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Kernel density estimation based on progressive type-II censoring

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Abstract

Progressive censoring is essential for researchers in industry as a mean to remove subjects before the final termination point in order to save time and reduce cost. Recently, kernel density estimation has been intensively investigated due to its asymptotic properties and applications. In this paper, we investigate the asymptotic properties of the kernel density estimators based on progressive type-II censoring and their application to hazard function estimation. A bias-adjusted kernel density estimator is also proposed. Our simulation indicates that the kernel density estimates under progressive type-II censoring is competitive compared with kernel density estimates under simple random sampling, depending on the censoring schemes. An example regarding failure times of aircraft windshields is used to illustrate the proposed methods.

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References

  • Alexandre, B. T. (2009). Introduction to nonparametric estimation. New York: Springer.

    MATH  Google Scholar 

  • Balakrishnan, N. (2007). Progressive censoring methodology: An appraisal. Test, 16(2), 211.

    MathSciNet  MATH  Google Scholar 

  • Balakrishnan, N., & Aggarwala, R. (2000). Progressive censoring: Theory, methods, and applications. New York: Springer.

    Google Scholar 

  • Balakrishnan, N., & Cramer, E. (2014). The art of progressive censoring. New York: Birkhauser.

    MATH  Google Scholar 

  • Balakrishnan, N., & Bordes, L. (2003). Nonparametric hazard rate estimation under progressive type-II censoring. Handbook of Statistics, 23, 227–249.

    Google Scholar 

  • Baxter, M. J., Beardah, C. C., & Westwood, S. (2000). Sample size and related issues in the analysis of lead isotope data. Journal of Archaeological Science, 27(10), 973–980.

    Google Scholar 

  • Berg, A., & Politis, D. (2009). CDF and survival function estimation with infinite-order kernels. Electronic Journal of Statistics, 3, 1436–1454.

    MathSciNet  MATH  Google Scholar 

  • Biswas, A., & Sundaram, R. (2006). Kernel survival function estimation based on doubly censored data. Communications in Statistics-Theory and Methods, 35(7), 1293–1307.

    MathSciNet  MATH  Google Scholar 

  • Blischke, W. R., & Murthy, D. P. (2000). Reliability: Modeling. Prediction and optimization. New York: Wiley.

  • Bordes, L. (2004). Nonparametric estimation under progressive censoring. Journal of Statistical Planning and Inference, 119(1), 171–189.

    MathSciNet  MATH  Google Scholar 

  • Bowman, A. W., & Azzalini, A. (1997). Applied smoothing techniques for data analysis: The kernel approach with S-Plus illustrations (Vol. 18). Oxford: OUP Oxford.

    MATH  Google Scholar 

  • Cramer, E. (2014). Extreme value analysis for progressively type-II censored order statistics. Communications in Statistics-Theory and Methods, 43(10–12), 2135–2155.

    MathSciNet  MATH  Google Scholar 

  • DiNardo, J., Fortin, N. M., & Lemieux, T. (1995). Labor market institutions and the distribution of wages, 1973–1992: A semiparametric approach (No. w5093). Cambridge: National Bureau of Economic Research.

  • Ferreyra, R. A., Podestá, G. P., Messina, C. D., Letson, D., Dardanelli, J., Guevara, E., et al. (2001). A linked-modeling framework to estimate maize production risk associated with ENSO-related climate variability in Argentina. Agricultural and Forest Meteorology, 107(3), 177–192.

    Google Scholar 

  • Fix, E., & Hodges, J. L. (1951). Discriminatory analysis-nonparametric discrimination: Consistency properties. Berkeley: California Univ Berkeley.

    MATH  Google Scholar 

  • Jeffrey, S. S. (1996). Smoothing methods in statistics. New York: Springer.

    MATH  Google Scholar 

  • Kamps, U. (1995). A concept of generalized order statistics. Journal of Statistical Planning and Inference, 48(1), 1–23.

    MathSciNet  MATH  Google Scholar 

  • Kamps, U. (1999). Order statistics, generalized. In Encyclopedia of Statistical Sciences.

  • Kim, K. D., & Heo, J. H. (2002). Comparative study of flood quantiles estimation by nonparametric models. Journal of Hydrology, 260(1), 176–193.

    Google Scholar 

  • Lejeune, M., & Sarda, P. (1992). Smooth estimators of distribution and density functions. Computational Statistics & Data Analysis, 14(4), 457–471.

    MathSciNet  MATH  Google Scholar 

  • Lin, C. T., & Balakrishnan, N. (2011). Asymptotic properties of maximum likelihood estimators based on progressive type-II censoring. Metrika, 74, 349–360.

    MathSciNet  MATH  Google Scholar 

  • Nadaraya, E. A. (1974). On the integral mean square error of some nonparametric estimates for the density function. Theory of Probability & Its Applications, 19(1), 133–141.

    Google Scholar 

  • Ng, H. K. T., Chan, P. S., & Balakrishnan, N. (2002). Estimation of parameters from progressively censored data using EM algorithm. Computational Statistics & Data Analysis, 39(4), 371–386.

    MathSciNet  MATH  Google Scholar 

  • Ng, H. K. T., Chan, P. S., & Balakrishnan, N. (2004). Optimal progressive censoring plan for Weibull distribution. Technometrics, 46, 470–481.

    MathSciNet  Google Scholar 

  • Paulsen, O., & Heggelund, P. (1996). Quantal properties of spontaneous EPSCs in neurones of the guinea-pig dorsal lateral geniculate nucleus. The Journal of Physiology, 496(Pt 3), 759.

    Google Scholar 

  • Parzen, E. (1962). On estimation of a probability density function and mode. The Annals of Mathematical Statistics, 33(3), 1065–1076.

    MathSciNet  MATH  Google Scholar 

  • Polansky, A. M., & Baker, E. R. (2000). Multistage plug-in bandwidth selection for kernel distribution function estimates. Journal of Statistical Computation and Simulation, 65(1–4), 63–80.

    MathSciNet  MATH  Google Scholar 

  • Ramlau-Hansen, H. (1983). Smoothing counting process intensities by means of kernel functions. The Annals of Statistics, 1, 453–466.

    MathSciNet  MATH  Google Scholar 

  • Rosenblatt, M. (1971). Curve estimates. The Annals of Mathematical Statistics, 42(6), 1815–1842.

    MathSciNet  MATH  Google Scholar 

  • Samawi, H., Rochani, H., Yin, J., Linder, D., & Vogel, R. (2018). Notes on kernel density based mode estimation using more efficient sampling designs. Computational Statistics, 33, 1–20.

    MathSciNet  MATH  Google Scholar 

  • See, C., & Chen, J. (2008). Inequalities on the variance of convex functions of random variables. Journal of inequalities in pure and applied mathematics, 9(3), 1–5.

    MathSciNet  Google Scholar 

  • Segal, M. R., & Wiemels, J. L. (2002). Clustering of translocation breakpoints. Journal of the American Statistical Association, 97(457), 66–76.

    MathSciNet  MATH  Google Scholar 

  • Simonoff, J. S. (2012). Smoothing methods in statistics. New York: Springer.

    MATH  Google Scholar 

  • Singh, R. S. (1977). Applications of estimators of a density and its derivatives to certain statistical problems. Journal of the Royal Statistical Society Series B (Methodological), 39, 357–363.

    MathSciNet  MATH  Google Scholar 

  • Silverman, B. W. (1986). Density estimation for statistics and data analysis (Vol. 26). New York: CRC Press.

    MATH  Google Scholar 

  • Tortosa-Ausina, E. (2002). Financial costs, operating costs, and specialization of Spanish banking firms as distribution dynamics. Applied Economics, 34(17), 2165–2176.

    Google Scholar 

  • Viveros, R., & Balakrishnan, N. (1994). Interval estimation of parameters of life from progressively censored data. Technometrics, 36(1), 84–91.

    MathSciNet  MATH  Google Scholar 

  • Wand, M. P., & Jones, M. C. (1995). Kernel smoothing. London: Chapman & Hall.

    MATH  Google Scholar 

  • Wolfgang, H., Marlene, M., Stefan, S., & Axel, W. (2004). Nonparametric and semiparametric models. Berlin: Springer.

    MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, The University of Jordan, Amman, Jordan

    Amal Helu

  2. Jiann-Ping Hsu College of Public Health, Georgia Southern University, Statesboro, USA

    Hani Samawi, Haresh Rochani, Jingjing Yin & Robert Vogel

Authors
  1. Amal Helu
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  2. Hani Samawi
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  3. Haresh Rochani
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  4. Jingjing Yin
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  5. Robert Vogel
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Correspondence to Amal Helu.

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Helu, A., Samawi, H., Rochani, H. et al. Kernel density estimation based on progressive type-II censoring. J. Korean Stat. Soc. 49, 475–498 (2020). https://doi.org/10.1007/s42952-019-00022-y

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