scholarly journals Failure rate properties of parallel systems

2020 ◽  
Vol 52 (2) ◽  
pp. 563-587
Author(s):  
Idir Arab ◽  
Milto Hadjikyriakou ◽  
Paulo Eduardo Oliveira
Keyword(s):  
Failure Rate ◽  
Random Variables ◽  
Stochastic Ordering ◽  
Parallel Systems ◽  
Iteration Procedure ◽  
Residual Lifetime ◽  
Hereditary Property ◽  
Suitable Function ◽  
Hereditary Properties ◽  
New Criterion

AbstractWe study failure rate monotonicity and generalised convex transform stochastic ordering properties of random variables, with an emphasis on applications. We are especially interested in the effect of a tail-weight iteration procedure to define distributions, which is equivalent to the characterisation of moments of the residual lifetime at a given instant. For the monotonicity properties, we are mainly concerned with hereditary properties with respect to the iteration procedure providing counterexamples showing either that the hereditary property does not hold or that inverse implications are not true. For the stochastic ordering, we introduce a new criterion, based on the analysis of the sign variation of a suitable function. This criterion is then applied to prove ageing properties of parallel systems formed with components that have exponentially distributed lifetimes.

scholarly journals ITERATED FAILURE RATE MONOTONICITY AND ORDERING RELATIONS WITHIN GAMMA AND WEIBULL DISTRIBUTIONS

2018 ◽  
Vol 33 (1) ◽  
pp. 64-80 ◽  
Author(s):  
Idir Arab ◽  
Paulo Eduardo Oliveira
Keyword(s):  
Explicit Expression ◽  
Failure Rate ◽  
Random Variables ◽  
Stochastic Ordering ◽  
Higher Order ◽  
Distribution Functions ◽  
Weibull Distributions ◽  
Stochastic Orderings ◽  
Definition Of ◽  
Relative Convexity

Stochastic ordering of random variables may be defined by the relative convexity of the tail functions. This has been extended to higher order stochastic orderings, by iteratively reassigning tail-weights. The actual verification of stochastic orderings is not simple, as this depends on inverting distribution functions for which there may be no explicit expression. The iterative definition of distributions, of course, contributes to make that verification even harder. We have a look at the stochastic ordering, introducing a method that allows for explicit usage, applying it to the Gamma and Weibull distributions, giving a complete description of the order of relations within each of these families.

scholarly journals Stochastic Comparisons of Some Distances between Random Variables

Mathematics ◽  
2021 ◽  
Vol 9 (9) ◽  
pp. 981
Author(s):  
Patricia Ortega-Jiménez ◽  
Miguel A. Sordo ◽  
Alfonso Suárez-Llorens
Keyword(s):  
Financial Risk ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Stochastic Ordering ◽  
Random Variable ◽  
Distribution Functions ◽  
Stochastic Comparisons ◽  
Stochastic Orderings ◽  
Absolute Value ◽  
The Difference

The aim of this paper is twofold. First, we show that the expectation of the absolute value of the difference between two copies, not necessarily independent, of a random variable is a measure of its variability in the sense of Bickel and Lehmann (1979). Moreover, if the two copies are negatively dependent through stochastic ordering, this measure is subadditive. The second purpose of this paper is to provide sufficient conditions for comparing several distances between pairs of random variables (with possibly different distribution functions) in terms of various stochastic orderings. Applications in actuarial and financial risk management are given.

Minimal Characteristic Algebras for Rectangular k-Normal Identities

2011 ◽  
Vol 18 (04) ◽  
pp. 611-628
Author(s):  
K. Hambrook ◽  
S. L. Wismath
Keyword(s):  
Hereditary Property ◽  
Fixed Type ◽  
Hereditary Properties

A characteristic algebra for a hereditary property of identities of a fixed type τ is an algebra [Formula: see text] such that for any variety V of type τ, we have [Formula: see text] if and only if every identity satisfied by V has the property p. This is equivalent to [Formula: see text] being a generator for the variety determined by all identities of type τ which have property p. Płonka has produced minimal (smallest cardinality) characteristic algebras for a number of hereditary properties, including regularity, normality, uniformity, biregularity, right- and leftmost, outermost, and external-compatibility. In this paper, we use a construction of Płonka to study minimal characteristic algebras for the property of rectangular k-normality. In particular, we construct minimal characteristic algebras of type (2) for k-normality and rectangularity for 1 ≤ k ≤ 3.

Distances on circles, toruses and spheres

1978 ◽  
Vol 15 (1) ◽  
pp. 136-143 ◽  
Author(s):  
B. W. Silverman
Keyword(s):  
Weak Convergence ◽  
Empirical Distribution ◽  
Random Variables ◽  
Suitable Function ◽  
Distribution Process

Families of heavily dissociated random variables are defined and discussed. These include families of the form g(Yi, Yj) for some suitable function g of two arguments and independent uniformly distributed random variables Y1, Y2, ··· on the circle, the torus or the sphere. The weak convergence of the empirical distribution process is discussed. The particular case of distances between pairs of observations on the circle is considered in greater detail.

scholarly journals Statistical properties and different methods of estimation of transmuted Rayleigh distribution

2017 ◽  
Vol 40 (1) ◽  
pp. 165-203 ◽  
Author(s):  
Sanku Dey ◽  
Enayetur Raheem ◽  
Saikat Mukherjee
Keyword(s):  
Stochastic Ordering ◽  
Real Data ◽  
Rayleigh Distribution ◽  
Statistical Properties ◽  
Point Of View ◽  
Strength Parameter ◽  
Residual Lifetime ◽  
Mean Deviation ◽  
Unknown Parameters ◽  
Anderson Darling

This article addresses the various properties and different methods of estimation of the unknown parameters of the Transmuted Rayleigh (TR) distribution from the frequentist point of view. Although, our main focus is on estimation from frequentist point of view,  yet, various mathematical and statistical properties of the TR distribution (such as quantiles, moments, moment generating function, conditional moments,  hazard rate, mean residual lifetime, mean past lifetime,  mean deviation about mean and median, the stochastic ordering,  various entropies, stress-strength parameter  and order statistics) are derived.  We briefly describe different frequentist methods of estimation approaches, namely, maximum likelihood estimators, moments estimators, L-moment estimators, percentile based estimators, least squares estimators, method of maximum product of spacings,  method of Cram\'er-von-Mises, methods of Anderson-Darling and right-tail Anderson-Darling and compare them using extensive numerical simulations. Monte Carlo simulations are performed to compare the performances of the proposed methods of estimation for both small and large samples. Finally, the potentiality of the model is analyzed by means of two real data sets which is further illustrated by obtaining bias and standard error of the estimates and the bootstrap percentile confidence intervals using bootstrap resampling.

scholarly journals RELATIONS BETWEEN STOCHASTIC ORDERINGS AND GENERALIZED STOCHASTIC PRECEDENCE

2015 ◽  
Vol 29 (3) ◽  
pp. 329-343 ◽  
Author(s):  
Emilio De Santis ◽  
Fabio Fantozzi ◽  
Fabio Spizzichino
Keyword(s):  
Random Variables ◽  
Stochastic Ordering ◽  
Stochastic Independence ◽  
Stochastic Orderings ◽  
Stochastic Dependence ◽  
Natural Concept ◽  
Decisions Under Risk ◽  
Stochastic Precedence ◽  
Special Classes

The concept of stochastic precedence between two real-valued random variables has often emerged in different applied frameworks. In this paper, we analyze several aspects of a more general, and completely natural, concept of stochastic precedence that also had appeared in the literature. In particular, we study the relations with the notions of stochastic ordering. Such a study leads us to introducing some special classes of bivariate copulas. Motivations for our study can arise from different fields. In particular, we consider the frame of Target-Based Approach in decisions under risk. This approach has been mainly developed under the assumption of stochastic independence between “Prospects” and “Targets”. Our analysis concerns the case of stochastic dependence.

scholarly journals Stochastic Order Relations Among Parallel Systems from Weibull Distributions

2015 ◽  
Vol 52 (01) ◽  
pp. 102-116 ◽  
Author(s):  
Nuria Torrado ◽  
Subhash C. Kochar
Keyword(s):  
Likelihood Ratio ◽  
Hazard Rate ◽  
Stochastic Order ◽  
Random Variables ◽  
Parallel Systems ◽  
Parallel System ◽  
Weibull Distributions ◽  
Scale Parameters ◽  
Order Relations ◽  
Stochastic Order Relations

Let X λ1 , X λ2 , …, X λ n be independent Weibull random variables with X λ i ∼ W(α, λ i ), where λ i > 0 for i = 1, …, n. Let X n:n λ denote the lifetime of the parallel system formed from X λ1 , X λ2 , …, X λ n . We investigate the effect of the changes in the scale parameters (λ1, …, λ n ) on the magnitude of X n:n λ according to reverse hazard rate and likelihood ratio orderings.

Research on the Hereditary Properties of the Cartesian Product Operation of Graphs

2012 ◽  
Vol 241-244 ◽  
pp. 2802-2806
Author(s):  
Hua Dong Wang ◽  
Bin Wang ◽  
Yan Zhong Hu
Keyword(s):  
Euler Characteristic ◽  
Cartesian Product ◽  
Hamilton Cycle ◽  
Hereditary Property ◽  
Constant Property ◽  
Product Graph ◽  
Product Operation ◽  
Graph Operation ◽  
Hereditary Properties

This paper defined the hereditary property (or constant property) concerning graph operation, and discussed various forms of the hereditary property under the circumstance of Cartesian product graph operation. The main conclusions include: The non-planarity and Hamiltonicity of graph are hereditary concerning the Cartesian product, but planarity of graph is not, Euler characteristic and non-hamiltonicity of graph are not hereditary as well. Therefore, when we applied this principle into practice, we testified that Hamilton cycle does exist in hypercube.

SOME EXTENSIONS OF THE RESIDUAL LIFETIME AND ITS CONNECTION TO THE CUMULATIVE RESIDUAL ENTROPY

2011 ◽  
Vol 26 (1) ◽  
pp. 129-146 ◽  
Author(s):  
Stella Kapodistria ◽  
Georgios Psarrakos
Keyword(s):  
Random Variables ◽  
Distribution Functions ◽  
Residual Entropy ◽  
Residual Lifetime ◽  
Numerical Examples ◽  
Tail Distribution ◽  
Cumulative Residual Entropy ◽  
Recursive Formulas ◽  
Cumulative Residual

In this article we present a sequence of random variables with weighted tail distribution functions, constructed based on the relevation transform. For this sequence, we prove several recursive formulas and connections to the residual entropy through the unifying framework of the Dickson–Hipp operator. We also give some numerical examples to evaluate our results.

Sign in / Sign up

Export Citation Format

Share Document