In this paper, we focus on stochastic comparisons of extreme order statistics stemming from multiple-outlier scale models with dependence. Archimedean copula is used to model dependence structure among nonnegative random variables. Sufficient conditions are obtained for comparison of the largest order statistics in the sense of the usual stochastic, reversed hazard rate, star and Lorenz orders. The smallest order statistics are also compared with respect to the usual stochastic, hazard rate, star and Lorenz orders. To illustrate the theoretical establishments, some examples are provided.
Whether an extreme observation is an outlier or not, depends strongly on the corresponding tail behaviour of the underlying distribution. We develop an automatic, data-driven method to identify extreme tail behaviour that deviates from the intermediate and central characteristics. This allows for detecting extreme outliers or sets of extreme data that show less spread than the bulk of the data. To this end we extend a testing method proposed in Bhattacharya et al 2019 for the specific case of heavy tailed models, to all max-domains of attraction. Consequently we propose a tail-adjusted boxplot…
In this paper we compare the minimums of two independent and heterogeneous samples each following Kumaraswamy-G distribution with the same and the different parent distribution functions. The comparisons are carried out with respect to usual stochastic ordering and hazard rate ordering with majorized shape parameters of the distributions. The likelihood ratio ordering between the minimum order statistics is established for heterogeneous multiple outlier Kumaraswamy-G random variables with the same parent distribution function
We introduce a trimmed version of the Hill estimator for the index of a heavy-tailed distribution, which is robust to perturbations in the extreme order statistics. In the ideal Pareto setting, the estimator is essentially finite-sample efficient among all unbiased estimators with a given strict upper break-down point. For general heavy-tailed models, we establish the asymptotic normality of the estimator under second order regular variation conditions and also show it is minimax rate-optimal in the Hall class of distributions. We also develop an automatic, data-driven method for the choice of…
Dispersive order is a type of variability order for comparing the variability in probability distributions. Star order compares the skewness of probability distributions. This work considers dispersive and star orders of extreme order statistics from dependent random variables following the proportional odds (PO) model. The joint distribution of the random variables is modeled with Archimedean copula. Numerical examples are provided to illustrate the findings.
For a bivariate random vector (X,Y), symmetry conditions are presented that yield stochastic orderings among |X|, |Y|, |max(X,Y)|, and | min(X, Y)|. Partial extensions of these results for multivariate random vectors (X1,...,Xn) are also given.
Let $X_1, X_2,\ldots, X_n$ (resp. $Y_1, Y_2,\ldots, Y_n$) be independent random variables such that $X_i$ (resp. $Y_i$) follows generalized exponential distribution with shape parameter $θ_i$ and scale parameter $λ_i$ (resp. $δ_i$), $i=1,2,\ldots, n$. Here it is shown that if $\left(λ_1, λ_2,\ldots,λ_n\right)$ is $p$-larger than (resp. weakly supermajorizes) $\left(δ_1,δ_2,\ldots,δ_n\right)$, then $X_{n:n}$ will be greater than $Y_{n:n}$ in usual stochastic order (resp. reversed hazard rate order). That no relation exists between $X_{n:n}$ and $Y_{n:n}$, under same condition, in terms of likel…
Outliers due to technical errors in water-quality data from in situ sensors can reduce data quality and have a direct impact on inference drawn from subsequent data analysis. However, outlier detection through manual monitoring is unfeasible given the volume and velocity of data the sensors produce. Here, we proposed an automated framework that provides early detection of outliers in water-quality data from in situ sensors caused by technical issues.The framework was used first to identify the data features that differentiate outlying instances from typical behaviours. Then statistical transfo…
Similar to the generalized extreme value (GEV) family, the generalized extreme value distributions under power normalization are introduced by Roudsari (1999) and Barakat et al. (2013). In this article, we study the asymptotic behavior of GEV laws under power normalization and derive expressions for the kth moments, entropy, ordering in dispersion, rare event estimation and application of real data set. We also show that, under some conditions, the Shannon entropy and variance of GEV families are ordered.
The task of estimation of the tails of probability distributions having small samples seems to be still opened and almost unsolvable. The paper tries to make a step in filling this gap. In 2017 Jordanova et al. introduce six new characteristics of the heaviness of the tails of theoretical distributions. They rely on the probability to observe {\color{blue}mild or} extreme outliers. The main their advantage is that they always exist. This work presents some new properties of these characteristics. Using them six distribution sensitive estimators of the extremal index are defined. A brief simula…
A popular data-driven method for choosing the bandwidth in standard kernel regression is cross-validation. Even when there are outliers in the data, robust kernel regression can be used to estimate the unknown regression curve [Robust and Nonlinear Time Series Analysis. Lecture Notes in Statist. (1984) 26 163--184]. However, under these circumstances standard cross-validation is no longer a satisfactory bandwidth selector because it is unduly influenced by extreme prediction errors caused by the existence of these outliers. A more robust method proposed here is a cross-validation method that d…
We propose a simple multiple outlier identification method for parametric location-scale and shape-scale models when the number of possible outliers is not specified. The method is based on a result giving asymptotic properties of extreme z-scores. Robust estimators of model parameters are used defining z-scores. An extensive simulation study was done for comparing of the proposed method with existing methods. For the normal family, the method is compared with the well known Davies-Gather, Rosner's, Hawking's and Bolshev's multiple outlier identification methods. The choice of an upper limit f…
In this paper, we have obtained conditions on parameters that result in dispersive ordering and star ordering among two unequal sets of random variables from Proportional hazard rate and Proportional reversed hazard rate family of distributions. The n-independent random variables under observation belonging to a multiple-outlier model are given as corollary. The conditions obtained involve simple inequalities among the parameters and class of life distribution corresponding to ageing. Some stochastic ordering results for sample minimum and maximum with dependency based on Archimedean copula ha…
We study the characteristics of the Pickands' dependence function for bivariate extreme distribution for minima, BEVM, when considering the stochastics ordering of the two variables. The existing Pickand's dependence function terminologies and theories are modified to suit the dependence functions of extreme cases. We successful implement and apply these functions to our simulate extreme data
