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Chapter 4: Algorithm Overview 54smoothing as applied to uni-variate and multi-variate datasets. Papers that discuss theinformation bound bandwidth selection methods are [Wu and Lin, 1996] and [Jones etal., 1996].The bandwidth selection for the process of density estimation is important to assert theaccuracy of the density estimate. The choice of the bandwidth at least theoretically canbe derived to minimize the mean integrated square error between the actual densityand the computed density. Some methods that are used for this purpose are Distribution Scale Methods, Cross Validation Methods, Plug-In Methods, and Bootstrap methods.In the next few paragraphs, we will very briefly discuss the rationale behind theseobjective methods for bandwidth selection. Assume that f is the actual density of thedata and fˆ is the estimated density. The process of bandwidth selection is aimed atminimizing the integrated mean square error between the actual and the estimateddensity. The integrated mean square error is defined as the expected value of theintegrated square error and is given by Equation 4.13.MSE{f ( x;h )}122= n− {( K h * f )( x ) − ( K h * f ) ( x )} + {( K h * f )( x ) − f ( x )} . (4.13)The Mean Integrated Square Error ( MISE ) is the integral of the mean squared errorthat can be simplified as shown below.2MISE{MISE{ fˆ ( x,h )}12=MSE{ fˆ ( x;h )}dx = E{fˆ ( x;h ) − f ( x )} dx22{(Kh* f )( x ) −(Kh* f ) ( x )}dx +{(Kh* f )( x ) −fˆ (.;h )} = n− f ( x )} dx = −− + − − + 2 (4.14)MISE is the sum of the integrated square bias and the integrated variance and henceminimization of that error is effectively the tradeoff between the bias and variance.The closed form solution that is derived for the optimal bandwidth by minimizing theMISE is the h opt in Equation 4.15.h opt= n(2K(z)dz 222z K(z)dz)( f "( x)dx)1/ 5(4.15)