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This work is devoted to the study of rates of convergence of the empirical measures \\mu_{n} = \\frac {1}{n} \\sum_{k=1}^n \\delta_{X_k}, n \\geq 1, over a sample (X_{k})_{k \\geq 1} of independent identically distributed real-valued random variables towards the common distribution \\mu in Kantorovich transport distances W_p. The focus is on finite range bounds on the expected Kantorovich distances \\mathbb{E}(W_{p}(\\mu_{n},\\mu )) or \\big [ \\mathbb{E}(W_{p}^p(\\mu_{n},\\mu )) \\big ]^1/p in terms of moments and analytic conditions on the measure \\mu and its distribution function. The study describes a variety of rates, from the standard one \\frac {1}{\\sqrt n} to slower rates, and both lower and upper-bounds on \\mathbb{E}(W_{p}(\\mu_{n},\\mu )) for fixed n in various instances. Order statistics, reduction to uniform samples and analysis of beta distributions, inverse distribution functions, log-concavity are main tools in the investigation. Two detailed appendices collect classical and some new facts on inverse distribution functions and beta distributions and their densities necessary to the investigation.

A wide selection of classical and recent tests for exponentiality are discussed and compared. The classical procedures include the statistics of Kolmogorov-Smirnov and Cramér-von Mises, a statistic based on spacings, and a method involving the score function. Among the most recent approaches emphasized are methods based on the empirical Laplace transform and the empirical characteristic function, a method based on entropy as well as tests of the Kolmogorov-Smirnov and Cramér-von Mises type that utilize a characterization of exponentiality via the mean residual life function. We also propose a new goodness-of-fit test utilizing a novel characterization of the exponential distribution through its characteristic function. The finite-sample performance of the tests is investigated in an extensive simulation study.

In multiple change-point problems, different data segments often follow different distributions, for which the changes may occur in the mean, scale or the entire distribution from one segment to another. Without the need to know the number of change-points in advance, we propose a nonparametric maximum likelihood approach to detecting multiple change-points. Our method does not impose any parametric assumption on the underlying distributions of the data sequence, which is thus suitable for detection of any changes in the distributions. The number of change-points is determined by the Bayesian information criterion and the locations of the change-points can be estimated via the dynamic programming algorithm and the use of the intrinsic order structure of the likelihood function. Under some mild conditions, we show that the new method provides consistent estimation with an optimal rate. We also suggest a prescreening procedure to exclude most of the irrelevant points prior to the implementation of the nonparametric likelihood method. Simulation studies show that the proposed method has satisfactory performance of identifying multiple change-points in terms of estimation accuracy and computation time.

This chapter contains sections titled: Fuzzy valued empirical distribution function Fuzzy empirical fractiles Smoothed empirical distribution function Problems

The purpose of this paper is estimating the dependence function of multivariate extreme values copulas. Different nonparametric estimators are developed in the literature assuming that marginal distributions are known. However, this assumption is unrealistic in practice. To overcome the drawbacks of these estimators, we substituted the extreme value marginal distribution by the empirical distribution function. Monte Carlo experiments are carried out to compare the performance of the Pickands, Deheuvels, Hall-Tajvidi, Zhang and Gudendorf-Segers estimators. Empirical results showed that the empirical distribution function improved the estimators’ performance for different sample sizes.

We explore various estimators for the parameters of a pair-copula construction (PCC), among those the stepwise semiparametric (SSP) estimator, designed for this dependence structure. We present its asymptotic properties, as well as the estimation algorithm for the two most common types of PCCs. Compared to the considered alternatives, that is, maximum likelihood, inference functions for margins and semiparametric estimation, SSP is in general asymptotically less efficient. As we show in a few examples, this loss of efficiency may however be rather low. Furthermore, SSP is semiparametrically efficient for the Gaussian copula. More importantly, it is computationally tractable even in high dimensions, as opposed to its competitors. In any case, SSP may provide start values, required by the other estimators. It is also well suited for selecting the pair-copulae of a PCC for a given data set.

Recently, it has been suggested that microgrids (MGs) can improve the resilience of distribution systems. However, predictions about future faults are uncertain. This makes calculating the exact value of the benefits of system resilience enhancement close to impossible at the time of MG planning. Therefore, this paper proposes a framework for MG planning, which focuses on resilience estimation. To consider the uncertainties of future failure events, the proposed method for estimating the resilience utilized the Monte Carlo simulation. In addition, an optimal scenario was estimated using a cost–benefit analysis and constraints on the expected value of resilience enhancement. In the case study, an actual MG installation at D-university was evaluated to obtain the optimal MG planning scenario. The results show that the capacity and installation locations of the distributed generators (DGs) impact the resilience enhancement. The proposed method can effectively derive the optimal MG planning scenario by evaluating the possibility of future operations based on the segmentation of both the system configuration and type of DG to improve the resilience of distribution systems.

The range of Lesser Prairie-Chickens (Tympanuchus pallidicinctus) spans 4 unique ecoregions along 2 distinct environmental gradients. The Sand Shinnery Oak Prairie ecoregion of the Southern High Plains of New Mexico and Texas is environmentally isolated, warmer, and more arid than the Short-Grass, Sand Sagebrush, and Mixed-Grass Prairie ecoregions in Colorado, Kansas, Oklahoma, and the northeast panhandle of Texas. Weather is known to influence Lesser Prairie-Chicken nest survival in the Sand Shinnery Oak Prairie ecoregion; regional variation may also influence nest microclimate and, ultimately, survival during incubation. To address this question, we placed data loggers adjacent to nests during incubation to quantify temperature and humidity distribution functions in 3 ecoregions. We developed a suite of a priori nest survival models that incorporated derived microclimate parameters and visual obstruction as covariates in Program MARK. We monitored 49 nests in Mixed-Grass, 22 nests in Sand Shinnery Oak, and 30 nests in Short-Grass ecoregions from 2010 to 2014. Our findings indicated that (1) the Sand Shinnery Oak Prairie ecoregion was hotter and drier during incubation than the Mixed- and Short-Grass ecoregions; (2) nest microclimate varied among years within ecoregions; (3) visual obstruction was positively associated with nest survival; but (4) daily nest survival probability decreased by 10% every half-hour when temperature was greater than 34°C and vapor pressure deficit was less than −23 mmHg during the day (about 0600–2100 hours). Our major finding confirmed microclimate thresholds for nest survival under natural conditions across the species' distribution, although Lesser Prairie-Chickens are more likely to experience microclimate conditions that result in nest failures in the Sand Shinnery Oak Prairie ecoregion. The species would benefit from identification of thermal landscapes and management actions that promote cooler, more humid nest microclimates.

In response to the growing need for flexible parametric models for skewed and heavy-tailed data, this paper introduces a novel goodness-of-fit test for the Skew-t distribution, a widely used flexible parametric probability distribution. Traditional methods often fail to capture the complex behavior of data in fields such as engineering, public health, and the social sciences. Our proposed test, based on energy statistics, provides practitioners with a robust and powerful tool for assessing the suitability of the Skew-t distribution for their data. We present a comprehensive methodological evaluation, including a comparative study that highlights the advantages of our approach over traditional tests. The results of our simulation studies demonstrate a significant improvement in power, leading to more reliable inference. To further showcase the practical utility of our method, we apply the proposed test to three real-world datasets, offering a valuable contribution to both the theoretical and applied aspects of statistical modeling for non-normal data.