arXiv subjects

Javier Cárcamo

Publications and source records attributed to Javier Cárcamo.

7 recordsLinked to original sources

arXiv2025

Bootstrap tests for almost goodness-of-fit

We introduce the \textit{almost goodness-of-fit} test, a procedure to assess whether a (parametric) model provides a good representation of the probability distribution generating the observed sample. Specifically, given a distribution function $F$ and a parametric family $\mathcal{G}=\{ G(\boldsymbolθ) : \boldsymbolθ \in Θ\}$, we consider the testing problem \[ H_0: \| F - G(\boldsymbolθ_F) \|_p \geq ε\quad \text{vs} \quad H_1: \| F - G(\boldsymbolθ_F) \|_p < ε, \] where $ε>0$ is a margin of error and $G(\boldsymbolθ_F)$ denotes a representative of $F$ within the parametric class. The approximate model is determined via an M-estimator of the parameters. %The objective is the approximate validation of a distribution or an entire parametric family up to a pre-specified threshold value. The methodology also quantifies the percentage improvement of the proposed model relative to a non-informative (constant) benchmark. The test statistic is the $\mathrm{L}^p$-distance between the empirical distribution function and that of the estimated model. We present two consistent, easy-to-implement, and flexible bootstrap schemes to carry out the test. The performance of the proposal is illustrated through simulation studies and analysis and real-data applications.

stat.ME↗

arXiv2024

On uniqueness of the set of k-means

We provide necessary and sufficient conditions for the uniqueness of the k-means set of a probability distribution. This uniqueness problem is related to the choice of k: depending on the underlying distribution, some values of this parameter could lead to multiple sets of k-means, which hampers the interpretation of the results and/or the stability of the algorithms. We give a general assessment on consistency of the empirical k-means adapted to the setting of non-uniqueness and determine the asymptotic distribution of the within cluster sum of squares (WCSS). We also provide statistical characterizations of k-means uniqueness in terms of the asymptotic behavior of the empirical WCSS. As a consequence, we derive a bootstrap test for uniqueness of the set of k-means. The results are illustrated with examples of different types of non-uniqueness and we check by simulations the performance of the proposed methodology.

math.ST↗

arXiv2024

A uniform kernel trick for high-dimensional two-sample problems

We use a suitable version of the so-called "kernel trick" to devise two-sample (homogeneity) tests, especially focussed on high-dimensional and functional data. Our proposal entails a simplification related to the important practical problem of selecting an appropriate kernel function. Specifically, we apply a uniform variant of the kernel trick which involves the supremum within a class of kernel-based distances. We obtain the asymptotic distribution (under the null and alternative hypotheses) of the test statistic. The proofs rely on empirical processes theory, combined with the delta method and Hadamard (directional) differentiability techniques, and functional Karhunen-Loève-type expansions of the underlying processes. This methodology has some advantages over other standard approaches in the literature. We also give some experimental insight into the performance of our proposal compared to the original kernel-based approach \cite{Gretton2007} and the test based on energy distances \cite{Szekely-Rizzo-2017}.

math.ST↗

arXiv2024

Tests for almost stochastic dominance

We introduce a 2-dimensional stochastic dominance (2DSD) index to characterize both strict and almost stochastic dominance. Based on this index, we derive an estimator for the minimum violation ratio (MVR), also known as the critical parameter, of the almost stochastic ordering condition between two variables. We determine the asymptotic properties of the empirical 2DSD index and MVR for the most frequently used stochastic orders. We also provide conditions under which the bootstrap estimators of these quantities are strongly consistent. As an application, we develop consistent bootstrap testing procedures for almost stochastic dominance. The performance of the tests is checked via simulations and the analysis of real data.

econ.EM↗

arXiv2021

Extremal points of Lorenz curves and applications to inequality analysis

We find the set of extremal points of Lorenz curves with fixed Gini index and compute the maximal $L^1$-distance between Lorenz curves with given values of their Gini coefficients. As an application we introduce a bidimensional index that simultaneously measures relative inequality and dissimilarity between two populations. This proposal employs the Gini indices of the variables and an $L^1$-distance between their Lorenz curves. The index takes values in a right-angled triangle, two of whose sides characterize perfect relative inequality-expressed by the Lorenz ordering between the underlying distributions. Further, the hypotenuse represents maximal distance between the two distributions. As a consequence, we construct a chart to, graphically, either see the evolution of (relative) inequality and distance between two income distributions over time or to compare the distribution of income of a specific population between a fixed time point and a range of years. We prove the mathematical results behind the above claims and provide a full description of the asymptotic properties of the plug-in estimator of this index. Finally, we apply the proposed bidimensional index to several real EU-SILC income datasets to illustrate its performance in practice.

econ.EM↗

arXiv2019

Directional differentiability for supremum-type functionals: statistical applications

We show that various functionals related to the supremum of a real function defined on an arbitrary set or a measure space are Hadamard directionally differentiable. We specifically consider the supremum norm, the supremum, the infimum, and the amplitude of a function. The (usually non-linear) derivatives of these maps adopt simple expressions under suitable assumptions on the underlying space. As an application, we improve and extend to the multidimensional case the results in \cite{Raghavachari} regarding the limiting distributions of Kolmogorov-Smirnov type statistics under the alternative hypothesis. Similar results are obtained for analogous statistics associated with copulas. We additionally solve an open problem about the Berk-Jones statistic proposed by \cite{Jager-Wellner-2004}. Finally, the asymptotic distribution of maximum mean discrepancies over Donsker classes of functions is derived.

math.ST↗

arXiv2018

New distance measures for classifying X-ray astronomy data into stellar classes

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.

math.ST↗