On Strong Consistency of Density Estimates

Let X₁, X₂, ⋯, Xₙ, be a sample of n independent observations of a random variable X with distribution F(x) = F(x₁, ⋯, xₘ) or Rᵐ and Lebesgue density f(x) = f(x₁, ⋯, xₘ). To estimate the density $f(x)$ consider estimates of the form {equation*} {1} f_n(x) = n⁻¹ ∑^nⱼ₌₁ K_n(x, X_j), K_n(x, X_j) = h_n⁻ᵐK(h_n⁻¹(x - X_j));{equation*} where K(u) = K(u₁, ⋯, uₘ) is a real-valued Borel-measurable function on Rᵐ such that {equation*} {2} K(u) is a density on R^m{equation*} {equation*} {3} uε R^m K(u) < ∞{equation*} {equation*} {4} \|u\|^m K(u) → 0 as \|u\|^2 = ∑^mᵢ₌₁ u_i^2 → ∞{equation*} and ₙ\ is a sequence of numbers such that {equation*} {5} h_n > 0, n = 1, 2, ⋯; limn→∞ h_n = 0 and limn→∞ nh_n^m = ∞.{equation*} Such density estimates have been shown to be weakly consistent (that is, fₙ(x) → f(x) in probability as n → ∞) on the continuity set, $C(f),$ of the density $f(x)$ by Parzen [4] for $m = 1$ and by Cacoullos [1] for $m > 1$. In Theorem 1, we state conditions under which strong consistency (that is, fₙ(x) → f(x) with probability one as n → ∞) of such estimates obtains. Theorem 2 gives conditions under which uniform (in x) strong consistency of the estimates (1) is valid. In this respect, our results are very similar in the case $m = 1$ to those of Nadaraya [4], although the method of proof and conditions imposed are different. Theorem 3 concerns the estimation of the unique mode of the density $f(x)$ when it exists.