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Generalized order statistics (GOSs) are often adopted as a tool for providing a unified approach to several stochastic models dealing with ordered random variables. In this contribution, we first recall various useful results based on the notion of total positivity. Then, some stochastic comparisons between spacings of GOSs from one sample, as well as two samples, are developed under the more general assumptions on the parameters of the model. Specifically, the given results deal with the likelihood ratio order, the hazard rate order and the mean residual life order. Finally, an application is demonstrated for sequential systems.

The propagation of fake news in online social networks nowadays is becoming a critical issue. Consequently, many mathematical models have been proposed to mimic the related time evolution. In this work, we first consider a deterministic model that describes rumor propagation and can be viewed as an extended logistic model. In particular, we analyze the main features of the growth curve, such as the limit behavior, the inflection point, and the threshold-crossing-time, through fixed boundaries. Then, in order to study the stochastic counterparts of the model, we consider two different stochastic processes: a time non-homogeneous linear pure birth process and a lognormal diffusion process. The conditions under which the means of the processes are identical to the deterministic curve are discussed. The first-passage-time problem is also investigated both for the birth process and the lognormal diffusion process. Finally, in order to study the variability of the stochastic processes introduced so far, we perform a comparison between their variances.

Aiming to construct a simple stochastic model able to describe systems alternating due to state-dependent dichotomous noise, we consider a generalized telegraph process whose sample-paths fluctuates around the zero state. Indeed, the latter process describes the motion of a particle on the real line, which is characterized by constant velocities and state-dependent intensities that vanish when the motion is toward the origin. This assumption allows to adopt an approach based on renewal theory to obtain formal expressions of the forward and backward transition densities of the process. The special case when certain random times of the motion possess gamma distribution leads to closed-form expressions of the transition densities, given in terms of the generalized Mittag-Leffler function. We also analyze a first-passage-time problem for the considered process in the presence of two constant boundaries.

We consider an extension of the mean inactivity time based on a non-negative weight function. We show various properties of the new notion, and relate it to various functions of interest in reliability theory and information measures, such as the dynamic cumulative entropy, the past entropy, the varentropy, and the weighted cumulative entropy. Moreover, based on the comparison of weighted mean inactivity times, we introduce and study a new stochastic order and compare it with other suitable orders. We also discuss some results about the variance of transformed random variables and the weighted generalized cumulative entropy. Then, we investigate certain connections with the location-independent riskier order. Finally, we pinpoint several characterizations and preservation properties of the new stochastic order under shock models, random maxima, and notions of renewal theory.

The aim of this research is to identify an efficient model to describe the fluctuations around the trend of the soil temperatures monitored in the volcanic caldera of the Campi Flegrei area in Naples (Italy). This study focuses on the data concerning the temperatures in the mentioned area through a seven-year period. The research is initially finalized to identify the deterministic component of the model given by the seasonal trend of the temperatures, which is obtained through an adapted regression method on the time series. Subsequently, the stochastic component from the time series is tested to represent a fractional Brownian motion (fBm). An estimation based on the periodogram of the data is used to estabilish that the data series follows an fBm motion rather than fractional Gaussian noise. An estimation of the Hurst exponent H of the process is also obtained. Finally, an inference test based on the detrended moving average of the data is adopted in order to assess the hypothesis that the time series follows a suitably estimated fBm.

We propose a generalization of the alternating Poisson process from the point of view offractional calculus. We consider the system of differential equations governing the state probabilitiesof the alternating Poisson process and replace the ordinary derivative with the fractional derivative (inthe Caputo sense). This produces a fractional 2-state point process. We obtain the probability massfunction of this process in terms of the (two-parameter) Mittag-Leffler function. Then we show thatit can be recovered also by means of renewal theory. We study the limit state probability, and certainproportions involving the fractional moments of the sub-renewal periods of the process. In conclusion,in order to derive new Mittag-Leffler-like distributions related to the considered process, we exploit atransformation acting on pairs of stochastically ordered random variables, which is an extension of theequilibrium operator and deserves interest in the analysis of alternating stochastic processes.

A shift-dependent information measure is favorable to handle in some specific applied contexts such as mathematical neurobiology and survival analysis. For this reason, the weighted differential entropy has been introduced in the literature. In accordance with this measure, we propose the weighted generalized cumulative residual entropy as well. Despite existing apparent similarities between these measures, however, there are quite substantial and subtle differences between them because of their different metrics. In this paper, particularly, we show that the proposed measure is equivalent to the generalized cumulative residual entropy of the cumulative weighted random variable. Thus, we first provide expressions for the variance and the new measure in terms of the weighted mean residual life function and then elaborate on some characteristics of such measures, including equivalent expressions, stochastic comparisons, bounds, and connection with the excess wealth transform. Finally, we also illustrate some applications of interest in system reliability with reference to shock models and random minima.

A suitable replacement model for random lifetimes is extended to the context of past lifetimes. At a fixed time u an item is planned to be replaced by another one having the same age but a different lifetime distribution. We investigate the past lifetime of this system, given that at a larger time t the system is found to be failed. Subsequently, we perform some stochastic comparisons between the random lifetimes of the single items and the doubly truncated random variable that describes the system lifetime. Moreover, we consider the relative ratio of improvement evaluated at x ∈ ( u , t ) , which is finalized to measure the goodness of the replacement procedure. The characterization and the properties of the differential entropy of the system lifetime are also discussed. Finally, an example of application to the firing activity of a stochastic neuronal model is provided.

The generalized cumulative residual entropy is a recently defined dispersion measure. In this paper, we obtain some further results for such a measure, in relation to the generalized cumulative residual entropy and the variance of random lifetimes. We show that it has an intimate connection with the non-homogeneous Poisson process. We also get new expressions, bounds and stochastic comparisons involving such measures. Moreover, the dynamic version of the mentioned notions is studied through the residual lifetimes and suitable aging notions. In this framework we achieve some findings of interest in reliability theory, such as a characterization for the exponential distribution, various results on k-out-of-n systems, and a connection to the excess wealth order. We also obtain similar results for the generalized cumulative entropy, which is a dual measure to the generalized cumulative residual entropy.

We consider the logistic growth model and analyze its relevant properties, such as the limits, the monotony, the concavity, the inflection point, the maximum specific growth rate, the lag time, and the threshold crossing time problem. We also perform a comparison with other growth models, such as the Gompertz, Korf, and modified Korf models. Moreover, we focus on some stochastic counterparts of the logistic model. First, we study a time-inhomogeneous linear birth-death process whose conditional mean satisfies an equation of the same form of the logistic one. We also find a sufficient and necessary condition in order to have a logistic mean even in the presence of an absorbing endpoint. Then, we obtain and analyze similar properties for a simple birth process, too. Then, we investigate useful strategies to obtain two time-homogeneous diffusion processes as the limit of discrete processes governed by stochastic difference equations that approximate the logistic one. We also discuss an interpretation of such processes as diffusion in a suitable potential. In addition, we study also a diffusion process whose conditional mean is a logistic curve. In more detail, for the considered processes we study the conditional moments, certain indices of dispersion, the first-passage-time problem, and some comparisons among the processes.

We investigate an extension of the spike train stochastic model based on the conditionalintensity, in which the recovery function includes an interaction between several excitatoryneural units. Such function is proposed as depending both on the time elapsed since thelast spike and on the last spiking unit. Our approach, being somewhat related to thecompeting risks model, allows to obtain the general form of the interspike distribution andof the probability of consecutive spikes from the same unit. Various results are finally presentedin the two cases when the free firing rate function (i) is constant, and (ii) has a sinusoidal form.

In this paper, we discuss the cumulative measure of inaccuracy in k-lower record values and study characterization results of dynamic cumulative inaccuracy. We also present some properties of the proposed measures, and the empirical cumulative measure of inaccuracy in k-lower record values. We prove a central limit theorem for the empirical cumulative measure of inaccuracy under exponentially distributed populations. Finally, we analyze the mutual information for measuring the degree of dependency between lower record values, and we show that it is distribution-free.

We consider a suitable replacement model for random lifetimes, in which at a fixed time an item is replaced by another one having the same age but different lifetime distribution. We focus first on stochastic comparisons between the involved random lifetimes, in order to assess conditions leading to an improvement of the system. Attention is also given to the relative ratio of improvement, which is proposed as a suitable index finalized to measure the goodness of the replacement procedure. Finally, we provide various results on the dynamic differential entropy of the lifetime of the improved system.

We consider a time-non-homogeneous double-ended queue subject to catastrophes and repairs. The catastrophes occur according to a non-homogeneous Poisson process and lead the system into a state of failure. Instantaneously, the system is put under repair, such that repair time is governed by a time-varying intensity function. We analyze the transient and the asymptotic behavior of the queueing system. Moreover, we derive a heavy-traffic approximation that allows approximating the state of the systems by a time-non-homogeneous Wiener process subject to jumps to a spurious state (due to catastrophes) and random returns to the zero state (due to repairs). Special attention is devoted to the case of periodic catastrophe and repair intensity functions. The first-passage-time problem through constant levels is also treated both for the queueing model and the approximating diffusion process. Finally, the goodness of the diffusive approximating procedure is discussed.

The International Conference ``BIOCOMP2012 - Mathematical Modeling and Computational Topics in Biosciences'',was held in Vietri sul Mare (Italy), June 4-8, 2012. It was dedicated to the Memory of Professor Luigi M. Ricciardi(1942-2011), who was a visionary and tireless promoter of the 3 previous editions of the BIOCOMP conference series.We thought that the best way to honor his memory was to continue the BIOCOMP program. Over the years, thisconference promoted scientific activities related to his wide interests and scientific expertise, which ranged invarious areas of applications of mathematics, probability and statistics to biosciences and cybernetics, alsowith emphasis on computational problems. We are pleased that many of his friends and colleagues, as well asmany other scientists, were attracted by the goals of this recent event and offered to contribute to its success.For more information please click the “Full Text” above.