Joaquín Muñoz-García

Goodness-of-fit tests based on empirical characteristic functions

A class of goodness-of-fit tests based on the empirical characteristic function is studied. They can be applied to continuous as well as to discrete or mixed data with any arbitrary fixed dimension. The tests are consistent against any fixed alternative for suitable choices of the weight function involved in the definition of the test statistic. The bootstrap can be employed to estimate consistently the null distribution of the test statistic. The goodness of the bootstrap approximation and the power of some tests in this class for finite sample sizes are investigated by simulation.

Outliers : a formal approach

Summary We propose an outline which enables us to analyse in a generic way the errors which can affect the experimental observations. From this we state a definition for the term outlier, and typify the problems of Identification Techniques of Outliers. This allows us to construct a general framework in which to carry out a qualitative analysis of data which when applied to the Analysis of Outliers presents certain advantages with respect to the classical approach.

A comparison of classification models to identify the Fragile X Syndrome

The main models of machine learning are briefly reviewed and considered for building a classifier to identify the Fragile X Syndrome (FXS). We have analyzed 172 patients potentially affected by FXS in Andalusia (Spain) and, by means of a DNA test, each member of the data set is known to belong to one of two classes: affected, not affected. The whole predictor set, formed by 40 variables, and a reduced set with only nine predictors significantly associated with the response are considered. Four alternative base classification models have been investigated: logistic regression, classification trees, multilayer perceptron and support vector machines. For both predictor sets, the best accuracy, considering both the mean and the standard deviation of the test error rate, is achieved by the support vector machines, confirming the increasing importance of this learning algorithm. Three ensemble methods – bagging, random forests and boosting – were also considered, amongst which the bagged versions of support vector machines stand out, especially when they are constructed with the reduced set of predictor variables. The analysis of the sensitivity, the specificity and the area under the ROC curve agrees with the main conclusions extracted from the accuracy results. All of these models can be fitted by free R programs.

Identification of outlier bootstrap samples

We define a variation of Efrons method II based on the outlier bootstrap sample concept. A criterion for the identification of such samples is given, with which a variation in the bootstrap sample generation algorithm is introduced. The results of several simulations are analyzed in which, in comparison with Efrons method II, a higher degree of closeness to the estimated quantities can be observed.

Bootstrapping parameter estimated degenerate U and V statistics

Let X1,X2,...,Xn be independent random vectors with common distribution function F and let be a parametric family of distributions. Let Tn([theta])=Tn(X1,X2,...,Xn;[theta]) be a degree-2 V statistic and let be a consistent estimator of [theta]. Several test statistics for testing the composite null hypothesis has the form . Typically, the null distribution of depends on the unknown value of [theta]. The purpose of this paper is to show that the bootstrap can be used to approximate the null distribution of this type of statistics. We also give similar results for statistics , with Wn([theta]) a degree-2 U statistic.

The Frechet's metric as a measure of influence in multivariate linear models with random errors elliptically distributed

In general, the influence diagnostics are based on two aspects: the perturbation of the model and the comparison of the results. The most used scheme of perturbation is the omission of the cases and, generally, as a pattern of comparison, a metric is used. A sample version of the Frechets metric is proposed to quantify the deviation between the sample distributions of the statistics. This is the way that a generic influence measure is obtained which can be applied to a great deal of statistical techniques. In particular, for linear multivariate models with random errors elliptically distributed, it is applied on the BLUE of the estimable linear functions.

Two classes of divergence statistics for testing uniform association

The problem of testing uniform association in cross-classifications having ordered categories is considered. Two families of test statistics, both based on divergences between certain functions of the observed data, are studied and compared. Our theoretical study is based on asymptotic properties. For each family, two consistent approximations to the null distribution of the test statistic are studied: the asymptotic null distribution and a bootstrap estimator; all the tests considered are consistent against fixed alternatives; finally, we do a local power study. Surprisingly, both families detect the same local alternatives. The finite sample performance of the tests in these two classes is numerically investigated through some simulation experiments. In the light of the obtained results, some practical recommendations are given.

Influence analysis on discriminant coordinates

Discriminant analysis (DA), particularly Discriminant Coordinates (DC), is broadly applied in the scientific literature and included in many statistical software packages. DC is used to analyze biomedical data, especially for differential diagnosis on the basis of laboratory profiles. Articles handling influence analysis in DA can be found in the literature; however, this topic has been scarcely touched upon in DC. In this article, the case-deletion approach is followed to introduce a perturbation in the data and influence measures are proposed to assess the effect on three statistics of interest: the transformation matrix, canonical directions, and configuration, of the sample centroids.

Cressie and Read power-divergences as influence measures for logistic regression models

A sample version of the power-divergence measures of Cressie and Read is proposed for the influence analysis in the logistic regression model. Influence measures are obtained by quantifying the deviation between the sample distribution of an estimate obtained with all the observations and the sample distribution of the same estimate obtained without any observation. In particular, this approach is applied to three estimates of the model: the MLE of regression coefficients vector, the probabilities vector and the linear predictor of a future case. Some examples are considered to clarify the usefulness of the introduced diagnostics.

Bootstrapping the sample median

If F is a univariate ditribution function having a positive derivative continuous in a neighborhood of its median μ, the bootstrap distribution of is consistent (Bickel and Freedman (1981), Singh (1981)), where mn is the sample median. Nevertheless, to ensure the consistency of the bootstrap estimator of the asymptotic variance of Zn we need further conditions (Ghosh et al. (1984), Babu (1986)). In this paper we show that if the distribution of Zn is estimated through the method proposed by Munoz-Garcia et al. (1997), these additonal conditions can be dropped to ensure such consistency.