Javier Cárcamo

Tests for the Second Order Stochastic Dominance Based on L -Statistics

We use some characterizations of convex and concave-type orders to define discrepancy measures useful in two testing problems involving stochastic dominance assumptions. The results are connected with the mean value of the order statistics and have a clear economic interpretation in terms of the expected cumulative resources of the poorest (or richest) in random samples. Our approach mainly consists of comparing the estimated means in ordered samples of the involved populations. The test statistics we derive are functions of L-statistics and are generated through estimators of the mean order statistics. We illustrate some properties of the procedures with simulation studies and an empirical example.

Tests for zero-inflation and overdispersion: A new approach based on the stochastic convex order

A new methodology to detect zero-inflation and overdispersion is proposed, based on a comparison of the expected sample extremes among convexly ordered distributions. The method is very flexible and includes tests for the proportion of structural zeros in zero-inflated models, tests to distinguish between two ordered parametric families and a new general test to detect overdispersion. The performance of the proposed tests is evaluated via some simulation studies. For the well-known fetal lamb data, the conclusion is that the zero-inflated Poisson model should be rejected against other more disperse models, but the negative binomial model cannot be rejected.

An extremal property of Bernstein operators

We establish a strong version of a known extremal property of Bernstein operators, as well as several characterizations of a related specific class of positive polynomial operators.

On Certain Best Constants for Bernstein-Type Operators

We discuss the generalized version of a best-constant problem raised by Z. Li in a note which recently appeared in Journal of Approximation Theory. Some best constants for well known Bernstein-type operators are obtained.

Tests for Stochastic Orders and Mean Order Statistics

The aim of this article is to emphasize the fact, not observed previously in the literature, that many discrepancy measures used in tests related to different stochastic orders can be expressed as expectations of order statistics. In this way, we provide a new meaning to the corresponding test statistics which allows us to understand better, and potentially improve, the testing procedures. As illustration, we consider tests to detect overdispersion with respect to a specific probability model. In this setting, a test for the Weibull distribution is discussed in detail.

A Test for Convex Dominance With Respect to the Exponential Class Based on an

We consider the problem of testing if a non-negative random variable is dominated, in the convex order, by the exponential class. Under the null hypothesis, the variable is harmonic new better than used in expectation (HNBUE), a well-known class of ageing distributions in reliability theory. As a test statistic, we propose the L1 norm of a suitable distance between the empirical and the exponential distributions, and we completely determine its asymptotic properties. The practical performance of our proposal is illustrated with simulation studies, which show that the asymptotic test has a good behavior and power, even for small sample sizes. Finally, three real data sets are analyzed.

L^1

Given a classifier, we describe a general method to construct a simple linear classification rule. This rule, called the tangent classifier, is obtained by computing the tangent hyperplane to the separation boundary of the groups (generated by the initial classifier) at a certain point. When applied to a quadratic region, the tangent classifier has a neat closed-form expression. We discuss various examples and the application of this new linear classifier in two situations under which standard rules may fail: when there is a fraction of outliers in the training sample and when the dimension of the data is large in comparison with the sample size.

Distance

The classification of the X-ray sources into classes (such as extragalactic sources, background stars,...) is an essential task in astronomy. Typically, one of the classes corresponds to extragalactic radiation, whose photon emission behaviour is well characterized by a homogeneous Poisson process. We propose to use normalized versions of the Wasserstein and Zolotarev distances to quantify the deviation of the distribution of photon interarrival times from the exponential class. Our main motivation is the analysis of a massive dataset from X-ray astronomy obtained by the Chandra Orion Ultradeep Project (COUP). This project yielded a large catalog of 1616 X-ray cosmic sources in the Orion Nebula region, with their series of photon arrival times and associated energies. We consider the plug-in estimators of these metrics, determine their asymptotic distributions, and illustrate their finite-sample performance with a Monte Carlo study. We estimate these metrics for each COUP source from three different classes. We conclude that our proposal provides a striking amount of information on the nature of the photon emitting sources. Further, these variables have the ability to identify X-ray sources wrongly catalogued before. As an appealing conclusion, we show that some sources, previously classified as extragalactic emissions, have a much higher probability of being young stars in Orion Nebula.

The Tangent Classifier

The problem of establishing Hadamard-type inequalities for convex functions on d-dimensional convex bodies (d ≥ 2) translates into the problem of finding appropriate majorants of the involved random vector for the usual convex order. In this work, we use a stochastic approach based on the Brownian motion to establish a multidimensional version of the classical Hadamard inequality. The main result is closely related to the Dirichlet problem and is applied to obtain inequalities for harmonic functions on general convex bodies.