Assume that,${\mathbf { X}}={\mathcal { S}}(\theta _{0},{\mathbf { Y}})$, are$n \ i.i.d$. observations in$\mathbb {R}^{p}$, with cdf$F_{\theta _{0}}$of unknown mathematical form;$\mathcal { S}$is a black-box,$p\ge 1$. Parameter$\theta _{0}$is unknown, element of metric space$(\Theta , d_{\Theta })$which can be infinite dimensional.Yis random, with distribution independent of$\theta _{0}$. For each$\theta ^{*}$in a discretization,$\Theta ^{*}$, of$\Theta , \ F_{\theta ^{*}}$is “learned” using independent${\mathbf { X}}^{*}_{j}(\theta ^{*})={\mathcal { S}}(\theta ^{*}, {\mathbf { Y}}), 1\le j \le N_{rep}$. Kolmogorov distance,$d_{K}$, and the empirical cdfs of the observed$\bf x$and of the$N_{rep}~{\mathbf { X}}^{*}_{j}(\theta ^{*}), \theta ^{*} \in \Theta ^{*}$, are used to introduce for$\theta _{0}:~a$) the minimum matching distance estimator (MMDE), andb) the brand new, maximum matching support probability estimator (MMSPE). Under two mild assumptions, uniform consistency for both estimators is established. Their common upper error rate in probability depends on Kolmogorov’s entropy of$\Theta $, is not affected bypwhen$\Theta \subseteq \mathbb {R}^{m}$, and establishes that MMSPE is competitive. Both estimators have theoretical advantages over competitors, and complement each other in numerical applications presented for$p=1, 2$, because of$d_{K}$’s computational cost for largep, relaxed with recent algorithms. MMSPE obtained using, instead of$d_{K}$, the Euclidean sup-norm distance of$\bf x$from each of the$N_{rep}~{\mathbf { X}}^{*}_{j} (\theta ^{*}), \theta ^{*} \in \Theta ^{*}$, provides an approximation for the MLE when$N_{rep}$is large, indicating MLE’s limitations for largep. MMDE and MMSPE obtained using Wolfowitz’s half-spaces distance and matching tool the empirical measure instead of the empirical cdf, have a computational advantage for largep.

The inherent bias pathology of the maximum likelihood estimation method is confirmed for models with unknown parameters θ and ψ when maximum likelihood estimate (MLE) ψ̂ is function of MLE θ̂. To reduce ψ̂'s bias the likelihood equation to be solved for ψ is updated using the model for the data Y in it. For various models with ψ a shape parameter model updated (MU) MLE, ψ̂MU, reduces ψ̂'s bias. For the Pareto model ψ̂MUreduces in addition ψ̂'s variance. The results explain the difference that puzzled Fisher, between biased ψ̂ and the unbiased estimate he obtained for two models with the abandoned two-stage procedure used in MUMLE's implementation. ψ̂MUis also obtained with the minimum message length method thus motivating the use of priors in frequentist inference.