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Abstract
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One of the main issues in statistics is appropriate estimation of statistical functions including regression function that is of great importance. In this study, our purpose is estimation of unknown function $ f (x_{i}) $ in regression equation $ y_{i}=f(x_{i}) e_{i}, i=1,2,...,n $, using wavelet shrinkage method and investigating the asymptotic convergence of the estimator under space $ L ^ {P} $. In regression equation, $ y_{i} $ is observation. The first chapter of this thesis,is devoted to the definition of the wavelet and in the second chapter, the wavelet shrinkage method for estimation of functions is introduced. In the third chapter, asymptotical behavior of the wavelet estimator for regression function is examined. In the fourth chapter, estimation of the $f (x) $ for an irregular design with correlated errors is calculated. In the fifth chapter, efficiency of the wavelet shrinkage method under different thresholds is examined using an example and finally,in the simulation section, volatility function for the data from ''GBI''company in USA is estimated using wavelet shrinkage method.
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