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On generalized conditional cumulative past inaccuracy measure

Amit Ghosh, Chanchal Kundu (2018)

Applications of Mathematics

The notion of cumulative past inaccuracy (CPI) measure has recently been proposed in the literature as a generalization of cumulative past entropy (CPE) in univariate as well as bivariate setup. In this paper, we introduce the notion of CPI of order α and study the proposed measure for conditionally specified models of two components failed at different time instants, called generalized conditional CPI (GCCPI). Several properties, including the effect of monotone transformation and bounds of GCCPI...

On identifiability of mixtures of independent distribution laws

Mikhail Kovtun, Igor Akushevich, Anatoliy Yashin (2014)

ESAIM: Probability and Statistics

We consider representations of a joint distribution law of a family of categorical random variables (i.e., a multivariate categorical variable) as a mixture of independent distribution laws (i.e. distribution laws according to which random variables are mutually independent). For infinite families of random variables, we describe a class of mixtures with identifiable mixing measure. This class is interesting from a practical point of view as well, as its structure clarifies principles of selecting...

On lower bounds for the variance of functions of random variables

Faranak Goodarzi, Mohammad Amini, Gholam Reza Mohtashami Borzadaran (2021)

Applications of Mathematics

In this paper, we obtain lower bounds for the variance of a function of random variables in terms of measures of reliability and entropy. Also based on the obtained characterization via the lower bounds for the variance of a function of random variable X , we find a characterization of the weighted function corresponding to density function f ( x ) , in terms of Chernoff-type inequalities. Subsequently, we obtain monotonic relationships between variance residual life and dynamic cumulative residual entropy...

On Measure Concentration of Vector-Valued Maps

Michel Ledoux, Krzysztof Oleszkiewicz (2007)

Bulletin of the Polish Academy of Sciences. Mathematics

We study concentration properties for vector-valued maps. In particular, we describe inequalities which capture the exact dimensional behavior of Lipschitz maps with values in ℝ k . To this end, we study in particular a domination principle for projections which might be of independent interest. We further compare our conclusions with earlier results by Pinelis in the Gaussian case, and discuss extensions to the infinite-dimensional setting.

On measure-preserving transformations and doubly stationary symmetric stable processes

A. Gross, A. Weron (1995)

Studia Mathematica

In a 1987 paper, Cambanis, Hardin and Weron defined doubly stationary stable processes as those stable processes which have a spectral representation which is itself stationary, and they gave an example of a stationary symmetric stable process which they claimed was not doubly stationary. Here we show that their process actually had a moving average representation, and hence was doubly stationary. We also characterize doubly stationary processes in terms of measure-preserving regular set isomorphisms...

On Mieshalkin-Rogozin theorem and some properties of the second kind beta distribution

Włodzimierz Krysicki (2000)

Discussiones Mathematicae Probability and Statistics

The decomposition of the r.v. X with the beta second kind distribution in the form of finite (formula (9), Theorem 1) and infinity products (formula (17), Theorem 2 and form (21), Theorem 3) are presented. Next applying Mieshalkin - Rogozin theorem we receive the estimation of the difference of two c.d.f. F(x) and G(x) when sup|f(t) - g(t)| is known, improving the result of Gnedenko - Kolmogorov (formulae (23) and (24)).

On monotone dependence functions of the quantile type

Andrzej Krajka, Dominik Szynal (1995)

Applicationes Mathematicae

We introduce the concept of monotone dependence function of bivariate distributions without moment conditions. Our concept gives, among other things, a characterization of independent and positively (negatively) quadrant dependent random variables.

On preservation under univariate weighted distributions

Salman Izadkhah, Mohammad Amini, Gholam Reza Mohtashami Borzadaran (2015)

Applications of Mathematics

We derive some new results for preservation of various stochastic orders and aging classes under weighted distributions. The corresponding reversed preservation properties as straightforward conclusions of the obtained results for the direct preservation properties, are developed. Damage model of Rao, residual lifetime distribution, proportional hazards and proportional reversed hazards models are discussed as special weighted distributions to try some of our results.

On Probability Distribution Solutions of a Functional Equation

Janusz Morawiec, Ludwig Reich (2005)

Bulletin of the Polish Academy of Sciences. Mathematics

Let 0 < β < α < 1 and let p ∈ (0,1). We consider the functional equation φ(x) = pφ (x-β)/(1-β) + (1-p)φ(minx/α, (x(α-β)+β(1-α))/α(1-β)) and its solutions in two classes of functions, namely ℐ = φ: ℝ → ℝ|φ is increasing, φ | ( - ∞ , 0 ] = 0 , φ | [ 1 , ∞ ) = 1 , = φ: ℝ → ℝ|φ is continuous, φ | ( - ∞ , 0 ] = 0 , φ | [ 1 , ∞ ) = 1 . We prove that the above equation has at most one solution in and that for some parameters α,β and p such a solution exists, and for some it does not. We also determine all solutions of the equation in ℐ and we show the exact connection...

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