Strong uniform consistency of the frequency polygon density estimator for stable non-anticipative stochastic processes. (English) Zbl 1524.62149
Summary: The author establishes a new mathematical expression for the Frequency Polygon. He uses it to prove the strong uniform consistency of the Frequency Polygon marginal density estimator for non-anticipative stationary stochastic processes which are stable in the sense of W. B. Wu [Proc. Natl. Acad. Sci. USA 102, No. 40, 14150–14154 (2005; Zbl 1135.62075)]. He gives examples of several time series models for which this result is relevant.
MSC:
| 62G07 | Density estimation |
| 62G20 | Asymptotic properties of nonparametric inference |
| 62M10 | Time series, auto-correlation, regression, etc. in statistics (GARCH) |
Keywords:
density estimation; frequency polygon; strong uniform consistency; dependent stochastic processes; time seriesCitations:
Zbl 1135.62075References:
| [1] | Adams, R.; Fournier, J., Sobolev Spaces (2003), Academic Press · Zbl 1098.46001 |
| [2] | Carbon, M.; Garel, B.; Tran, L., Frequency polygons for weakly dependent processes, Statist. Probab. Lett., 33, 1-13 (1997) · Zbl 0901.62057 |
| [3] | Diaconis, P.; Freedman, D., Iterated random functions, SIAM Rev., 41, 1, 45-76 (1999) · Zbl 0926.60056 |
| [4] | Kang, S.; Abbasi, R.; Wang, H.; Xu, L.; Wang, X.; Tang, Y., Asymptotic behaviour of frequency polygon under \(\phi\) mixing samples, Comm. Statist. Theory Methods, 47, 21, 5369-5377 (2018) · Zbl 1508.62086 |
| [5] | Parzen, E., On estimation of a probability density function and mode, Ann. Math. Stat., 33, 1065-1076 (1962) · Zbl 0116.11302 |
| [6] | Révész, P., On empirical density function, Period. Math. Hungar., 2, 85-110 (1972) · Zbl 0246.62063 |
| [7] | Rosenblatt, M., Remarks on some nonparametric estimates of a density function, Ann. Math. Stat., 27, 832-837 (1956) · Zbl 0073.14602 |
| [8] | Scott, D., Frequency polygons: Theory and application, J. Amer. Statist. Assoc., 80, 390, 348-354 (1985) · Zbl 0573.62036 |
| [9] | Stone, C., Optimal uniform rate of convergence for nonparametric estimators of a density function or its derivatives, (Recent Advances in Statistics (1983), Academic Press), 393-406 · Zbl 0591.62031 |
| [10] | Taylor, M., Partial Differential Equation I. Basic Theory (1996), Springer · Zbl 0869.35001 |
| [11] | Wang, W.; Huang, H.; Wu, Y.; Chen, K., On the uniform consistency of frequency polygons for \(\rho^-\)-mixing samples, J. Math. Inequal., 15, 3, 1287-1298 (2021) · Zbl 1476.62072 |
| [12] | Wu, W., Nonlinear system theory: Another look at dependence, Proc. Nat. Acad. Sci. USA, 102, 14150-14154 (2005) · Zbl 1135.62075 |
| [13] | Wu, W., Oscillation of empirical distribution functions under dependence, (IMS Lecture Notes-Monograph Series, High Dimensional Probability (2006), Institute of Mathematical Statistics), 53-61 · Zbl 1126.60020 |
| [14] | Xing, G.; Yang, S.; Liang, X., On the uniform consistency of frequency polygons for \(\psi \)-mixing samples, J. Korean Stat. Soc., 44, 179-186 (2015) · Zbl 1343.60029 |
| [15] | Yang, X., Frequency polygon estimation of density function for dependent samples, J. Korean Stat. Soc., 44, 530-537 (2015) · Zbl 1327.62232 |
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