Wavelet Density Estimation of Censoring Data and Evaluate of Mean Integral Square Error with Convergence Ratio and Empirical Distribution of Given Estimator

Affiliation(s)

Department of Statistics, College of Science, Persian Gulf University, Bushehr, Iran.

Department of Statistics, College of Science, Persian Gulf University, Bushehr, Iran.

ABSTRACT

Wavelet has rapid development in the current mathematics new areas. It also has a double meaning of theory and application. In signal and image compression, signal analysis, engineering technology has a wide range of applications. In this paper, we use wavelet method, for estimating the density function for censoring data. We evaluate the mean integrated squared error, convergence ratio of given estimator. Also, we obtain empirical distribution of given estimator and verify the conclusion by two simulation examples.

Wavelet has rapid development in the current mathematics new areas. It also has a double meaning of theory and application. In signal and image compression, signal analysis, engineering technology has a wide range of applications. In this paper, we use wavelet method, for estimating the density function for censoring data. We evaluate the mean integrated squared error, convergence ratio of given estimator. Also, we obtain empirical distribution of given estimator and verify the conclusion by two simulation examples.

Cite this paper

Afshari, M. (2014) Wavelet Density Estimation of Censoring Data and Evaluate of Mean Integral Square Error with Convergence Ratio and Empirical Distribution of Given Estimator.*Applied Mathematics*, **5**, 2062-2072. doi: 10.4236/am.2014.513200.

Afshari, M. (2014) Wavelet Density Estimation of Censoring Data and Evaluate of Mean Integral Square Error with Convergence Ratio and Empirical Distribution of Given Estimator.

References

[1] Harr, A. (1910) Zur Theorie der Orthogonalen Funktionen. Mathematische Annalen, 69, 331-371.

[2] Daubechies, I. (1988) Orthogonal Bases of Compactly Supported Wavelets. Communication in Pure and Applied Mathematics, 41, 909-996.

[3] Antoniadis, A. (1996) Smoothing Noisy Data with Tapered Coiflets Series. Scandinavian Journal of Statistics, 23, 313-330.

[4] Afshari, M. (2013) A Fast Wavelet Algorithm for Analyzing of Signal Processing and Empirical Distribution of Wavelet Coefficients with Numerical Example and Simulation. Communication of Statistics-Theory and Methods, 42, 4156-4169.

[5] Afshari, M. (2014) Estimation of Hazard Function for Censoring Random Variable by Using Wavelet Decomposition and Evaluate of MISE, AMSE With Simulation. Journal of Data Analysis and Information Processing, 2, 1-5.

http://dx.doi.org/10.4236/jdaip.2014.21001

[6] Afshari, M. (2008) Wavelet-Kernel Estimation of Regression Function for Uniformly Mixing Process. Word Applied Sciences Journal, 4, 605-609.

[7] Donoha, D.L. and Johnstone, I.M. (1994) Ideal Spatial Adaptation by Wavelet Shrinkage. Biometrika Journal, 81, 425-455.

http://dx.doi.org/10.1093/biomet/81.3.425

[8] Kerkyacharian, G. and Picard, D. (1993) Density Estimation Bykernel and Probability. McGraw-Hill Science, New York, 327-336.

[9] Mallat, S.G. (1989) A Theory for Multiresolution Signal Decomposition: The Wavelet Representation. Transformations on Pattern Analysis and Machine Intelligence, 11, 674-693.

[10] Meyer, Y. (1990) On de lettes et operateurs. Hermann, Paris.

[11] Hall, P. and Patil, P. (1995) Formula for Mean Integrated Squarederror of Non-Linear Wavelet Based Density Estimators. Annals of Statistics, 23, 905-928.

http://dx.doi.org/10.1214/aos/1176324628

[12] Antoniadis, A., Gregoire, G. and Nason, P. (1999) Density and Hazard Rate Estimation for Right Censored Data Using Wavelet Methods. Journal of Royal Statistical Society Series B, 23, 313-330.

[13] Vidakovik, B. (1999) Statistical Modeling by Wavelets. Wiley, New York.

http://dx.doi.org/10.1002/9780470317020

[1] Harr, A. (1910) Zur Theorie der Orthogonalen Funktionen. Mathematische Annalen, 69, 331-371.

[2] Daubechies, I. (1988) Orthogonal Bases of Compactly Supported Wavelets. Communication in Pure and Applied Mathematics, 41, 909-996.

[3] Antoniadis, A. (1996) Smoothing Noisy Data with Tapered Coiflets Series. Scandinavian Journal of Statistics, 23, 313-330.

[4] Afshari, M. (2013) A Fast Wavelet Algorithm for Analyzing of Signal Processing and Empirical Distribution of Wavelet Coefficients with Numerical Example and Simulation. Communication of Statistics-Theory and Methods, 42, 4156-4169.

[5] Afshari, M. (2014) Estimation of Hazard Function for Censoring Random Variable by Using Wavelet Decomposition and Evaluate of MISE, AMSE With Simulation. Journal of Data Analysis and Information Processing, 2, 1-5.

http://dx.doi.org/10.4236/jdaip.2014.21001

[6] Afshari, M. (2008) Wavelet-Kernel Estimation of Regression Function for Uniformly Mixing Process. Word Applied Sciences Journal, 4, 605-609.

[7] Donoha, D.L. and Johnstone, I.M. (1994) Ideal Spatial Adaptation by Wavelet Shrinkage. Biometrika Journal, 81, 425-455.

http://dx.doi.org/10.1093/biomet/81.3.425

[8] Kerkyacharian, G. and Picard, D. (1993) Density Estimation Bykernel and Probability. McGraw-Hill Science, New York, 327-336.

[9] Mallat, S.G. (1989) A Theory for Multiresolution Signal Decomposition: The Wavelet Representation. Transformations on Pattern Analysis and Machine Intelligence, 11, 674-693.

[10] Meyer, Y. (1990) On de lettes et operateurs. Hermann, Paris.

[11] Hall, P. and Patil, P. (1995) Formula for Mean Integrated Squarederror of Non-Linear Wavelet Based Density Estimators. Annals of Statistics, 23, 905-928.

http://dx.doi.org/10.1214/aos/1176324628

[12] Antoniadis, A., Gregoire, G. and Nason, P. (1999) Density and Hazard Rate Estimation for Right Censored Data Using Wavelet Methods. Journal of Royal Statistical Society Series B, 23, 313-330.

[13] Vidakovik, B. (1999) Statistical Modeling by Wavelets. Wiley, New York.

http://dx.doi.org/10.1002/9780470317020