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Introduction and Motivation
Published in Corain Livio, Arboretti Rosa, Bonnini Stefano, Ranking of Multivariate Populations, 2017
Corain Livio, Arboretti Rosa, Bonnini Stefano
This notion of multivariate stochastic ordering generalizes the univariate one and simply indicates that one random vector is more likely than another to take on larger values. The relationship between multivariate and univariate stochastic ordering can be established by Theorem 1.1 (Davidov and Peddada, 2011).
Distributionally robust optimization with multivariate second-order stochastic dominance constraints with applications in portfolio optimization
Published in Optimization, 2023
Shuang Wang, Liping Pang, Hua Guo, Hongwei Zhang
However, in portfolio problems, decisions are often drawn from multidimensional attributes and, in this situation, decision-makers rely on multidimensional utility. Thus, it is necessary to consider the multivariate extension of stochastic dominance (see [8–15] for the one-stage problems and [16,17] for the two-stage problems). There are mainly two ways to introduce the corresponding concept about the MSSD constraints. One is to compare the expectations of multi-dimensional utility functions [8,10,12], while the other is to follow the univariate concept after scalarization [9,11,13] which is the majority in the existing studies. In this paper, we adopt the latter method to deal with the MSSD constraints. Linear scalarization is a common method. Consider random vectors X and Y in ; Dentcheva and Ruszczyński [8] define a stochastic ordering relation between these vectors by requiring the stochastic dominance relation for each coordinate . They also use the notion of positive linear stochastic dominance for a stochastic dominance constraint. Homem-de-Mello and Mehrotra [14] and Hu et al. [15] use a generalized version of this notion where the scalarization vector is selected in a more general set. Many works on the transformation of multivariate stochastic dominance are also based on the linear scalarization function [13–15,18]. Specially, Noyan and Rudolf [19] introduce a more general kind of scalarization function, named min-biaffine functions.
Image based statistical process monitoring via partial first order stochastic dominance
Published in Quality Engineering, 2022
Panagiotis Tsiamyrtzis, Marco Luigi Giuseppe Grasso, Bianca Maria Colosimo
In this study, we suggest the use of the empirical cumulative distribution function (ecdf) of pixel values of each frame, to summarize the process signature in a frame. The ecdf computation is combined with an image pre-processing operation aimed at limiting the analysis of ecdf patterns in consecutive frames to the foreground region only, where relevant process dynamics occur. Instead of deriving or modeling a reference ecdf during phase I, the proposed approach aims at estimating a benchmark that will demarcate the IC behavior. Thus, we will establish what is the region of ecdf’s IC behavior and utilize the partial first order stochastic dominance (PFOSD) properties to test (in-line during phase II) whether the process has moved from the IC to the OOC state (for an extensive bibliographic review on the concept of stochastic dominance refer to Bawa (1982)). Stochastic dominance is a form of stochastic ordering between random variables. More specifically, a random variable X is first order stochastic dominant over another random variable if its cumulative distribution function is lower than the cumulative distribution function of Y over the entire domain. PFOSD is a relaxation of the first order stochastic dominance concept as it entails a dominance over a predetermined subset of the original domain only. In our proposed approach, the random variable of interest is the pixel intensity. Comparing ecdf curves of pixel intensities corresponding to different video frames can be viewed as a PFOSD problem, where the term “partial” refers to the restriction of the range of pixel intensity values to foreground intensities only.
Two-Parameter Logistic-Exponential Distribution: Some New Properties and Estimation Methods
Published in American Journal of Mathematical and Management Sciences, 2020
Sajid Ali, Sanku Dey, Muhammad Hussain Tahir, Muhammad Mansoor
Stochastic ordering is an important tool in reliability theory and finance to assess the comparative behavior of the systems. Let X1 and X2 be two random variables having cumulative distribution functions, survival functions, and probability density functions and and and and respectively. The random variable X1 is said to be smaller than X2 in the following ordering as:stochastic order (denoted by ) if for all x;likelihood ratio order (denoted by ) if is decreasing in hazard rate order (denoted by ) if is decreasing in reversed hazard rate order (denoted by ) if is decreasing in