Modelfree4

# Modelfree4 - Tukeys Prediction Tukeys Prediction European...

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Tukey’s Prediction

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Tukey’s Prediction, European Translation
cos sin . iz e z i z = +

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Unbiased k th Bolt { } ( 29 { } 1 1 1 1 0 1 ˆ E E exp 2 exp 2 . n k j j n k j B n ikX n f x ikx dx B π - = - =     = -     = - =
Basic Subdivision of Series

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Wahba (1981) "raw" Fourier series density estimator 1, ; 0, k k b k m b k m = ≤ = ( 29 { } R ˆ ˆ exp 2 . m k k m f x B ikx π =- =
( 29 { } ( 29 ( 29 { } { } 2 2 R 0 1 2 2 2 2 1 1 ˆ ˆ ˆ ˆ ˆ ˆ 1 ˆ ˆ 1 2 cos2 sin 2 . m ikx ikx k k k m ikx ikx ikx ikx k k k m k k k f x B B e B e C e e iS e e C kx S kx π - - = - - = = = + + = + + + - = + -

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Estimator of Density Derivative ( 29 ( 29 { } ( 29 { } 0 ˆ ˆ ˆ exp 2 ˆ 2 exp 2 . m k k m m k k m k d d f x f x B ikx dx dx ik B ikx π =- =- = = =
sample Fourier coefficients ( 29 1 0 ˆ ˆ xf x dx μ= { } 1 1 0 0 0 0 ˆ ˆ ˆ exp 2 . m

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Modelfree4 - Tukeys Prediction Tukeys Prediction European...

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