A Polynomial Algorithm for Density Estimation

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.