This paper studies the kernel-type distribution estimator based on asymptotically almost negatively associated (AANA, for short) samples. The rate of uniformly strong consistency is established under some mild conditions. As applications, the uniformly strong convergence rates of kernel-type density estimator and kernel-type hazard rate estimator are also obtained. Some Monte Carlo simulations are presented to illustrate the finite sample performance of the kernel method. Finally, a real data analysis of Alibaba stock returns data is used to illustrate the usefulness of the proposed methodology.

References

• Azzalini, A. (1981). A note on the estimation of a distribution function and quantiles by a kernel method. Biometrika, 68(1), 326–328.
• Cai, Z. (1993). Uniform strong convergence and rates for the kernel estimators of a distribution function and a regression function under weakly dependent assumptions. Chinese Journal of Applied Probability and Statistics, 9(1), 11–17.
• Cai, Z., & Roussas, G. G. (1992). Uniform strong estimation under α-mixing, with rates. Statistics & Probability Letters, 15(1), 47–55.
• Cai, Z., & Roussas, G. G. (1999). Berry-Esseen bounds for smooth estimator of a distribution function under association. Journal of Nonparametric Statistics, 11(1-3), 79–106.
• Chandra, T. K., & Ghosal, S. (1996a). Extensions of the strong law of large numbers of Marcinkiewicz and Zygmund for dependent variables. Acta Mathematica Hungarica, 71(4), 327–336.
• Chandra, T. K., & Ghosal, S. (1996b). The strong law of large numbers for weighted averages under dependence assumptions. Journal of Theoretical Probability, 9(3), 797–809. https://doi.org/10.1007/BF02214087
• Chen, Z. Y., Wang, X. J., & Hu, S. H. (2015). Strong laws of large numbers for weighted sums of asymptotically almost negatively associated random variables. Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas. RACSAM, 109(1), 135–152.
• Ghorai, J. K., & Susarla, V. (1990). Kernel estimation of a smooth distribution function based on censored data. Metrika, 37(2), 71–86. https://doi.org/10.1007/BF02613509
• Györfi, L., Härdle, W., Sarda, P., & Vieu, P. (1989). Nonparametric Curve Estimation from Time Series. Springer.
• Jabbari, H., Azarnoosh, H. A., & Fakoor, V. (2009). Almost sure convergence of two-dimensional distribution function under negative association. Journal of Applied Probability and Statistics, 4(2), 157–166.
• Joag-Dev, K., & Proschan, F. (1983). Negative association of random variables with applications. The Annals of Statistics, 11(1), 286–295. https://doi.org/10.1214/aos/1176346079
• Li, Q., & Rcine, J. S. (2007). Nonparametric Econometrics: Theory and Practice. Princeton University Press.
• Li, Y. M., Wei, C., & Yang, S. C. (2010). The recursive kernel distribution function estimator based on negatively and positively associated sequences. Communications in Statistics–Theory and Methods, 39(20), 3585–3595. https://doi.org/10.1080/03610920903427750
• Li, X. Q., Zhang, L., Shen, Y., & Chen, Z. Y. (2023). Asymptotic approximations of random ratio model based on AANA sequences. Communications in Statistics-Simulation and Computation, 52(8), 3796–3819. https://doi.org/10.1080/03610918.2021.1948573
• Li, Y. M., & Zhou, Y. (2020). The Kaplan-Meier estimator and hazard estimator for censored END survival time observations. Communications in Statistics–Theory and Methods, 49(11), 2690–2702. https://doi.org/10.1080/03610926.2019.1580737
• Liu, R., & Yang, L. (2008). Kernel estimation of multivariate cumulative distribution function. Journal of Nonparametric Statistics, 20(8), 661–677. https://doi.org/10.1080/10485250802326391
• Loftsgarden, D. O., & Quesenberry, C. P. (1965). A nonparametric estimate of a multivariate density function. The Annals of Mathematical Statistics, 36(3), 1049–1051. https://doi.org/10.1214/aoms/1177700079
• Mikael, L. (2001). A model for stock return distribution. International Journal of Finance & Economics, 6(2), 159–169. https://doi.org/10.1002/ijfe.v6:2
• Peligrad, M. (1992). Properties of uniform consistency of the kernel estimators of density and of regression functions under dependence assumptions. Stochastics and Stochastic Reports, 40, 147–168. https://doi.org/10.1080/17442509208833786
• Roussas, G. G. (1995). Asymptotic normality of a smooth estimate of a random field distribution function under association. Statistics & Probability Letters, 24(1), 77–90. https://doi.org/10.1016/0167-7152(94)00151-W
• Shen, A. T., & Wang, X. H. (2016). Kaplan-Meier estimator and hazard estimator for censored negatively superadditive dependent data. Statistics, 50(2), 377–388.
• Shen, A. T., & Wu, R. C. (2014). Strong convergence for sequences of asymptotically almost negatively associated random variables. Stochastics, 86(2), 291–303. https://doi.org/10.1080/17442508.2013.775289
• Swanepoel, J. W. H. (1988). Mean integrated squared error properties and optimal kernels when estimating a distribution function. Communications in Statistics–Theory and Methods, 17(11), 3785–3799. https://doi.org/10.1080/03610928808829835
• Wang, Y. B., Yan, J. G., & Cheng, F. Y. (2003). The strong law of large numbers and the iterated logarithm for product sums of NA and AANA random variables. Southeast Asian Bulletin of Mathematics, 27(2), 369–384.
• Watson, G. S., & Leadbetter, M. R. (1964). Hazard analysis II. Sankhyā: The Indian Journal of Statistics, Series A, 26, 101–116.
• Winter, B. B. (1979). Convergence rate of perturbed empirical distribution functions. Journal of Applied Probability, 16(1), 163–173. https://doi.org/10.2307/3213384
• Wu, Y., & Wang, X. J. (2019). On consistency of the nearest neighbor estimator of the density function and its applications. Acta Mathematica Sinica (English Series), 35(5), 703–720. https://doi.org/10.1007/s10114-019-8099-9
• Wu, Y., Yu, W., & Wang, X. J. (2022). Strong representations of the Kaplan-Meier estimator and hazard estimator with censored widely orthant dependent data. Computational Statistics, 37(1), 383–402. https://doi.org/10.1007/s00180-021-01125-z
• Yamato, H. (1973). Uniform convergence of an estimator of a distribution function. Bulletin of Mathematical Statistics, 15(3–4), 69–78. https://doi.org/10.5109/13073
• Yang, S. C. (2003). Consistency of nearest neighbor estimator of density function for negative associated samples. Acta Mathematicae Applicatae Sinica (English Series), 26(3), 385–394.
• Yuan, D., & An, J. (2009). Rosenthal type inequalities for asymptotically almost negatively associated random variables and applications. Science in China Series A: Mathematics, 52(9), 1887–1904. https://doi.org/10.1007/s11425-009-0154-z
• Yuan, D., & An, J. (2012). Laws of large numbers for Cesàro alpha-integrable random variables under dependence condition AANA or AQSI. Acta Mathematica Sinica (English Series), 28(6), 1103–1118. https://doi.org/10.1007/s10114-012-0033-3

To cite this article: Shipeng Wu, Yi Wu, Wenzhi Yang & Xuejun Wang (09 Apr 2025): The uniformly strong consistency of kernel-type distribution estimator under asymptotically almost negatively associated samples, Statistical Theory and Related Fields, DOI: 10.1080/24754269.2025.2484980

To link to this article: https://doi.org/10.1080/24754269.2025.2484980