Optimal Convergence Properties of Variable Knot, Kernel, and Orthogonal Series Methods for Density Estimation

Let Wₚ⁽ᵐ⁾(M) = : f(ν) abs. cont., ν = 0, 1,⋯, m - 1, f⁽ᵐ⁾ ∈ Lₚ, \|f⁽ᵐ⁾\|ₚ M\, where \|·\|ₚ is the norm in Lₚ, m is a positive integer and p is a real number, p 1. Let ̂ₙ(x)\, n = 1, 2,⋯ be any sequence of estimates of a density at the point x where f̂ₙ(x) depends on n independent observations from some density f ∈ Wₚ⁽ᵐ⁾(M). It is shown that if _f∈ Wₚ⁽ᵐ⁾ (M) Ef(f(x) - f̂ₙ(x))² = bₙn-φ(m, p+ε), where φ(m, p) = (2m - 2/p)/(2m + 1 - 2/p), and ε > 0, then there exists a D₀ > 0 such that bₙ D₀ for infinitely many n. Thus the best possible mean square convergence rate for a density estimate, which is uniform over Wₚ⁽ᵐ⁾(M), is not better than n-φ(m,p+ε) for arbitrarily small ε. The following types of density estimates are shown to have mean square error at a point bounded above by Dn-φ(m,p), provided that a certain parameter, usually depending on $m, p$ and M, is chosen optimally: the polynomial algorithm, kernel-type estimates, certain orthogonal series estimates, and the ordinary histogram. D's for each method are given.