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. 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.
 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.
 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.
 J. Heckman, “Selection Bias and Self-Selection,” The New Palgrave: A Dictionary of Economics, MacMillan Press, Stockton, 1985, pp. 287-296.
 R. Royall, “Statistical Evidence,” A Likelihood Paradigm, Chapman and Hall, London, 1997.
 R. Royall, “On the Probability of Observing Misleading Statistical Evidence,” Journal of the American Statistical Association, Vol. 95, No. 451, 2000, pp. 760-780.
 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
 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.
 P. Doukhan, “Mixing Properties and Examples,” Lecture Notes in Statistics 85, Springer Verlag, New York, 1994.
 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
 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
 Y. Meyer, “Wavelets and Operators,” Cambridge University Press, Cambridge, 1992.
 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.
 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.
 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.
 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