A Note on Spline Estimator of Unknown Probability Density Function

ABSTRACT

In the present paper as estimation of unknown pdf derivative of a spline function is suggested. It is studied its some statistical properties which are used to approximate maximal deviation of the spline estimation from pdf with maximum of nonstationary gaussian process.

In the present paper as estimation of unknown pdf derivative of a spline function is suggested. It is studied its some statistical properties which are used to approximate maximal deviation of the spline estimation from pdf with maximum of nonstationary gaussian process.

Cite this paper

nullM. Muminov and K. Soatov, "A Note on Spline Estimator of Unknown Probability Density Function,"*Open Journal of Statistics*, Vol. 1 No. 3, 2011, pp. 157-160. doi: 10.4236/ojs.2011.13019.

nullM. Muminov and K. Soatov, "A Note on Spline Estimator of Unknown Probability Density Function,"

References

[1] N. B. Smirnov, “On Construction of a Confidence Interval for the Probability Density Function,” Soviet Reports, Vol. 74, 1959, pp. 1189-1191.

[2] P. J. Bikel and M. Rosenblatt, “On Some Global Measures of the Deviations of Density Functions Estimates,” The Annals of Statistics, Vol. 1, No. 6, 1973, pp. 1071- 1095.

[3] M. Rosenblatt, “On the maximal deviation of k-dimen- sional density estimates”, Annals of Probability, Vol. 4, No. 6, 1976, pp. 1009-1015. doi:10.1214/aop/1176995945

[4] M. S. Muminov and Sh. A. Khashimov, “On Limit Distribution of the Maximal Deviation of Spline Density Estimators,” FAN, Tashkent, 1986.

[5] M. S. Muminov, “On a Limit Distribution of the Maximal Level of Empirical Distribution Density and the Regression Function. I,” Theory Probability and Its Application, Vol. 55, No. 3, 2010, pp. 582-590.

[6] M. S. Muminov, “On a Limit Distribution of the Maximal Level of Empirical Distribution Density and the Regression Function. II,” Theory Probability and Its Application, Vol. 56, No. 1, 2011, pp. 162-173.

[7] V. D. Konakov and V. I. Piterbarg, “On the Convergence Rate of Maximal Deviations Distribution for Kernel Regression Estimates,” Journal of Multivariate Annalysis, Vol. 15, No. 3, 1984, pp. 279-294. doi:10.1016/0047-259X(84)90053-8

[8] V. D. Konakov and V. I. Piterbarg, “High Level Excursions of Gaussian Fields and the Weakly Optimal Choice of the Smoothing Parameter. I,” Mathenatical Methods of Statistics, Vol. 4, 1995, pp. 481-434.

[9] V. D. Konakov and V. I. Piterbarg, “High Level Excursions of Gaussian Fields and the Weakly Optimal Choice of the Smoothing Parameter. II,” Mathenatical Methods of Statistics, Vol. 1, 1997, pp. 112-124.

[10] K. S. Lii and M. Rosenblatt, “Asymptotic Behavior of a Spline of a Density Function,” Computters & Mathematics with Applications, No. 1, 1975, pp. 223-235.

[11] M. S. Muminov, “On Statistical Estimation of the Probability Density Function by LineFunctions,” Ph.D. Thesis, Tashkent, p. 110.

[12] M. S. Muminov, “On Approximating the Probability of a Large Excursion a Nonstationary Gaussian Process,” Siberian Mathematical Journal, Vol. 51, No. 1, 2010, pp. 175-195. doi:10.1007/s11202-010-0015-6

[13] K. S. Lii, “A Global Measure of a Spline Density Estimate,” The Annals of Statistics, Vol. 6, No. 5, 1978, pp. 1138-1148. doi:10.1214/aos/1176344316

[14] Y. Komlos, P. Major and G. Tusnady, “An Approximation of Partial Sums of Independent RV’s and the Sample DF. I,” Probability Theory and Related Fields, Vol. 32, No. 1-2, 1975, pp.111-131.

[15] G. Lamperty, “Probability,” Nauka, Moscow, 1973.

[16] G. Cramér, “The Mathematical Method in Statistics,” Mir, Moscow, 1976.

[17] A. V. Skorohod, “The Random Processes with Indepen- dent Increments,” Nauka, Moskov, 1964.

[1] N. B. Smirnov, “On Construction of a Confidence Interval for the Probability Density Function,” Soviet Reports, Vol. 74, 1959, pp. 1189-1191.

[2] P. J. Bikel and M. Rosenblatt, “On Some Global Measures of the Deviations of Density Functions Estimates,” The Annals of Statistics, Vol. 1, No. 6, 1973, pp. 1071- 1095.

[3] M. Rosenblatt, “On the maximal deviation of k-dimen- sional density estimates”, Annals of Probability, Vol. 4, No. 6, 1976, pp. 1009-1015. doi:10.1214/aop/1176995945

[4] M. S. Muminov and Sh. A. Khashimov, “On Limit Distribution of the Maximal Deviation of Spline Density Estimators,” FAN, Tashkent, 1986.

[5] M. S. Muminov, “On a Limit Distribution of the Maximal Level of Empirical Distribution Density and the Regression Function. I,” Theory Probability and Its Application, Vol. 55, No. 3, 2010, pp. 582-590.

[6] M. S. Muminov, “On a Limit Distribution of the Maximal Level of Empirical Distribution Density and the Regression Function. II,” Theory Probability and Its Application, Vol. 56, No. 1, 2011, pp. 162-173.

[7] V. D. Konakov and V. I. Piterbarg, “On the Convergence Rate of Maximal Deviations Distribution for Kernel Regression Estimates,” Journal of Multivariate Annalysis, Vol. 15, No. 3, 1984, pp. 279-294. doi:10.1016/0047-259X(84)90053-8

[8] V. D. Konakov and V. I. Piterbarg, “High Level Excursions of Gaussian Fields and the Weakly Optimal Choice of the Smoothing Parameter. I,” Mathenatical Methods of Statistics, Vol. 4, 1995, pp. 481-434.

[9] V. D. Konakov and V. I. Piterbarg, “High Level Excursions of Gaussian Fields and the Weakly Optimal Choice of the Smoothing Parameter. II,” Mathenatical Methods of Statistics, Vol. 1, 1997, pp. 112-124.

[10] K. S. Lii and M. Rosenblatt, “Asymptotic Behavior of a Spline of a Density Function,” Computters & Mathematics with Applications, No. 1, 1975, pp. 223-235.

[11] M. S. Muminov, “On Statistical Estimation of the Probability Density Function by LineFunctions,” Ph.D. Thesis, Tashkent, p. 110.

[12] M. S. Muminov, “On Approximating the Probability of a Large Excursion a Nonstationary Gaussian Process,” Siberian Mathematical Journal, Vol. 51, No. 1, 2010, pp. 175-195. doi:10.1007/s11202-010-0015-6

[13] K. S. Lii, “A Global Measure of a Spline Density Estimate,” The Annals of Statistics, Vol. 6, No. 5, 1978, pp. 1138-1148. doi:10.1214/aos/1176344316

[14] Y. Komlos, P. Major and G. Tusnady, “An Approximation of Partial Sums of Independent RV’s and the Sample DF. I,” Probability Theory and Related Fields, Vol. 32, No. 1-2, 1975, pp.111-131.

[15] G. Lamperty, “Probability,” Nauka, Moscow, 1973.

[16] G. Cramér, “The Mathematical Method in Statistics,” Mir, Moscow, 1976.

[17] A. V. Skorohod, “The Random Processes with Indepen- dent Increments,” Nauka, Moskov, 1964.