Preface to First Edition - lib

142 DENSITY ESTIMATIONestimation would be reduced to estimating the parameters of the assumeddistribution. More commonly, however, we wish to allow the data to speak forthemselves and so one of a variety of non-parametric estimation proceduresthat are now available might be used. Density estimation is covered in detailin several books, including Silverman (1986), Scott (1992), Wand and Jones(1995) and Simonoff (1996). One of the most popular classes of proceduresis the kernel density estimators, which we now briefly describe for univariateand bivariate data.8.2.1 Kernel Density EstimatorsFrom the definition of a probability density, if the random X has a density f,1f(x) = lim P(x − h < X < x + h). (8.1)h→0 2hFor any given h a naïve estimator of P(x − h < X < x + h) is the proportionof the observations x 1 , x 2 , ...,x n falling in the interval (x − h, x + h), that isˆf(x) = 12hnn∑I(x i ∈ (x − h, x + h)), (8.2)i=1i.e., the number of x 1 , ...,x n falling in the interval (x − h, x + h) divided by2hn. If we introduce a weight function W given by⎧1⎨2|x| < 1W(x) =⎩0 elsethen the naïve estimator can be rewritten asˆf(x) = 1 n∑ni=1(1 x −h W xih). (8.3)Unfortunately this estimator is not a continuous function and is not particularlysatisfactory for practical density estimation. It does however leadnaturally to the kernel estimator defined byˆf(x) = 1hnn∑( ) x − xiKhi=1(8.4)where K is known as the kernel function and h as the bandwidth or smoothingparameter. The kernel function must satisfy the condition∫ ∞−∞K(x)dx = 1.Usually, but not always, the kernel function will be a symmetric density function,for example, the normal. Three commonly used kernel functions are© 2010 by Taylor and Francis Group, LLC