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Journal Article

# A Polynomial Algorithm for Density Estimation

Grace Wahba
The Annals of Mathematical Statistics
Vol. 42, No. 6 (Dec., 1971), pp. 1870-1886
Stable URL: http://www.jstor.org/stable/2240113
Page Count: 17

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## Abstract

An algorithm for density estimation based on ordinary polynomial (Lagrange) interpolation is studied. Let Fn(x) be n/(n + 1) times the sample c.d.f. based on n order statistics, t1, t2, ⋯ tn, from a population with density f(x). It is assumed that f(v) is continuous, v = 0, 1, 2,⋯, r, r = m - 1, and f(m) ∈ L2(-∞, ∞). Fn(x) is first locally interpolated by the mth degree polynomial passing through Fn(tikn ), Fn(t(i+1)kn ),⋯ Fn(t(i+m)kn ), where kn is a suitably chosen number, depending on n. The density estimate is then, locally, the derivative of this interpolating polynomial. If kn = O(n(2m-1)/(2m)), then it is shown that the mean square convergence rate of the estimate to the true density is O(n-(2m-1)/(2m)). Thus these convergence rates are slightly better than those obtained by the Parzen kernel-type estimates for densities with r continuous derivatives. If it is assumed that f(m) is bounded, and kn = O(n2m/(2m+1)), then it is shown that the mean square convergence rates are O(n-2m/(2m+1)), which are the same as those of the Parzen estimates for m continuous derivatives. An interesting theorem about Lagrange interpolation, concerning how well a function can be interpolated knowing only its integral at nearby points, is also demonstrated.

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