M.D. Jiménez-Gamero

Generalized derivative and π-derivative for set-valued functions

In this paper we study the generalized derivative and the @p-derivative for interval-valued functions. We show the connections between these derivatives. Some illustrative examples and applications to interval differential equations and fuzzy functions are presented.

Calculus for interval-valued functions using generalized Hukuhara derivative and applications

This paper is devoted to studying differential calculus for interval-valued functions by using the generalized Hukuhara differentiability, which is the most general concept of differentiability for interval-valued functions. Conditions, examples and counterexamples for limit, continuity, integrability and differentiability are given. Special emphasis is set to the class F(t)=C.g(t), where C is an interval and g is a real function of a real variable. Here, the emphasis is placed on the fact that F and g do not necessarily share their properties, underlying the extra care that must be taken into account when dealing with interval-valued functions. Two applications of the obtained results are presented. The first one determines a Delta method for interval valued random elements. In the second application a new procedure to obtain solutions to an interval differential equation is introduced. Our results are relevant to fuzzy set theory because the usual fuzzy arithmetic, extension functions and (mathematical) analysis are done on @a-cuts, which are intervals.

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.

The extension principle and a decomposition of fuzzy sets

We give an algorithm to decompose a fuzzy interval u. Using this decomposition and the multilinearization of a univariate function f, we obtain an approximation of the fuzzy interval , where is obtained from f by applying the extension principle. We provide approximation bounds. Some numeric illustration is provided.

Fuzzy quasilinear spaces and applications

We introduce the concept of fuzzy quasilinear space and fuzzy quasilinear operator. Moreover we state some properties and give results which extend to the fuzzy context some results of linear functional analysis.

An approximation to the extension principle using decomposition of fuzzy intervals

This article presents a proposal for the decomposition of large ranges of uncertainty associated with the Cartesian product of two fuzzy intervals. We compute an approximation of the fuzzy set obtained by applying extension principle to a real function by means of this decomposition and piecewise linearization of the function. It is proved the efficiency of the method via convergence in D-metric. We also extend the method to the case when the function has more than two arguments. Some examples are included to illustrate the proposed method and to compare it with others.

Minimum ϕ-divergence estimation in misspecified multinomial models

The consequences of model misspecification for multinomial data when using minimum [phi]-divergence or minimum disparity estimators to estimate the model parameters are considered. These estimators are shown to converge to a well-defined limit. Two applications of the results obtained are considered. First, it is proved that the bootstrap consistently estimates the null distribution of certain class of test statistics for model misspecification detection. Second, an application to the model selection test problem is studied. Both applications are illustrated with numerical examples.

Bias Correction in the Type I Generalized Logistic Distribution

Four strategies for bias correction of the maximum likelihood estimator of the parameters in the Type I generalized logistic distribution are studied. First, we consider an analytic bias-corrected estimator, which is obtained by deriving an analytic expression for the bias to order n −1; second, a method based on modifying the likelihood equations; third, we consider the jackknife bias-corrected estimator; and fourth, we consider two bootstrap bias-corrected estimators. All bias correction estimators are compared by simulation. Finally, an example with a real data set is also presented.

Bootstrapping divergence statistics for testing homogeneity in multinomial populations

We consider the problem of testing the equality of @n (@n>=2) multinomial populations, taking as test statistic a sample version of an f-dissimilarity between the populations, obtained by the replacement of the unknown parameters in the expression of the f-dissimilarity among the theoretical populations, by their maximum likelihood estimators. The null distribution of this test statistic is usually approximated by its limit, the asymptotic null distribution. Here we study another way to approximate it, the bootstrap. We show that the bootstrap yields a consistent distribution estimator. We also study by simulation the finite sample performance of the bootstrap distribution and compare it with the asymptotic approximation. From the simulations it can be concluded that it is worth calculating the bootstrap estimator, because it is more accurate than the approximation yielded by the asymptotic null distribution which, in addition, cannot always be exactly computed.

Influence diagnostics in regression with complex designs through conditional bias

One of the areas of Statistics in which the influence analysis has been widely studied is the multiple linear regression model. Nevertheless, the influence diagnostics proposed in this context cannot be applied to regression in complex survey, under randomized inference, since the i.i.d. case does not incorporate any probability weighting or population structure, such as clustering, stratification or measures of size into the analysis.In this paper we introduce some influence diagnostics in regression in complex survey. They are built on the conditional bias concept (Moreno-Rebollo et al., 1999). We emphasize the similarities and differences of the proposed measures with respect to the existing ones for the i.i.d. case.