Probability Statistics

An introduction to probability theory and its applications, by William Feller

By William Feller

Appropriate for self examine Use genuine examples and genuine facts units that would be conventional to the viewers advent to the bootstrap is integrated – it is a smooth technique lacking in lots of different books

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6 (b) estimates based on the canonical versions of these kernels and equal bandwidths are plotted. In this case the estimates are almost identical. 21), we only need to choose K to minimise C(K00 ). However, because of the scale invariance of C(K), the optimal K is the one that minimises C(K) subject to j K(x)dx = 1, j xK(x)dx = 0, j x K(x)dx = 2 and a2 < oo K(x) 2 0 for all x. The solution can be shown to be (Hodges and Lehmann, 1956). 22) t This kernel is often called the Epanechnikov kernel since its optimality properties in the density estimation setting were first described by Epanechnikov (1969).

Hodges (1951). This report was published out of historical interest as Fix and Hodges (1989). See also Silverman and Jones (1989). Akaike (1954) also contains some of these basic ideas. However, it was Rosenblatt (1956) and Parzen (1962) that provided the stimulus for considerable further interest in kernel methodology. Recent books on kernel density estimation include Silverman (1986), Hiirdle (1990a) and Scott (1992). 3 Detailed studies of the kernel density estimator with respect to the MIAE criterion, or expected L 1 norm, are given in Devroye and Gyorfi (1985) and Devroye (1987).

G. Gasser and Muller, 1979). , and K equal to the biweight kernel. UNIVARIATE KERNEL DENSITY ESTIMATION 48 \ 0 C\i Oi ···········a= -·-a= ---a= -a= c j~ ~ co "Co c . 17. KL(u;a) based on the biweight kernel for a= (solid curve}, ~ (dashed curve}, ~ (dotted curve} and 1 (dotdashed curve). 18 shows a kernel density estimate of the exponential density with KL(·; a) used near the boundary, with K equal to the biweight kernel. 5. 18. Kernel density estimate based on a sample of size 1000 from the exponential density with KL(·; a) used near the left boundary at x = 0.

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