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Chapter 4: Algorithm Overview 55The problem with using this closed form solution is the dependence of the optimalbandwidth on the second derivative of the density function f that we are trying tocompute. By using the Gaussian kernel for our implementation we have ensured thedifferentiability of the estimated density and also justified the reason for not choosingthe naïve estimator or its rugged counterparts.Two popular but quick and simple bandwidth selectors are based on the normal scalerule and maximum smoothing principle. For example, an easy approach would be tomake use of a standard family of distributions to assign a value to the doublederivative term. In Equation 4.12 we assume normal density and compute the secondderivative. This method can lead to gross errors in cases when the data is notdistributed the way it was assumed.f2−52−5"(x)dx = σ φ"(x)dx ≈ 0.212σ(4.16)The rationale behind the principle of cross validation is to use the same dataset toextract data points partially as a construction set and a training set. A model is fitassuming the correctness of the training dataset and is tested for accuracy with theconstruction dataset. The error in the estimate is minimized by defining a cost functionof the error. Based on the construction of the cost function, methods are named as leastsquares cross validation, biased cross validation and likelihood cross validation. Moreadvanced bandwidth selectors are the plug-in and the bootstrap methods that “plug-in”estimates of the unknown quantities that appear in the formulae for asymptoticallyoptimal bandwidth. Bootstrap methods make use of a pilot bandwidth to initialize thedensity estimation process and improve the pilot bandwidth based on the data. InEquation 4.15, we show the plug-in method of bandwidth selection. Plug-in methodsinvolve the estimation of the integrated squared density derivatives called functionals.15 243R(K ) = =2 =2ĥσ where R( K ) K( t ) dt, µ 2(K )t K( t ) dt235µ2(K ) n[ ](4.17)and σ = med j | X j − medi( X i ) | is the absolute deviation. (4.18)We discuss implementation issues in the next chapter. Our next building block is theinformation measure on the accurate density of curvature estimated using thebandwidth optimized kernel density estimators.