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We regard high degree stop-loss transforms as iterated equilibrium distributions to establish comparison criteria with respect to star-shape and convex transform orders. As an application, we derive monotonicity properties for exceedance probabilities, providing information on how likely these distributions are to be far from the extreme values of their support. [ABSTRACT FROM AUTHOR]

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 to detect deviations from the IOR property and establish their consistency. The performance of these tests is also evaluated through simulations. [ABSTRACT FROM AUTHOR]

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 a 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. [ABSTRACT FROM AUTHOR]

The star-shaped ordering between probability distributions is a common way to express aging properties. A well-known criterion was proposed by Saunders and Moran [(1978). On the quantiles of the gamma and F distributions. Journal of Applied Probability 15(2): 426–432], to order families of distributions depending on one real parameter. However, the lifetime of complex systems usually depends on several parameters, especially when considering heterogeneous components. We extend the Saunders and Moran criterion characterizing the star-shaped order when the multidimensional parameter moves along a given direction. A few applications to the lifetime of complex models, namely parallel and series models assuming different individual components behavior, are discussed. [ABSTRACT FROM AUTHOR]

High order stop-loss transforms provide a risk measure that enables some flexibility on the weight given to high or low values of the risk. We interpret stop-loss transforms as iterated distributions and prove a recursive representation for risks expressed as convolutions. We apply this to the case of gamma distributions with integer shape parameter, the Erlang distributions, proving that high order stop-loss transforms are equivalent to the tails of the exponential distribution. The latter result is also extended to general gamma distributions. Furthermore, we prove that this equivalence to exponential tails does not hold in general, by proving that the stop-loss transform for a Weilbull distribution degenerates, unless, of course, in the exponential case. [ABSTRACT FROM AUTHOR]

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 characterization 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. [ABSTRACT FROM AUTHOR]

We study the problem of comparing ageing patterns of lifetimes of k-out-of-n systems with i.i.d. components. 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 components 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. 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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this article, we deal with an iteration method for approximating a fixed point of a contraction mapping using 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 domain for the fixed point. Moreover, some numerical examples are considered. [ABSTRACT FROM AUTHOR]

In this article, we study the algorithm of Kiefer–Wolfowitz underquasi-associated random errors. We establish the complete convergence and obtain an exponential bound. Additionally, we build a confidence interval for the minimum. Numerical examples are sketched out to confirm the theoretical results and show the accuracy of the algorithm. [ABSTRACT FROM AUTHOR]

Many mathematical and physical problems are led to find a root of a real functionf. This kind of equation is an inverse problem and it is difficult to solve it. Especially in engineering sciences, the analytical expression of the functionfis unknown to the experimenter, but it can be measured at each pointxkwithM(xk) as expected value and induced error ξk. The aim is to approximate the unique root θ under some assumptions on the functionfand errors ξk. We use a stochastic approximation algorithm that constructs a sequence (xk)k ⩾ 1. We establish the almost complete convergence of the sequence (xk)kto the exact root θ by considering the errors (ξk)kquasi-associated and we illustrate the method by numerical examples to show its efficiency. [ABSTRACT FROM AUTHOR]

In this article, we study a stochastic approximation algorithm that approximates the exact root θ of a functionMdefined in ℝdinto ℝd. The functionMcannot be known exactly, but only noisy measurements are available at each pointxnwith the error ξn. The sequence of noise (ξn)nis random; we treat both cases where it is independent and dependent and we establish the complete convergence of the approximated sequence of θ. [ABSTRACT FROM PUBLISHER]