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Idir Arab

Publications and source records attributed to Idir Arab.

16 recordsLinked to original sources
arXiv2026

Integral stochastic orders of $m$-generalized order statistics from transform-ordered nonparametric families

We provide sufficient conditions for comparing $m$-generalized order statistics with respect to the increasing concave, increasing convex, and star-shaped stochastic orders. These conditions allow us to rank classical order statistics, selected censored type-II order statistics, and records. They depend on both the parameters of the generalized order statistics and the underlying distribution. Rather than assuming a specific parametric form, we adopt a nonparametric approach and assume some stochastic transform-ordered property, that is, some suitable shape condition. This framework encompasses many relevant classes of distributions that are related, via transform order, to the generalized and the negative generalized Pareto distribution.

math.ST↗
arXiv2026

Stochastic Ordering of Dependent Systems under Transformation Models and Archimedean Copulas

We study stochastic ordering of system lifetimes with dependent and heterogeneous components whose marginal distributions are obtained through transformations of a common baseline. The dependence structure is modeled via Archimedean copulas, allowing for a unified treatment of several transformation-based models, including proportional hazard, proportional reversed hazard rate and proportional odds families. For parallel, series and $(n-k)$-out-of-$n$ systems, we derive conditions for stochastic dominance based on monotonicity of the transformation and structural properties of the copula generators, formulated through super-additivity and Schur-type arguments. The results provide tractable criteria that extend existing comparisons beyond independence and illustrate the combined effect of dependence and parameter heterogeneity on system reliability.

math.PR↗
arXiv2024

Convex combinations of random variables stochastically dominate the parent for a new class of heavy-tailed distributions

Stochastic dominance of a random variable by a convex combination of its independent copies has recently been shown to hold within the relatively narrow class of distributions with concave odds function, and later extended to broader families of distributions. A simple consequence of this surprising result is that the sample mean can be stochastically larger than the underlying random variable. We show that a key property for this stochastic dominance result to hold is the subadditivity of the cumulative distribution function of the reciprocal of the random variable of interest, referred to as the inverted distribution. By studying relations and inclusions between the different classes for which the stochastic dominance was proved to hold, we show that our new class can significantly enlarge the applicability of the result, providing a relatively mild sufficient condition.

math.PR↗
arXiv2024

Stochastic orders and shape properties for a new distorted proportional odds model

Building on recent developments in models focused on the shape properties of odds ratios, this paper introduces two new models that expand the class of available distributions while preserving specific shape characteristics of an underlying baseline distribution. The first model offers enhanced control over odds and log-odds functions, facilitating adjustments to skewness, tail behaviour, and hazard rates. The second model, with even greater flexibility, describes odds ratios as quantile distortions. This approach leads to an enlarged log-logistic family capable of capturing these quantile transformations and diverse hazard behaviours, including non-monotonic and bathtub-shaped rates. Central to our study are the shape relations described through stochastic orders; we establish conditions that ensure stochastic ordering both within each family and across models under various ordering concepts, such as hazard rate, likelihood ratio, and convex transform orders

math.ST↗
arXiv2022

Nonparametric inference about increasing odds rate distributions

To improve nonparametric estimates of lifetime distributions, we propose using the increasing odds rate (IOR) model as an alternative to other popular, but more restrictive, ``adverse ageing'' models, such as the increasing hazard rate one. This extends the scope of applicability of some methods for statistical inference under order restrictions, since the IOR model is compatible with heavy-tailed and bathtub distributions. We study a strongly uniformly consistent estimator of the cumulative distribution function of interest under the IOR constraint. Numerical evidence shows that this estimator often outperforms the classic empirical distribution function when the underlying model does belong to the IOR family. We also study two different tests, aimed at detecting deviations from the IOR property, and we establish their consistency. The performance of these tests is also evaluated through simulations.

stat.ME↗
arXiv2021

Transform orders and stochastic monotonicity of statistical functionals

In some inferential statistical methods, such as tests and confidence intervals, it is important to describe the stochastic behavior of statistical functionals, aside from their large sample properties. We study such behavior in terms of the usual stochastic order. For this purpose, we introduce a generalized family of stochastic orders, which is referred to as transform orders, showing that it provides a flexible framework for deriving stochastic monotonicity results. Given that our general definition makes it possible to obtain some well-known ordering relations as particular cases, we can easily apply our method to different families of functionals. These include some prominent inequality measures, such as the generalized entropy, the Gini index, and its generalizations. We also illustrate the applicability of our approach by determining the least favorable distribution, and the behavior of some bootstrap statistics, in some goodness-of-fit testing procedures.

math.ST↗
arXiv2020

Second-order stochastic comparisons of order statistics

We study the problem of comparing ageing patterns of the lifetime of k-out-of-n systems. Mathematically, this reduces to being able to decide about a stochastic ordering relationship between different order statistics. We discuss such relationships with respect to second-order stochastic dominance, obtaining characterizations through the verification of relative convexity with respect to a suitably chosen reference distribution function. We introduce a hierarchy of such reference functions leading to classes, each expressing different and increasing knowledge precision about the distribution of the component lifetimes. Such classes are wide enough to include popular families of distributions, such as, for example, the increasing failure rate distributions. We derive sufficient dominance conditions depending on the identification of the class which includes the component lifetimes. Concerning the conditions, as expected, relying on a larger class of distributions, meaning that we have less precise information about the components behavior, leads to the need of stronger assumptions. We discuss the applicability of this method and characterize a test for the relative convexity, as this notion plays a central role in the proposed approach.

stat.ME↗
arXiv2020

Convex transform order of Beta distributions with some consequences

The convex transform order is one way to make precise comparison between the skewness of probability distributions on the real line. We establish a simple and complete characterisation of when one Beta distribution is smaller than another according to the convex transform order. As an application, we derive monotonicity properties for the probability of Beta distributed random variables exceeding the mean or mode of their distribution. Moreover, we obtain a simple alternative proof of the mode-median-mean inequality for unimodal distributions that are skewed in a sense made precise by the convex transform order. This new proof also gives an analogous inequality for the anti-mode of distributions that have a unique anti-mode. Such inequalities for Beta distributions follow as special cases. Finally, some consequences for the values of distribution functions of Binomial distributions near to their means are mentioned.

math.PR↗
arXiv2019

Non-comparability with respect to the convex transform order with applications

In the literature of stochastic orders, one rarely finds results that can be considered as criteria for the non-comparability of random variables. In this paper, we provide results that enable researchers to use simple tools to conclude that two random variables are not comparable with respect to the convex transform order. The criteria are applied to prove the non-comparability of parallel systems with components that are either exponential, Weibull or Gamma distributed, providing a negative answer for a conjecture about comparability with respect to the convex transform order in a much broader scope than its initial statement.

math.PR↗
arXiv2019

On the limit behavior of iterated equilibrium distributions for the Gamma and Weibull families

In this paper, we study the evolution of iterated equilibrium distributions for the Gamma and Weibull families of distributions as the iteration step increases. We characterize their moments and the pointwise limit of the distribution functions corresponding to the iterated distributions. As a byproduct, we obtain approximations for higher order moments of the residual lifetime.

math.ST↗
arXiv2018

Failure Rate Properties of Parallel Systems

We study failure rate monotonicity and generalized convex transform stochastic ordering properties of random variables, with a concern on applications. We are especially interested in the effect of a tail weight iteration procedure to define distributions, which is equivalent to the characterization 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 counter-examples showing either that the hereditary property does not hold or that inverse implications are not true. For the stochastic ordering, we introduce a new criterium, based on the analysis of the sign variation of a suitable function. This criterium is then applied to prove ageing properties of parallel systems formed with components that have exponentially distributed lifetimes.

math.PR↗
arXiv2017

Iterated failure rate monotonicity and ordering relations within Gamma and Weibull distributions

Stochastic ordering of distributions 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 those 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 relations within each of those families.

math.ST↗
arXiv2017

Asymptotic results for certain weak dependent random variables

We consider a special class of weak dependent random variables with control on covariances of Lipschitz transformations. This class includes, but is not limited to, positively, negatively associated variables and a few other classes of weakly dependent structures. We prove a Strong Law of Large Numbers with a characterization of convergence rates, which is almost optimal, in the sense that it is arbitrarily close to the optimal rate for independent variables. Moreover, we prove an inequality comparing the joint distributions with the product distributions of the margins, similar to the well-known Newman's inequality for characteristic functions of associated variables. As a consequence, we prove a Central Limit Theorem together with its functional counterpart, and also the convergence of the empirical process for this class of weak dependent variables.

math.PR↗
arXiv2017

Strongly mixed random errors in Mann's iteration algorithm for a contractive real function

This work deals with the Mann's stochastic iteration algorithm under strong mixing random errors. We establish the Fuk-Nagaev's inequalities that enable us to prove the almost complete convergence with its corresponding rate of convergence. Moreover, these inequalities give us the possibility of constructing a confidence interval for the unique fixed point. Finally, to check the feasibility and validity of our theoretical results, we consider some numerical examples, namely a classical example from astronomy.

math.PR↗
arXiv2017

Exponential inequalities for Mann's iterative scheme with functional random errors

In this work, we deal with an iteration method for approximating a fixed point of a contraction mapping using the Mann's algorithm under functional random errors. We first show its almost complete convergence to the fixed point by mean of an exponential inequality and then we specify the induced rate of convergence. We finally build a confidence set for the fixed point.

math.PR↗