Consistency of Two Nonparametric Maximum Penalized Likelihood Estimators of the Probability Density Function

We study the consistency properties of a nonparametric estimator fₙ of a density function f on the real line, which is known as the "first MPLE of Good and Gaskins," and which is obtained by maximizing the likelihood functional multiplied by the roughness penality exp\- α ∫ (f'/f)² f\ with α > 0. Under modest assumptions on the density function f, and letting α = αₙ → ∞ and αₙ/n → 0 a.s. as n → ∞ we demonstrate the a.s. convergence of fₙ to f, with rates, in the Hellinger, L₁, L₂, R and Sobolev norms, as well as in integrated mean absolute deviation. Finally, the corresponding estimator for f supported on the half-line, is derived and the computational feasibility as well as the consistency properties of the estimator are indicated.