Nonparametric Recursive Kernel Type Eestimators for the Moment Generating Function Under Censored Data
AbstractWe are mainly concerned with kernel-type estimators for the moment-generating function in the present paper. More precisely, we establish the central limit theorem with the characterization of the bias and the variance for the nonparametric recursive kernel-type estimators for the moment-generating function under some mild conditions in the censored data setting. Finally, we investigate the methodology’s performance for small samples through a short simulation study.
Akaike, H. (1954). An approximation to the characteristic function. Ann. Inst. Statist. Math., Tokyo, 6, 127–132.
Bojanic, R. and Seneta, E. (1973). A unified theory of regularly Varying sequences. Math. Z., 134, 91–106.
Bouzebda, S. and El-hadjali, T. (2020). Uniform convergence rate of the kernel regression estimator adaptive to intrinsic dimension in presence of censored data. J. Nonparametr. Stat., 32(4), 864–914.
Bouzebda, S. and Nemouchi, B. (2020). Uniform consistency and uniform in bandwidth consistency for nonparametric regression estimates and conditional U-statistics involving functional data. J. Nonparametr. Stat., 32(2), 452–509.
Bouzebda, S. and Slaoui, Y. (2020). Nonparametric recursive method for kernel-type function estimators for censored data. J. Stoch.Anal., 1(3), Art. 4, 19.
Bouzebda, S. and Slaoui, Y. (2022). Nonparametric recursive method for moment generating function kernel-type estimators. Statist.Probab. Lett., page 109422.
Bouzebda, S. and Soukarieh, I. (2023a). Non-parametric conditional U-processes for locally stationary functional random fields under stochastic sampling design. Mathematics, 11(1), 1–70.
Bouzebda, S., Nezzal, A., and Zari, T. (2023b). Uniform consistency for functional conditional U-statistics using delta-sequences.Mathematics, 11(1), 1–39.
Clarkson, J. A. and Adams, C. R. (1933). On definitions of bounded Variation for functions of two Variables. Trans. Amer. Math. Soc.,35(4), 824–854.
Cs¨org˝o, S. and Welsh, A. H. (1989). Testing for exponential and Marshall-Olkin distributions. J. Statist. Plann. Inference, 23(3),287–300.
Deheuvels, P. and Einmahl, J. H. J. (2000). Functional limit laws for the increments of Kaplan-Meier product-limit processes and applications. Ann. Probab., 28(3), 1301–1335.
Devroye, L. (1987). A course in characteristic estimation, volume 14 of Progress in Probability and Statistics. Birkh¨auser Boston Inc., Boston, MA.
Duflo, M. (1997). Random iterative models. Collection Applications of Mathematics, Springer, Berlin.
Eggermont, P. P. B. and LaRiccia, V. N. (2001). Maximum penalized likelihood estimation. Vol. I. Springer Series in Statistics.Springer-Verlag, New York. characteristic estimation.
Epaneˇcnikov, V. A. (1969). Nonparametric estimation of a multidimensional probability density. Teor. Verojatnost. i Primenen., 14,156–162.
Epps, T. W., Singleton, K. J., and Pulley, L. B. (1982). A test of separate families of distributions based on the empirical moment generating function. Biometrika, 69(2), 391–399.
F¨oldes, A. and Rejt˝o, L. (1981). A LIL type result for the product limit estimator. Z. Wahrsch. Verw. Gebiete, 56(1), 75–86.
Galambos, J. and Seneta, E. (1973). Regularly Varying sequences. Proc. Amer. Math. Soc., 41, 110–116.
Gbur, E. E. and Collins, R. A. (1989). Estimation of the moment generating function. Comm. Statist. Simulation Comput., 18(3),1113–1134.
Henze, N. and Visagie, J. (2020). Testing for normality in any dimension based on a partial differential equation involving the moment generating function. Ann. Inst. Statist. Math., 72(5), 1109–1136.
Kaplan, E. L. and Meier, P. (1958). Nonparametric estimation from incomplete observations. J. Amer. Statist. Assoc., 53, 457–481.
Kushner and Yin (2003). Stochastic approximation and recursive algorithms and applications, Stoch. Model. Appl. Probab., 35 Springer.
Ljung, L. (1978). Strong convergence of a stochastic approximation algorithm, Ann. Statist. 6, 680–696.
Meintanis, S. G. (2007). A Kolmogorov-Smirnov type test for skew normal distributions based on the empirical moment generating function. J. Statist. Plann. Inference, 137(8), 2681–2688.
Mokkadem, A. and Pelletier, M. (2007). Compact law of the iterated logarithm for matrix-normalized sums of random vectors. Theory Probab. Appl., 52(4), 2459–2478.
Mokkadem, A. Pelletier, M. and Slaoui, Y. (2009a). The stochastic approximation method for the estimation of a multiVariate probability density. J. Statist. Plann. Inference, 139, 2459–2478.
Mokkadem, A. Pelletier, M. and Slaoui, Y. (2009b). Revisiting R´ev´esz’s stochastic approximation method for the estimation of a regression function. ALEA. Lat. Am. J. Probab. Math. Stat., 6, 63–114.
Nadaraya, E. (1964). On estimating regression. Theory of Probability & Its Applications, 9(1), 141–142.
Parzen, E. (1962). On estimation of a probability characteristic function and mode. Ann. Math. Statist., 33, 1065–1076.
Quandt, R. E. and Ramsey, J. B. (1978). Estimating mixtures of normal distributions and switching regressions. J. Amer. Statist. Assoc., 73(364), 730–752. With comments and a rejoinder by the authors.
Reid, N. (1988). Saddlepoint methods and statistical inference. Statist. Sci., 3(2), 213–238. With comments and a rejoinder by the author.
R´ev´esz, P. (1973). Robbins-Monro procedure in a Hilbert space and its application in the theory of learning processes. I. Studia Sci. Math. Hungar., 8, 391–398.
R´ev´esz, P. (1977). How to apply the method of stochastic approximation in the non-parametric estimation of a regression function.Math. Operationsforsch. Statist. Ser. Statist., 8(1), 119–126.
Robbins, H. and Monro, S. (1951). A stochastic approximation method. Anal. Math. Statist., 22, 400–407.
Rosenbaum, P. R. and Rubin, D. B. (1983). The central role of the propensity score in observational studies for causal effects. Biometrika, 70(1), 41–55.
Rosenblatt, M. (1956). Remarks on some nonparametric estimates of a characteristic function. Ann. Math. Statist., 27, 832–837.
Scott, D. W. (1992). Multivariate characteristic estimation. Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics. John Wiley & Sons Inc., New York. Theory, practice, and visualization, A Wiley-Interscience Publication.
Silverman, B. W. (1986). Density estimation for statistics and data analysis. Monographs on Statistics and Applied Probability. Chapman & Hall, London.
Slaoui, Y. (2014a). Bandwidth selection for recursive kernel density estimators defined by stochastic approximation method. Journal of Probability and Statistics, 2014, ID 739640, doi:10.1155/2014/739640.
Slaoui, Y. (2014b). The stochastic approximation method for the estimation of a distribution function. Math. Methods Statist. 23, 306–325.
Slaoui, Y. (2015). Plug-In order selector for recursive kernel regression estimators defined by stochastic approximation method. Stat. Neerl., 69, 483–509.
Soukarieh, I. and Bouzebda, S. (2023). Renewal type bootstrap for increasing degree U-process of a Markov chain. J. Multivariate Anal., 195, Paper No. 105143.
Yamato, H. (1972/73). Uniform convergence of an estimator of a distribution function. Bull. Math. Statist., 15(3-4), 69–78.
Wand, M. P. and Jones, M. C. (1995). Kernel smoothing, volume 60 of Monographs on Statistics and Applied Probability. Chapman and Hall Ltd., London.
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