1,720,975 research outputs found
On the fractional probabilistic Taylor's and mean value theorems
No full textIn order to develop certain fractional probabilistic analogues of Taylor’s theorem and mean value theorem, we introduce the nth-order fractional equilibrium distribution in terms of the Weyl fractional integral and investigate its main properties. Specifically, we show a characterization result by which the nth-order fractional equilibrium distribution is identical to the starting distribution if and only if it is exponential. The nth-order fractional equilibrium density is then used to prove a fractional probabilistic Taylor’s theorem based on derivatives of Riemann-Liouville type. A fractional analogue of the probabilistic mean value theorem is thus developed for pairs of nonnegative random variables ordered according to the survival bounded stochastic order. We also provide some related results, both involving the normalized moments and a fractional extension of the variance, and a formula of interest to actuarial science. In conclusion, we discuss the probabilistic Taylor’s theorem based on fractional Caputo derivatives
Competing risks driven by Mittag-Leffler distributions, under copula and time transformed exponential model
No full textWe consider a stochastic model for competing risks involving the Mittag-Leffler distribution, inspired by fractional random growth phenomena. We prove the independence between the time to failure and the cause of failure, and investigate some properties of the related hazard rates and ageing notions. We also face the general problem of identifying the underlying distribution of latent failure times when their joint distribution is expressed in terms of copulas and the time transformed exponential model. The special case concerning the Mittag-Leffler distribution is approached by means of numerical treatment. We finally adapt the proposed model to the case of a random number of independent competing risks. This leads to certain mixtures of Mittag-Leffler distributions, whose parameters are estimated through the method of moments for fractional moments
On a fractional alternating Poisson process
No full textWe propose a generalization of the alternating Poisson process from the point of view of fractional calculus.
We consider the system of differential equations governing the state probabilities of the alternating Poisson process
and replace the ordinary derivative with the fractional derivative (in the Caputo sense). This produces a
fractional 2-state point process. We obtain the probability mass function of this process
in terms of the (two-parameter) Mittag-Leffler function.
Then we show that it can be recovered also by means of renewal theory.
We study the limit state probability, and certain proportions 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 a transformation acting on
pairs of stochastically ordered random variables, which is an extension of the equilibrium operator
and deserves interest in the analysis of alternating stochastic processes
Fractional equilibrium distributions, fractional probabilistic Taylor’s and mean value theorems
No full textA fractional counting process and its connection with the Poisson process
No full textWe consider a fractional counting process with jumps of amplitude
, with , whose probabilities
satisfy a suitable system of fractional difference-differential equations.
We obtain the moment generating function and the probability law of the resulting process
in terms of generalized Mittag-Leffler functions. We also discuss two equivalent representations
both in terms of a compound fractional Poisson process and of a subordinator governed by
a suitable fractional Cauchy problem. The first occurrence time of a jump of fixed amplitude
is proved to have the same distribution as the waiting time of the first event of a classical
fractional Poisson process, this extending a well-known property of the Poisson process.
When we also express the distribution of the first passage time of the fractional
counting process in an integral form. Finally, we show that the ratios given by the powers of
the fractional Poisson process and of the counting process over their means tend to 1 in probability
Cumulative information generating function and generalized Gini functions
No full textWe introduce and study the cumulative information generating function, which
provides a unifying mathematical tool suitable to deal with classical and fractional
entropies based on the cumulative distribution function and on the survival function.
Specifically, after establishing its main properties and some bounds, we show that
it is a variability measure itself that extends the Gini mean semi-difference.
We also provide (i) an extension of such a measure, based on distortion functions,
and (ii) a weighted version based on a mixture distribution.
Furthermore, we explore some connections with the reliability of -out-of-
systems and with stress-strength models for multi-component systems.
Also, we address the problem of extending the cumulative information
generating function to higher dimensions
Fractional generalized cumulative entropy and its dynamic version
No full textFollowing the theory of information measures based on the cumulative distribution function, we propose the fractional generalized cumulative entropy, and its dynamic version. These entropies are particularly suitable to deal with distributions satisfying the proportional reversed hazard model. We study the connection with fractional integrals, and some bounds and comparisons based on stochastic orderings, that allow to show that the proposed measure is actually a variability measure. The investigation also involves various notions of reliability theory, since the considered dynamic measure is a suitable extension of the mean inactivity time. We also introduce the empirical generalized fractional cumulative entropy as a non-parametric estimator of the new measure. It is shown that the empirical measure converges to the proposed notion almost surely. Then, we address the stability of the empirical measure and provide an example of application to real data. Finally, a central limit theorem is established under the exponential distribution
On a jump-telegraph process driven by alternating fractional Poisson process
No full textThe basic jump-telegraph process with exponentially distributed interarrival times deserves interest in various applied fields such as financial modelling and queueing theory. Aiming to propose a more general setting, we analyse such stochastic process when the interarrival times separating consecutive velocity changes (and jumps) have generalized Mittag-Leffler distributions, and constitute the random times of a fractional alternating Poisson process. By means of renewal theory-based issues we obtain the forward and backward transition densities of the motion in series form, and prove their uniform convergence. Specific attention is then given to the case of jumps with constant size, for which we also obtain the mean of the process. Finally, we investigate the first-passage time of the process through a constant positive boundary, providing its formal distribution and suitable lower bounds
Competing risks and shock models governed by a generalized bivariate Poisson process
No full textWe propose a stochastic model for the failure times of items subject to two external random shocks occurring as events in an underlying bivariate counting process. This is a special formulation of the competing risks model, which is of interest in reliability theory and survival analysis. Specifically, we assume that a system, or an item, fails when the sum of the two types of shock reaches a critical random threshold.
In detail, the two kinds of shock occur according to a bivariate space-fractional Poisson process, which is a two-dimensional vector of independent homogeneous Poisson processes time-changed by an independent stable subordinator. Various results are given, such as analytic hazard rates, failure densities, the probability that the failure occurs due to a specific type of shock, and the survival function. Some special cases and aging notions related to the NBU characterization are also considered. In this way, we generalize certain results in the literature, which can be recovered when the underlying process reduces to the inhomogeneous Poisson process
Il laboratorio di statistica per l’Alternanza Scuola-Lavoro e per il Piano Lauree Scientifiche
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