We present a function ρ (F1, F2, t) which contains Matusita's affinity and expresses the affinity between moment generating functions. An interesting results is expressed through decomposition of this affinity ρ (F1, F2, t) when the functions considered are k-dimensional normal distributions. The same decomposition remains true for other families of distribution functions. Generalizations of these results are also presented.

We focus on stochastic comparisons of lifetimes of series and parallel systems consisting of independent and heterogeneous new Pareto type components. Sufficient conditions involving majorization type partial orders are provided to obtain stochastic comparisons in terms of various magnitude and dispersive orderings which include usual stochastic order, hazard rate order, dispersive order and right spread order. The usual stochastic order of lifetimes of series systems with possibly different scale...

Let (X, Y) be a random couple in S×T with unknown distribution P. Let (X1, Y1), …, (Xn, Yn) be i.i.d. copies of (X, Y), Pn being their empirical distribution. Let h1, …, hN:S↦[−1, 1] be a dictionary consisting of N functions. For λ∈ℝN, denote fλ:=∑j=1Nλjhj. Let ℓ:T×ℝ↦ℝ be a given loss function, which is convex with respect to the second variable. Denote (ℓ•f)(x, y):=ℓ(y; f(x)). We study the following penalized empirical risk minimization problem $${\widehat{\lambda }}^{\epsilon }:=\underset{\lambda \in {ℝ}^N}{argmin}\left[P_n(\ell •f_{\lambda })+\epsilon {\parallel \lambda \parallel }_{{\ell }_p}^p\right],$$ which is an empirical version of the problem $${\lambda }^{\epsilon }:=\underset{\lambda \in {ℝ}^N}{argmin}\left[P(\ell •f_{\lambda })+\epsilon {\parallel \lambda \parallel }_{{\ell }_p}^p\right]$$ (hereɛ≥0...

In this paper, we consider a birth–death process with generator $ℒ$ and reversible invariant probabilityπ. Given an increasing function ρ and the associated Lipschitz norm ‖⋅‖Lip(ρ), we find an explicit formula for ${\parallel (-ℒ)}^{-1}{\parallel }_{Lip\left(\rho \right)}$. As a typical application, with spectral theory, we revisit one variational formula of M. F. Chen for the spectral gap of $ℒ$ inL2(π). Moreover, by Lyons–Zheng’s forward-backward martingale decomposition theorem, we get convex concentration inequalities for additive functionals of birth–death...

The upper bounds of the uniform distance $\rho \left({\sum }_{k=1}^{\nu }{X}_k,{\sum }_{k=1}^{\nu }{\tilde{X}}_k\right)$ between two sums of a random number $\nu$ of independent random variables are given. The application of these bounds is illustrated by stability (continuity) estimating in models in queueing and risk theory.

Let $\lambda$ denote the failure rate function of the $d,f$. $F$ and let ${\lambda }_1$ denote the failure rate function of the mean residual life distribution. In this paper we characterize the distribution functions $F$ for which ${\lambda }_1=c\lambda$ and we estimate $F$ when it is only known that ${\lambda }_1/\lambda$ or ${\lambda }_1-c\lambda$ is bounded.

We study the fluctuations around non degenerate attractors of the empirical measure under mean field Gibbs measures. We prove that a mild change of the densities of these measures does not affect the central limit theorems. We apply this result to generalize the assumptions of [3] and [12] on the densities of the Gibbs measures to get precise Laplace estimates.

Axioms are given for positive comparative probabilities and plausibilities defined either on Boolean algebras or on arbitrary sets of events. These axioms allow to characterize binary relations representable by either standard or nonstandard measures (i. e. taking values either on the real field or on a hyperreal field). We also study relations between conditional events induced by preferences on conditional acts.

In this paper we show that the windings of geodesics around the cusps of a Riemann surface of a finite area, behave asymptotically as independent Cauchy variables.