Abstract
A technique to select fast blur coefficients of kernel functions is proposed to proceed with a nonparametric estimation of the probability density for a two-dimensional random variable having independent components. The distributions explored belong to the family of unimodal and symmetric densities of probability. The optimization feature is substantiated based on analyzing asymptotic expressions obtained for the mean square deviations of components of a two-dimensional random variable. Each component is characterized by the optimal blur coefficient of kernel functions which depends on the nonlinear probability density functional. The function describing the dependence on the coefficient of antikurtosis is obtained for a one-dimensional random variable. The effectiveness of the method proposed is confirmed by analytical studies.
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REFERENCES
M. Rudemo, ‘‘Empirical choice of histogram and kernel density estimators,’’ Scand. J. Stat. 9, 65–78 (1982).
A. W. Bowman, ‘‘A comparative study of some kernel-based non-parametric density estimators,’’ J. Stat. Comput. Simul. 21, 313–327 (1982). https://doi.org/10.1080/00949658508810822
P. Hall, ‘‘Large-sample optimality of least squares cross-validation in density estimation,’’ Ann. Stat. 11, 1156–1174 (1983).
M. Jiang and S. B. Provost, ‘‘A hybrid bandwidth selection methodology for kernel density estimation,’’ J. Stat. Comput. Simul. 84, 614–627 (2014). https://doi.org/10.1080/00949655.2012.721366
S. Dutta, ‘‘Cross-validation revisited,’’ Commun. Stat. Simul. Comput. 45, 472–490 (2016). https://doi.org/10.1080/03610918.2013.862275
N.-B. Heidenreich, A. Schindler, and S. Sperlich, ‘‘Bandwidth selection for kernel density estimation: a review of fully automatic selectors,’’ AStA Adv. Stat. Anal. 97, 403–433 (2013). https://doi.org/10.1007/s10182-013-0216-y
Q. Li and J. S. Racine, Nonparametric Econometrics: Theory and Practice (Princeton Univ. Press, Princeton, 2007).
R. P. W. Duin, ‘‘On the choice of smoothing parameters for Parzen estimators of probability density functions,’’ IEEE Trans. Comput. 25, 1175–1179 (1976). https://doi.org/10.1109/TC.1976.1674577
Z. I. Botev and D. P. Kroese, ‘‘Non-asymptotic bandwidth selection for density estimation of discrete data,’’ Methodol. Comput. Appl. Probab. 10, 435–451 (2008). https://doi.org/10.1007/s11009-007-9057-z
A. V. Lapko and V. A. Lapko, ‘‘Analysis of optimization methods for nonparametric estimation of the probability density with respect to the blur factor of kernel functions,’’ Meas. Tech. 60, 515–522 (2017). https://doi.org/10.1007/s11018-017-1228-x
A. G. Varzhapetyan and E. Yu. Mikhailova, ‘‘Methods for selecting the determining characteristics of nonparametric identification algorithms of models of complex system reliability by experimental data,’’ Vopr. Kibern. 94, 77–87 (1982).
B. W. Silverman, Density Estimation for Statistics and Data Analysis (Chapman and Hall, London, 1986).
S. Sheather and M. Jones, ‘‘A reliable data-based bandwidth selection method for kernel density estimation,’’ J. R. Stat. Soc. Ser. B. 53, 683–690 (1991). https://doi.org/10.1111/j.2517-6161.1991.tb01857.x
S. J. Sheather, ‘‘Density estimation,’’ Stat. Sci. 19, 588–597 (2004). https://doi.org/10.1214/088342304000000297
G. R. Terrell and D. W. Scott, ‘‘Oversmoothed nonparametric density estimates,’’ J. Am. Stat. Assoc. 80, 209–214 (1985).
M. C. Jones. J. S. Marron, and S. J. Sheather, ‘‘A brief survey of bandwidth selection for density estimation,’’ J. Am. Stat. Assoc. 91, 401–407 (1996).
D. W. Scott, Multivariate Density Estimation: Theory, Practice, and Visualization (Wiley, New Jersey, 2015). https://doi.org/10.1002/9780470316849
A. V. Lapko and V. A. Lapko, ‘‘Fast algorithm for choosing blur coefficients in multidimensional kernel probability density estimates,’’ Meas. Tech. 61, 979–986 (2019). https://doi.org/10.1007/s11018-019-01536-x
A. V. Lapko and V. A. Lapko, ‘‘Estimating the integral of the square of derivatives of symmetric probability densities of one-dimensional random variables,’’ Meas. Tech. 63, 171–176 (2020). https://doi.org/10.1007/s11018-020-01768-2
A. V. Lapko and V. A. Lapko, ‘‘Estimation of a nonlinear functional of probability density when optimizing nonparametric decision functions,’’ Meas. Tech. 64, 13–20 (2021). https://doi.org/10.1007/s11018-021-01889-2
A. V. Lapko and V. A. Lapko, ‘‘Analysis of the ratio of the standard deviations of the kernel estimate of the probability density with independent and dependent random variables,’’ Meas. Tech. 64, 166–171 (2021). https://doi.org/10.1007/s11018-021-01914-4
A. V. Lapko and V. A. Lapko, ‘‘Modified fast algorithm for the bandwidth selection of the kernel density estimation,’’ Optoelectron., Instrum. Data Process. 56, 566–572 (2020). https://doi.org/10.3103/S8756699020060102
A. V. Lapko and V. A. Lapko, ‘‘Fast algorithm for choosing kernel function blur coefficients in a nonparametric probability density estimate,’’ Meas. Tech. 61, 540–545 (2018). https://doi.org/10.1007/s11018-018-1463-9
A. V. Lapko and V. A. Lapko, ‘‘Nonparametric estimate of probability density of independent random variables,’’ Inf. Sist. Upr. 29 (3), 118–124 (2011).
V. A. Epanenchikov, ‘‘Non-parametric estimation of a multivariate probability density,’’ Theory Probab. Its Appl. 14, 153–158 (1969). https://doi.org/10.1137/1114019
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Lapko, A.V., Lapko, V.A. Quick Selecting Kernel Blur Coefficients to Estimate Probability Density for Independent Random Variables. Optoelectron.Instrument.Proc. 58, 24–29 (2022). https://doi.org/10.3103/S8756699022010071
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DOI: https://doi.org/10.3103/S8756699022010071