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A Note on Spline Estimator of Unknown Probability Density Function

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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.

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M. 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.

[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. |

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