ABSTRACT. In this paper, Ulam stability and data dependence for fractional differential equations with Caputo fractional derivative of order α are studied. We present four types of Ulam stability results for the fractional differential equation in the case of 0 < α < 1 and b = +∞ by virtue of the Henry-Gronwall inequality. Meanwhile, we give an interesting data dependence results for the fracti...

Recently, there has been much interest in designing and analyzing estimators for information theoretic functionals of probability density functions, under the assumption of independent and identically distributed (IID) data. However, estimators designed for IID data fail to capture relevant information when presented with time series data. We first present information theoretic functionals whic...

We aim at providing probabilistic explanations of equivalences, between conditions of positive dependence and of univariate ageing, that have been pointed out in the literature. To this purpose we consider bivariate survival functions F(x,y) and properties of them that are respectively invariant under transformations of the type F (φ(x),φ(y)) and ψ (

A variety of dependency formalisms have been studied for improving data quality. To treat these dependencies in a uniform framework, we propose a simple language, Quality Improving Dependencies (QIDs). We show that previous dependencies considered for data quality can be naturally expressed as QIDs, and that different enforcement mechanisms of QIDs yield various data repairing strategies.

Dependence structures for bivariate extremal events are analyzed using particular types of copula. Weak convergence results for copulas along the lines of the Pickands–Balkema– de Haan theorem provide limiting dependence structures for bivariate tail events. A characterization of these limiting copulas is also provided by means of invariance properties. The results obtained are applied to the c...

Demonstrates methods of detecting local dependence in binary data latent class models. Latent class models are applied to five repeated measures of voter turnout in the Dutch Parliamentary elections of 2006 and 2010 obtained from a probability sample of 9510 citizens. Modeling substantive local dependence as separate discrete latent variables while modeling nuisance dependencies as direct effec...

The authors show how the approach of Capéraà and Genest (Canad. J. Statist., 1990) can be used to order bivariate distributions with arbitrary marginals by their degree of dependence in the LTD (left-tail decreasing) or RTI (right-tail increasing) sense. Some properties of these new orderings are given, along with applications to Archimedean copulas, order statistics and compound random variables.

Operational risk data, when available, are usually scarce, heavy-tailed and possibly dependent. In this work, we introduce a model that captures such real-world characteristics and explicitly deals with heterogeneous pairwise and tail dependence of losses. By considering flexible families of copulas, we can easily move beyond modeling bivariate dependence among losses and estimate the total ris...