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 (t₈₊䂸), Fₙ (t (₈+₁) ₊䂸), Fₙ (t (₈+₌) ₊䂸), 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 (n^2m/ (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.
Grace Wahba (Wed,) studied this question.
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