.org/10.15446/rce.v38n2.51668

ACM

[1]
Hiabu, M., Martínez-Miranda, M.D., Nielsen, J.P., Spreeuw, J., Tanggaard, C. and Villegas, A.M. 2015. Global Polynomial Kernel Hazard Estimation. Revista Colombiana de Estadística. 38, 2 (Jul. 2015), 399–411. DOI:https://doi.org/10.15446/rce.v38n2.51668.

ACS

(1)
Hiabu, M.; Martínez-Miranda, M. D.; Nielsen, J. P.; Spreeuw, J.; Tanggaard, C.; Villegas, A. M. Global Polynomial Kernel Hazard Estimation. Rev. colomb. estad. 2015, 38, 399-411.

ABNT

HIABU, M.; MARTÍNEZ-MIRANDA, M. D.; NIELSEN, J. P.; SPREEUW, J.; TANGGAARD, C.; VILLEGAS, A. M. Global Polynomial Kernel Hazard Estimation. Revista Colombiana de Estadística, [S. l.], v. 38, n. 2, p. 399–411, 2015. DOI: 10.15446/rce.v38n2.51668. Disponível em: https://revistas.unal.edu.co/index.php/estad/article/view/51668. Acesso em: 22 jan. 2025.

Chicago

Hiabu, Munir, María Dolores Martínez-Miranda, Jens Perch Nielsen, Jaap Spreeuw, Carsten Tanggaard, and Andrés M. Villegas. 2015. “Global Polynomial Kernel Hazard Estimation”. Revista Colombiana De Estadística 38 (2):399-411. https://doi.org/10.15446/rce.v38n2.51668.

Harvard

Hiabu, M., Martínez-Miranda, M. D., Nielsen, J. P., Spreeuw, J., Tanggaard, C. and Villegas, A. M. (2015) “Global Polynomial Kernel Hazard Estimation”, Revista Colombiana de Estadística, 38(2), pp. 399–411. doi: 10.15446/rce.v38n2.51668.

IEEE

[1]
M. Hiabu, M. D. Martínez-Miranda, J. P. Nielsen, J. Spreeuw, C. Tanggaard, and A. M. Villegas, “Global Polynomial Kernel Hazard Estimation”, Rev. colomb. estad., vol. 38, no. 2, pp. 399–411, Jul. 2015.

MLA

Hiabu, M., M. D. Martínez-Miranda, J. P. Nielsen, J. Spreeuw, C. Tanggaard, and A. M. Villegas. “Global Polynomial Kernel Hazard Estimation”. Revista Colombiana de Estadística, vol. 38, no. 2, July 2015, pp. 399-11, doi:10.15446/rce.v38n2.51668.

Turabian

Hiabu, Munir, María Dolores Martínez-Miranda, Jens Perch Nielsen, Jaap Spreeuw, Carsten Tanggaard, and Andrés M. Villegas. “Global Polynomial Kernel Hazard Estimation”. Revista Colombiana de Estadística 38, no. 2 (July 1, 2015): 399–411. Accessed January 22, 2025. https://revistas.unal.edu.co/index.php/estad/article/view/51668.

Vancouver

1.
Hiabu M, Martínez-Miranda MD, Nielsen JP, Spreeuw J, Tanggaard C, Villegas AM. Global Polynomial Kernel Hazard Estimation. Rev. colomb. estad. [Internet]. 2015 Jul. 1 [cited 2025 Jan. 22];38(2):399-411. Available from: https://revistas.unal.edu.co/index.php/estad/article/view/51668

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