Wavelet Density Estimation and Statistical Evidences Role for a GARCH Model in the Weighted Distribution

ABSTRACT

We consider *n* observations from the GARCH-type model: *Z* = *UY*, where *U* and *Y* are independent random variables. We aim to estimate density function *Y* where *Y* have a weighted distribution. We determine a sharp upper bound of the associated mean integrated square error. We also make use of the measure of expected true evidence, so as to determine when model leads to a crisis and causes data to be lost.

Cite this paper

M. Abbaszadeh and M. Emadi, "Wavelet Density Estimation and Statistical Evidences Role for a GARCH Model in the Weighted Distribution,"*Applied Mathematics*, Vol. 4 No. 2, 2013, pp. 410-416. doi: 10.4236/am.2013.42061.

M. Abbaszadeh and M. Emadi, "Wavelet Density Estimation and Statistical Evidences Role for a GARCH Model in the Weighted Distribution,"

References

[1] M. Carrasco and X. Chen, “Mixing and Moment Properties of Various GARCH and Stochastic Volatility Models,” Econometric Theory, Vol. 18, No. 1, 2002, pp. 17-39. doi:10.1017/S0266466602181023

[2] S. T. Buckland, D. R. Anderson, K. P. Burnham and J. L. Laake, “Distance Sampling: Estimating Abundance of Biological Populations,” Chapman and Hall, London, 1993.

[3] D. Cox, “Some Sampling Problems in Technology,” In: N. L. Johnson and H. Smith Jr., Eds., New Developments in Survey Sampling, Wiley, New York, 1969, pp. 506-527.

[4] J. Heckman, “Selection Bias and Self-Selection,” The New Palgrave: A Dictionary of Economics, MacMillan Press, Stockton, 1985, pp. 287-296.

[5] R. Royall, “Statistical Evidence,” A Likelihood Paradigm, Chapman and Hall, London, 1997.

[6] R. Royall, “On the Probability of Observing Misleading Statistical Evidence,” Journal of the American Statistical Association, Vol. 95, No. 451, 2000, pp. 760-780. doi:10.1080/01621459.2000.10474264

[7] M. Emadi, J. Ahmadi and N. R. Arghami, “Comparing of Record Data and Random Observation Based on Statistical Evidence,” Statistical Papers, Vol. 48, No. 1, 2007, pp. 1-21. doi:10.1007/s00362-006-0313-z

[8] C. Chesneau and H. Doosti, “Wavelet Linear Density Estimation for a GARCH Model under Various Dependence Structures,” Journal of the Iranian Statistical Society, Vol. 11, No. 1, 2012, pp. 1-21.

[9] P. Doukhan, “Mixing Properties and Examples,” Lecture Notes in Statistics 85, Springer Verlag, New York, 1994.

[10] D. Modha and E. Masry, “Minimum Complexity Regression Estimation with Weakly Dependent Observations,” IEEE Transactions on Information Theory, Vol. 42, No. 6, 1996, pp. 2133-2145. doi:10.1109/18.556602

[11] A. Cohen, I. Daubechies, B. Jawerth and P. Vial, “Wavelets on the Interval and Fast Wavelet Transforms,” Applied and Computational Harmonic Analysis, Vol. 24, No. 1, 1993, pp. 54-81. doi:10.1006/acha.1993.1005

[12] Y. Meyer, “Wavelets and Operators,” Cambridge University Press, Cambridge, 1992.

[13] M. Abbaszadeh, C. Chesneau and H. Doosti, “Nonparametric Estimation of Density under Bias and Multiplicative Censoring via Wavelet Methods,” Statistics and Probability Letters, Vol. 82, No. 5, 2012, pp. 932-941. doi:10.1016/j.spl.2012.01.016

[14] W. H?rdle, G. Kerkyacharian, D. Picard and A. Tsybakov, “Wavelet, Approximation and Statistical Applications,” Lectures Notes in Statistics, Springer Verlag, New York, 1998, Vol. 129.

[15] M. Emadi and N. R. Arghami, “Some Measure of Support for Statistical Hypotheses,” Journal of Statical Theory and Applications, Vol. 2, No. 2, 2003, pp. 165-176.

[16] Y. Davydov, “The Invariance Principle for Stationary Processes,” Theory of Probability & Its Applications, Vol. 15, No. 3, 1970, pp. 498-509. doi:10.1137/1115050

[1] M. Carrasco and X. Chen, “Mixing and Moment Properties of Various GARCH and Stochastic Volatility Models,” Econometric Theory, Vol. 18, No. 1, 2002, pp. 17-39. doi:10.1017/S0266466602181023

[2] S. T. Buckland, D. R. Anderson, K. P. Burnham and J. L. Laake, “Distance Sampling: Estimating Abundance of Biological Populations,” Chapman and Hall, London, 1993.

[3] D. Cox, “Some Sampling Problems in Technology,” In: N. L. Johnson and H. Smith Jr., Eds., New Developments in Survey Sampling, Wiley, New York, 1969, pp. 506-527.

[4] J. Heckman, “Selection Bias and Self-Selection,” The New Palgrave: A Dictionary of Economics, MacMillan Press, Stockton, 1985, pp. 287-296.

[5] R. Royall, “Statistical Evidence,” A Likelihood Paradigm, Chapman and Hall, London, 1997.

[6] R. Royall, “On the Probability of Observing Misleading Statistical Evidence,” Journal of the American Statistical Association, Vol. 95, No. 451, 2000, pp. 760-780. doi:10.1080/01621459.2000.10474264

[7] M. Emadi, J. Ahmadi and N. R. Arghami, “Comparing of Record Data and Random Observation Based on Statistical Evidence,” Statistical Papers, Vol. 48, No. 1, 2007, pp. 1-21. doi:10.1007/s00362-006-0313-z

[8] C. Chesneau and H. Doosti, “Wavelet Linear Density Estimation for a GARCH Model under Various Dependence Structures,” Journal of the Iranian Statistical Society, Vol. 11, No. 1, 2012, pp. 1-21.

[9] P. Doukhan, “Mixing Properties and Examples,” Lecture Notes in Statistics 85, Springer Verlag, New York, 1994.

[10] D. Modha and E. Masry, “Minimum Complexity Regression Estimation with Weakly Dependent Observations,” IEEE Transactions on Information Theory, Vol. 42, No. 6, 1996, pp. 2133-2145. doi:10.1109/18.556602

[11] A. Cohen, I. Daubechies, B. Jawerth and P. Vial, “Wavelets on the Interval and Fast Wavelet Transforms,” Applied and Computational Harmonic Analysis, Vol. 24, No. 1, 1993, pp. 54-81. doi:10.1006/acha.1993.1005

[12] Y. Meyer, “Wavelets and Operators,” Cambridge University Press, Cambridge, 1992.

[13] M. Abbaszadeh, C. Chesneau and H. Doosti, “Nonparametric Estimation of Density under Bias and Multiplicative Censoring via Wavelet Methods,” Statistics and Probability Letters, Vol. 82, No. 5, 2012, pp. 932-941. doi:10.1016/j.spl.2012.01.016

[14] W. H?rdle, G. Kerkyacharian, D. Picard and A. Tsybakov, “Wavelet, Approximation and Statistical Applications,” Lectures Notes in Statistics, Springer Verlag, New York, 1998, Vol. 129.

[15] M. Emadi and N. R. Arghami, “Some Measure of Support for Statistical Hypotheses,” Journal of Statical Theory and Applications, Vol. 2, No. 2, 2003, pp. 165-176.

[16] Y. Davydov, “The Invariance Principle for Stationary Processes,” Theory of Probability & Its Applications, Vol. 15, No. 3, 1970, pp. 498-509. doi:10.1137/1115050