Related papers: Uniqueness of the maximum likelihood estimator for…

We characterize the existence of the maximum likelihood estimator for discrete exponential families. Our criterion is simple to apply as we show in various settings, most notably for exponential models of random graphs. As an application,…

We develop here several goodness-of-fit tests for testing the k-monotonicity of a discrete density, based on the empirical distribution of the observations. Our tests are non-parametric, easy to implement and are proved to be asymptotically…

This paper deals with multivariate Gaussian models for which the covariance matrix is a Kronecker product of two matrices. We consider maximum likelihood estimation of the model parameters, in particular of the covariance matrix. There is…

We present theoretical properties of the log-concave maximum likelihood estimator of a density based on an independent and identically distributed sample in $\mathbb{R}^d$. Our study covers both the case where the true underlying density is…

We prove asymptotic normality of the so-called maximum likelihood estimator of the extreme value index.

We give a general proof of the strong consistency of the Maximum Likelihood Estimator for the case of independent non-identically distributed (i.n.i.d) data, assuming that the density functions of the random variables follow a particular…

We consider maximum likelihood estimation of finite mixture of uniform distributions. We prove that maximum likelihood estimator is strongly consistent, if the scale parameters of the component uniform distributions are restricted from…

The three-parameter generalized extreme value distribution arises from classical univariate extreme value theory and is in common use for analyzing the far tail of observed phenomena. Curiously, important asymptotic properties of…

The maximum-likelihood method for quantum estimation is reviewed and applied to the reconstruction of density matrix of spin and radiation as well as to the determination of several parameters of interest in quantum optics.

Shape constrained densities are encountered in many nonparametric estimation problems. The classes of monotone or convex (and monotone) densities can be viewed as special cases of the classes of k-monotone densities. A density g is said to…

We study nonparametric maximum likelihood estimation for two classes of multivariate distributions that imply strong forms of positive dependence; namely log-supermodular (MTP$_2$) distributions and log-$L^\#$-concave (LLC) distributions.…

We study the asymptotic behavior of the Maximum Likelihood and Least Squares Estimators of a $k$-monotone density $g_0$ at a fixed point $x_0$ when $k>2$. We find that the $j$th derivative of the estimators at $x_0$ converges at the rate…

We propose two least-squares estimators of a discrete probability under the constraint of k-monotony and study their statistical properties. We give a characterization of these estimators based on the decomposition on a spline basis of…

We show the uniqueness for the measure of maximal entropy for regular automorphisms of C^k.

We show that algebraic varieties with maximum likelihood degree one are exactly the images of reduced A-discriminantal varieties under monomial maps with finite fibers. The maximum likelihood estimator corresponding to such a variety is…

Maximum likelihood estimation in statistics leads to the problem of maximizing a product of powers of polynomials. We study the algebraic degree of the critical equations of this optimization problem. This degree is related to the number of…

We settle a conjecture by Coons and Sullivant stating that the maximum likelihood (ML) degree of a facial submodel of a toric model is at most the ML degree of the model itself. We discuss the impact on the ML degree from observing zeros in…

Let X_1, ..., X_n be independent and identically distributed random vectors with a log-concave (Lebesgue) density f. We first prove that, with probability one, there exists a unique maximum likelihood estimator of f. The use of this…

Consider a parametrized family of general hidden Markov models, where both the observed and unobserved components take values in a complete separable metric space. We prove that the maximum likelihood estimator (MLE) of the parameter is…

Bogdan et al. established a new criterion to determine the existence of a maximum likelihood estimator in discrete exponential families. It uses the notion of the set of uniqueness, which allows to apply the problem to the Ising model from…