Estimation in Discrete Parameter Models

In some estimation problems, especially in applications dealing with information theory, signal processing and biology, theory provides us with additional information allowing us to restrict the parameter space to a finite number of points. In this case, we speak of discrete parameter models. Even though the problem is quite old and has interesting connections with testing and model selection, asymptotic theory for these models has hardly ever been studied. Therefore, we discuss consistency, asymptotic distribution theory, information inequalities and their relations with efficiency and superefficiency for a general class of m-estimators

The United States COVID-19 Forecast Hub dataset

Academic researchers, government agencies, industry groups, and individuals have produced forecasts at an unprecedented scale during the COVID-19 pandemic. To leverage these forecasts, the United States Centers for Disease Control and Prevention (CDC) partnered with an academic research lab at the University of Massachusetts Amherst to create the US COVID-19 Forecast Hub. Launched in April 2020, the Forecast Hub is a dataset with point and probabilistic forecasts of incident cases, incident hospitalizations, incident deaths, and cumulative deaths due to COVID-19 at county, state, and national, levels in the United States. Included forecasts represent a variety of modeling approaches, data sources, and assumptions regarding the spread of COVID-19. The goal of this dataset is to establish a standardized and comparable set of short-term forecasts from modeling teams. These data can be used to develop ensemble models, communicate forecasts to the public, create visualizations, compare models, and inform policies regarding COVID-19 mitigation. These open-source data are available via download from GitHub, through an online API, and through R packages

On embedding Gretl in a Python module

Additional functionalities can be developed for Gretl either directly in the main C code or with the Gretl scripting language. We illustrate through an example how it would be possible to wrap the C source of Gretl with SWIG to create an interface to Python that makes use of the matrix library NumPy. Such an interface would make it easier for users to extend Gretl since it would allow for developing and distributing Gretl extensions as Python modules

The Asymptotic Distribution of Quadratic Discrepancies

In Numerical Analysis, several discrepancies have been introduced to test that a sample of n points in the unit hypercube [0,1]d comes from a uniform distribution. An outstanding example is given by Hickernell\u2019s generalized Lp 12discrepancies, that constitute a generalization of the Kolmogorov-Smirnov and the Cram\ue9r-von Mises statistics. These discrepancies can be used in numerical integration by Monte Carlo and quasi-Monte Carlo methods, design of experiments, uniformity and goodness of fit tests. In this paper, after having recalled some necessary asymptotic results derived in companion papers, we show that the case of L2 12discrepancies is more convenient to handle and we provide a new computational approximation of their asymptotic distribution. As an illustration, we show that our algorithm is able to recover the tabulated asymptotic distribution of the Cram\ue9r-von Mises statistic. The results so obtained are very general and can be applied with minor modifications to other discrepancies, such as the diaphony, the weighted spectral test, the Fourier discrepancy and the class of chi-square tests

Computational aspects of discrepancies for equidistribution on the hypercube

In this paper, we study the asymptotic statistical properties of some discrepancies defined on the unit hypercube, originally introduced in Numerical Analysis to assess the equidistribution of low-discrepancy sequences. We show that they have highly desirable properties. Nevertheless, it turns out that the limiting distribution is an (infinite) weighted sum of chi-squared random variables. This raises some problems concerning the approximation of the asymptotic distribution. These issues are considered in detail: several solutions are proposed and compared, and bounds for the approximation error are discussed

Bootstrap confidence sets for the Aumann mean of a random closed set

The objective is to develop a reliable method to build confidence sets for the Aumann mean of a random closed set as estimated through the Minkowski empirical mean. First, a general definition of the confidence set for the mean of a random set is provided. Then, a method using a characterization of the confidence set through the support function is proposed and a bootstrap algorithm is described, whose performance is investigated in Monte Carlo simulations

Statistical Properties of Generalized Discrepancies and Related Quantities

When testing that a sample of n points in the unit hypercube [0,1]d comes from a uniform distribution, the Kolmogorov-Smirnov and the Cram\ue9r-von Mises statistics are simple and well-known procedures. To encompass these measures of uniformity, Hickernell (1996, 1998) introduced the so-called generalized Lp 12discrepancies. These discrepancies can be used in numerical integration by Monte Carlo and quasi-Monte Carlo methods, design of experiments, uniformity testing and goodness of fit tests. The aim of this paper is to derive the strong and weak asymptotic properties of these statistics

Confidence Sets for the Aumann Mean of a Random Closed Set

The objective of this paper is to develop a set of reliable methods to build confidence sets for the Aumann mean of a random closed set estimated through the Minkowski empirical mean. In order to do so, we introduce a procedure to build a confidence set based on Weil\u2019s result for the Hausdorff distance between the empirical and the Aumann means; then, we introduce a new procedure based on the support function

Stochastic Boundedness in Biological Models

In this article, we give sufficient conditions for stochastic boundedness in population models

Approximation of Stochastic Programming Problems

In Stochastic Programming, the aim is often the optimization of a criterion function that can be written as an integral or mean functional with respect to a probability measure P. When this functional cannot be computed in closed form, it is customary to approximate it through an empirical mean functional based on a random Monte Carlo sample. Several improved methods have been proposed, using quasi-Monte Carlo samples, quadrature rules, etc. In this paper, we propose a result on the epigraphical approximation of an integral functional through an approximate one. This result allows us to deal with Monte Carlo, quasi-Monte Carlo and quadrature methods. We propose an application to the epi-convergence of stochastic programs approximated through the empirical measure based on an asymptotically mean stationary (ams) sequence. Because of the large scope of applications of ams measures in Applied Probability, this result turns out to be relevant for approximation of stochastic programs through real data