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.