A new class of tests for multinormality with i.i.d. and garch data based on the empirical moment generating function

We generalize a recent class of tests for univariate normality that are based on the empirical moment generating function to the multivariate setting, thus obtaining a class of affine invariant, consistent and easy-to-use goodness-of-fit tests for multinormality. The test statistics are suitably weighted L2-statistics, and we provide their asymptotic behavior both for i.i.d. observations as well as in the context of testing that the innovation distribution of a multivariate GARCH model is Gaussian. We study the finite-sample behavior of the new tests, compare the criteria with alternative existing procedures, and apply the new procedure to a data set of monthly log returns.Ministerio de Economía y Competitividad (MINECO). Españ

A goodness-of-fit test for the multivariate Poisson distribution

Bivariate count data arise in several different disciplines and the bivariate Poisson distribution is commonly used to model them. This paper proposes and studies a computationally convenient goodness-of-fit test for this distribution, which is based on an empirical counterpart of a system of equations. The test is consistent against fixed alternatives. The null distribution of the test can be consistently approximated by a parametric bootstrap and by a weighted bootstrap. The goodness of these bootstrap estimators and the power for finite sample sizes are numerically studied. It is shown that the proposed test can be naturally extended to the multivariate Poisson distribution.Universidad del Bío-Bío (Chile)Ministerio de Educación (Chile)Ministerio de Economía y Competitivida

Una generalización de la métrica de Hausdorff sobre C(Rn)

En este trabajo hacemos una extensión de la métrica de Hausdorff H sobre C(Rn), el espacio de todos los conjuntos difusos cerrados en Rn, obteniendo una familia de métricas Df. Estudiamos algunas propiedades topológicas del espacio métrico (C(Rn), Df).Ministerio de Ciencia e InnovaciónComisión Nacional de Investigación Científica y Tecnológica (Chile)Fondo Nacional de Desarrollo Científico y Tecnológic

Computationally efficient goodness-of-fit tests for the error distribution in nonparametric regression

Several procedures have been proposed for testing goodness-of-fit to the error distribution in nonparametric regression models. The null distribution of the associated test statistics is usually approximated by means of a parametric bootstrap which, under certain conditions, provides a consistent estimator. This paper considers a goodness-of-fit test whose test statistic is an L2 norm of the difference between the empirical characteristic function of the residuals and a parametric estimate of the characteristic function in the null hypothesis. It is proposed to approximate the null distribution through a weighted bootstrap which also produces a consistent estimator of the null distribution but, from a computational point of view, is more efficient than the parametric bootstrap.Fundación CarolinaUniversidad Nacional de AsunciónUniversidad de SevillaMinisterio de Economía y Competitivida

Nonparametric estimation of the shape functions and related asymptotic results

Arriaza et al (Metrika 82:99–124, 2019) introduced the right and left shape functions, which enjoy interesting properties in terms of describing the global form of a distribution. This paper proposes and studies nonparametric estimators of those functions. The estimators involve nonparametric estimation of the quantile and density functions. Pointwise and uniform consistency are proved under general regularity assumptions, as well as the limit in law. Simulations are included to study the practical performance of the proposed estimators. The analysis of a real data set illustrates the methodology.Agencia Estatal de Investigación | Ref. PID2020-118101GB-I00Universidade de Vigo/CISU

Characterizations of multinormality and corresponding tests of fit, including for Garch models

We provide novel characterizations of multivariate normality that incorporate both the characteristic function and the moment generating function, and we employ these results to construct a class of affine invariant, consistent and easy-to-use goodness-of-fit tests for normality. The test statistics are suitably weighted L2-statistics, and we provide their asymptotic behavior both for i.i.d. observations as well as in the context of testing that the innovation distribution of a multivariate GARCH model is Gaussian. We also study the finite-sample behavior of the new tests and compare the new criteria with alternative existing tests

Reduced bootstrap for the median

In this paper we study a modified bootstrap that consists of only considering those bootstrap samples satisfying k1 ≤ νn ≤ k2, for some 1 ≤ k1 ≤ k2 ≤ n, where νn is the number of distinct original observations in the bootstrap sample. We call it reduced bootstrap, since it only uses a portion of the set of all possible bootstrap samples. We show that, under some conditions on k1 and k2, the reduced bootstrap consistently estimates the distribution and the variance of the sample median. Unlike the ordinary bootstrap, the reduced bootstrap variance estimator does not require conditions on the population generating the data to be a consistent estimator, but does rely an adequate choice of k1 and k2. Since several choices of k1 and k2 yield consistent estimators, we compare the finite sample performance of the corresponding estimators through a simulation study. The simulation study also considers consistent variance estimators proposed by other authors.Ministerio de Educación y Cienci

A Monte Carlo comparison of three consistent bootstrap procedures

Since bootstrap samples are simple random samples with replacement from the original sample, the information content of some bootstrap samples can be very low. To avoid this fact, some authors have proposed several variants of the classical bootstrap. In this paper we consider two of them: the sequential or Poisson bootstrap and the reduced bootstrap. Both of them, like ordinary bootstrap, can yield second order accurate distribution estimators, that is, the three bootstrap procedures are asymptotically equivalent. The question that naturally arises is which of them should be used in a practical situation, in other words, which of them should be used for finite sample sizes. To try to answer this question, we have carried out a simulation study. Although no method was found to exhibit best performance in all the considered situations, some recommendations are given.Ministerio de Educación y Cienci

Minimum penalized φ-divergence estimation under model misspecification

This paper focuses on the consequences of assuming a wrong model for multinomial data when using minimum penalized φ-divergence, also known as minimum penalized disparity estimators, to estimate the model parameters. These estimators are shown to converge to a well-defined limit. An application of the results obtained shows that a parametric bootstrap consistently estimates the null distribution of a certain class of test statistics for model misspecification detection. An illustrative application to the accuracy assessment of the thematic quality in a global land cover map is included.Ministerio de Economía y Competitivida