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Chapter 4: Algorithm Overview 50∞− ∞K(x) dx= 1 .(4.10)Analogous to the definition of the naive estimator, the kernel estimator with kernel Kis defined byfˆ ( x )1=nhni=1x − XK(hi)(4.11)While the naive estimator can be considered as a sum of boxes centered at theobservations, the kernel estimator is a sum of bumps placed at the observations. Thekernel function K determines the shape of the bumps while the window width hdetermines their width. It suffers inaccuracy with long tailed distributions because ofthe fixed bandwidth throughout the process of density estimation.The nearest neighbor class of estimators represents an attempt to adapt the amount ofsmoothing to the `local' density of data. The degree of smoothing is controlled by aninteger k, chosen to be considerably smaller than the sample size; typically k n 1/2 .Define the distance d(x, y) between two points on the line to be |x - y| in the usual way,and for each t define d 1 ( t ) ≤ d 2 ( t ) ≤ ... ≤ dn( t ) to be the distances, arranged in ascendingorder, from t to the points of the sample.The k th nearest neighbor density estimate is then defined byfˆ ( t )=2kndk( t.)(4.12)While the naive estimator is based on the number of observations falling in a box offixed width centered at the point of interest, the nearest neighbor estimate is inverselyproportional to the size of the box needed to contain a given number of observations.In the tails of the distribution, the distance d k (t) will be larger than in the main part ofthe distribution, and so the problem of under smoothing in the tails is reduced. Likethe naive estimator, to which it is related, the nearest choice neighbor estimate asdefined is not a smooth curve. The function d k (t) can easily be seen to be continuous,but its derivative will have a discontinuity. We would like to achieve stability with theinformation measure and since most of the surfaces that we are interested in havesmooth analytical parameterization, we are inclined to chose the continuous andsmooth looking kernel density estimate. We show how each of these methods estimatethe density of the same dataset in Figure 4.6. We have reproduced Figure 4.6 from[Silverman, 1986].