We establish global rates of convergence for the Maximum Likelihood Estimators (MLEs) of log-concave and s-concave densities on ℝ. The main finding is that the rate of convergence of the MLE in the Hellinger metric is no worse than n-2/5 when -1 < s < ∞ where s = 0 corresponds to the log-concave case. We also show that the MLE does not exist for the classes of s-concave densities with s < -1.

Abstract. Since Manski’s (1975) seminal work, the maximum score method for discrete choice models has been applied to various econometric problems. Kim and Pollard (1990) established the cube root asymptotics for the maximum score estimator. Since then, however, econometricians posed several open questions and conjectures in the course of generalizing the maximum score approach, such as (a) asy...

consider the linear system ax=b where the coefficient matrix a is an m-matrix. in the present work, it is proved that the rate of convergence of the gauss-seidel method is faster than the mixed-type splitting and aor (sor) iterative methods for solving m-matrix linear systems. furthermore, we improve the rate of convergence of the mixed-type splitting iterative method by applying a precondition...

consider the linear system ax=b where the coefficient matrix a is an m-matrix. in the present work, it is proved that the rate of convergence of the gauss-seidel method is faster than the mixed-type splitting and aor (sor) iterative methods for solving m-matrix linear systems. furthermore, we improve the rate of convergence of the mixed-type splitting iterative method by applying a precondition...

In this paper, we show a phenomenon where residual networks can be trained using an order of magnitude fewer iterations than is used with standard training methods, which we named “superconvergence.” One of the key elements of super-convergence is training with cyclical learning rates and a large maximum learning rate. Furthermore, we present evidence that training with large learning rates imp...

The `1-regularized maximum likelihood estimation problem has recently become a topic of great interest within the machine learning, statistics, and optimization communities as a method for producing sparse inverse covariance estimators. In this paper, a proximal gradient method (G-ISTA) for performing `1-regularized covariance matrix estimation is presented. Although numerous algorithms have be...

The `1-regularized maximum likelihood estimation problem has recently become a topic of great interest within the machine learning, statistics, and optimization communities as a method for producing sparse inverse covariance estimators. In this paper, a proximal gradient method (G-ISTA) for performing `1-regularized covariance matrix estimation is presented. Although numerous algorithms have be...

MaximumVariance Unfolding is one of the main methods for (nonlinear) dimensionality reduction. We study its large sample limit, providing specific rates of convergence under standard assumptions. We find that it is consistent when the underlying submanifold is isometric to a convex subset, and we provide some simple examples where it fails to be consistent.

We consider the Poisson equation −∆u = f with homogeneous Dirichlet boundary condition on a two-dimensional polygonal domain Ω with re-entrant angles. A multigrid method for the computation of singular solutions and stress intensity factors using piecewise linear functions is analyzed. When f ∈ L2(Ω), the rate of convergence to the singular solution in the energy norm is shown to be O(h), and t...