Search

kStatistics is a package in R that serves as a unified framework for estimating univariate and multivariate cumulants as well as products of univariate and multivariate cumulants of a random sample, using unbiased estimators with minimum variance. The main computational machinery of kStatistics is an algorithm for computing multi-index partitions. The same algorithm underlies the general-purpose multivariate Fa\`a di Bruno's formula, which has been therefore included in the last release of the package. This formula gives the coefficients of formal power series compositions as well as the partial derivatives of multivariable function compositions. One of the most significant applications of this formula is the possibility to generate many well-known polynomial families as special cases. So, in the package, there are special functions for generating very popular polynomial families, such as the Bell polynomials. However further families can be obtained, for suitable choices of the formal power series involved in the composition or when suitable symbolic strategies are employed. In both cases, we give examples on how to modify the R codes of the package to accomplish this task. Future developments are addressed at the end of the paper., Comment: In press

This note introduces a code snippet in R aiming to generate necklaces as well as bracelets. Among various uses, necklaces are useful tools to manage traces of products of random matrices. Functionality for necklaces and bracelets is provided with some examples of applications such as Lyndon words and de Bruijn sequences. The routines are collected in the Necklaces package available from the Comprehensive R Archive Network.

Multivariate subordinated L��vy processes are widely employed in finance for modeling multivariate asset returns. We propose to exploit non-linear dependence among financial assets through multivariate cumulants of these processes, for which we provide a closed form formula by using the multi-index generalized Bell polynomials. Using multivariate cumulants, we perform a sensitivity analysis, to investigate non-linear dependence as a function of the model parameters driving the dependence structure

In line with the increasing attention paid to deal with uncertainty in ordinal data models, we propose to combine Fuzzy models with \cub models within questionnaire analysis. In particular, the focus will be on \cub models' uncertainty parameter and its interpretation as a preliminary measure of heterogeneity, by introducing membership, non-membership and uncertainty functions in the more general framework of Intuitionistic Fuzzy Sets. Our proposal is discussed on the basis of the Evaluation of Orientation Services survey collected at University of Naples Federico II., Comment: 10 pages, invited contribution at SIS2016 (Salerno, Italy), in SIS2016 proceedings

A very simple closed-form formula for Sheppard's corrections is recovered by means of the classical umbral calculus. By means of this symbolic method, a more general closed-form formula for discrete parent distributions is provided and the generalization to the multivariate case turns to be straightforward. All these new formulae are particularly suited to be implemented in any symbolic package., Comment: In press on Math. Meth. in Stat. MAPLE procedures available on MAPLE Application Center http://www.maplesoft.com/applications/

We consider the first-crossing-time problem through a constant boundary for a Wiener process perturbed by random jumps driven by a counting process. On the base of a sample-path analysis of the jump-diffusion process we obtain explicit lower bounds for the first-crossing-time density and for the first-crossing-time distribution function. In the case of the distribution function, the bound is improved by use of processes comparison based on the usual stochastic order. The special case of constant jumps driven by a Poisson process is thoroughly discussed., 12 pages, 4 figures

Analysis of daily rainfall data, and subsequent modelling of some derived variables concerning rainfall, is fundamental in different areas such as agricultural, ecological, and engineering disciplines. A way of studying the alternance of consecutive rainy days (wet spells) and no-rainy days (dry spells) is through the interarrival time (IT), which is the time elapsed between two consecutives rainy days. If we suppose that IT observations are independent and identically distributed (i.i.d.), ITs are usually modelled through a renewal processes. The simplest renewal process is the Bernoulli process with ITs geometrically distributed. The need to suppose a non-constant probability of rain brings to more sophisticated models. Previous works [Agnese et al. (2014), Baiamonte et al. (2019)] have successfully proposed the three-parameter family of the Hurwitz-Lerch-Zeta distribution (HLZD), which represents a forward step with respect to other commonly used IT distributions. In [Agnese et al. (2022)], a second successfully reached goal was to show that the HLZD is also suitable to model the rainfall depth, h. In literature, rainfall depths are more frequently treated as continuous, despite sometimes these models fail to account for the time discreteness of the sampled process. Indeed, daily rainfall depth measurements are usually carried out by automatic-counting how many times a small bucket corresponding to 0.2 mm is filled. Due to the abundance of ties in the data, the variable depth h is well suited to be considered discrete. We present results involving data never considered in literature and consisting of measures sampled along 60-70 years at 7 different stations. These stations represent different climates from the rainfall characteristics point of view and let us to infer about the great handiness of the HLZD within rainfall modelling. Current research is addressed to modelling further rainfall variables related to IT and h, such as wet and dry spells, and the cumulative rainfall depth in a wet spell. Furthermore, given the remarkable performance of the HLZD family of distributions in the univariate modelling, we aim at modelling the dependence structure between IT and h, exploiting possibly new methodological advances in the subject, such as discrete copulas. References Agnese, G. Baiamonte, E. Di Nardo, S. Ferraris, and T. Martini (2022). Modelling the frequency of interarrival times and rainfall depths with the Poisson Hurwitz-Lerch zeta distribution. Fractal and Fractional, 6(9). Agnese, G. Baiamonte, and C. Cammalleri (2014). Modelling the occurrence of rainy daysunder a typical Mediterranean climate. Advances in Water Resources, 64:62–76. G. Baiamonte, L. Mercalli, D. C. Berro, C. Agnese, and S. Ferraris (2019). Modelling the frequency distribution of inter-arrival times from daily precipitation time-series in north-west Italy. Hydrology Research, 50(1):339–357.

In dealing with veracity of data analytics, fuzzy methods are more and more relying on probabilistic and statistical techniques to underpin their applicability. Conversely, standard statistical models usually disregard to take into account the inherent fuzziness of choices and this issue is particularly worthy of note in customers' satisfaction surveys, since there are different shades of evaluations that classical statistical tools fail to catch. Given these motivations, the paper introduces a model-based fuzzy analysis of questionnaire with sound statistical foundation, driven by the design of a hybrid method that sets in between fuzzy evaluation systems and statistical modelling. The proposal is advanced on the basis of \cub mixture models to account for uncertainty in ordinal data analysis and moves within the general framework of Intuitionistic Fuzzy Set theory to allow membership, non-membership, vagueness and accuracy assessments. Particular emphasis is given to defuzzification procedures that enable uncertainty measures also at an aggregated level. An application to a survey run at the University of Naples Federico II about the evaluation of Orientation Services supports the efficacy of the proposal., Comment: 22 pages, 9 figures, 8 Tables