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

Abstract

We study the consistency properties of a nonparametric estimator $f_n$ 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 $\exp\{- \alpha \int (f'/f)^2 f\}$ with $\alpha > 0$. Under modest assumptions on the density function $f$, and letting $\alpha = \alpha_n \rightarrow \infty$ and $\alpha_n/n \rightarrow 0$ a.s. as $n \rightarrow \infty$ we demonstrate the a.s. convergence of $f_n$ to $f$, with rates, in the Hellinger, $L_1, L_2, \sup_{\mathbb{R}}$ 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.