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This paper provides an introductory overview of a portion of distribution theory which is currently under intense development. The starting point of this topic has been the so-called skew-normal distribution, but the connected area is becoming increasingly broad, and its branches include now many extensions, such as the skew-elliptical families, and some forms of semiparametric formulations, extending the relevance of the field much beyond the original theme of 'skewness'. The final part of the paper illustrates connections with various areas of application, including selective sampling, models for compositional data, robust methods, some problems in econometrics, non-linear time series, especially in connection with financial data, and more.

Given the reality of anthropogenic global warming, it is tempting to seek an anthropogenic component in any recent change in the statistics of extreme weather. This paper cautions that such efforts may, however, lead to wrong conclusions if the distinctively skewed and heavy-tailed aspects of the probability distributions of daily weather anomalies are ignored or misrepresented. Departures of several standard deviations from the mean, although rare, are far more common in such a distinctively non-Gaussian world than they are in a Gaussian world. This further complicates the problem of detecting changes in tail probabilities from historical records of limited length and accuracy. A possible solution is to exploit the fact that the salient non-Gaussian features of the observed distributions are captured by so-called stochastically generated skewed (SGS) distributions that include Gaussian distributions as special cases. SGS distributions are associated with damped linear Markov processes perturbed by asymmetric stochastic noise and as such represent the simplest physically based prototypes of the observed distributions. The tails of SGS distributions can also be directly linked to generalized extreme value (GEV) and generalized Pareto (GP) distributions. The Markov process model can be used to provide rigorous confidence intervals and to investigate temporal persistence statistics. The procedure is illustrated for assessing changes in the observed distributions of daily wintertime indices of large-scale atmospheric variability in the North Atlantic and North Pacific sectors over the period 1872–2011. No significant changes in these indices are found from the first to the second half of the period.

Leaf size (i.e., leaf surface area and leaf dry mass) profoundly affects a variety of biological carbon, water and energy processes. Therefore, the remarkable variability in individual leaf size and its trade-off with total leaf number in a plant have particularly important implications for understanding the adaption strategy of plants to environmental changes. The various leaf sizes of plants growing in the same habitat are expected to have distinct abilities of thermal regulation influencing leaf water loss and shedding heat. Here, we sampled 16 tree species co-occurring in a temperate forest in northeastern China to quantify the variation of leaf, stomata and twigs traits, and to determine the relationships of leaf size with leaf number and leaf water loss. We examined the right-skewed distributions of leaf size, leafing intensity, stomatal size and stomatal density across species. Leafing intensity was significantly negatively correlated with leaf size, accounting for 4 and 12% of variation in leaf area and leaf mass, respectively. Species was the most important factor in explaining the variation in leaf size (conditional of 0.92 for leaf area and 0.82 for leaf mass). Leaf area and mass significantly increased with increasing diameter of twigs. Leaf water loss was strongly negatively correlated with leaf area and leaf mass during the first four hours of the measurement. Leaf area and leaf mass accounted for 38 and 30% of variation in total leaf water loss, respectively. Leaf water loss rate ( ) was significantly different among tree species and markedly linearly decreased with increasing leaf area and leaf mass for simple-leaved tree species. In conclusion, the existence of a cross-species trade-off between the size of individual leaves and the number of leaves per yearly twig unit was confirmed in that temperate forest. There was strongly negative correlation between leaf water loss and leaf size across tree species, which provides evidences for leaf size in leaf temperature regulation in dry environment with strong radiation. The size-dependent leaf water relation is of central importance to recognize the functional role of leaf size in a changing climate including rapid changes in air temperature and rainfall.

Only a few scientists are able to publish a substantial number of papers every year; most of the scientists have an output of only a few publications or no publications at all. Several theories (e.g., the “sacred spark” theory) have been proposed in the past to explain these productivity differences that are complementary and focus on different aspects in the publication process. This study is intended to introduce a research program for studying productivity differences in science (skewed distributions of scientists’ productivity). The program is based on the Anna Karenina Principle (AKP). The AKP states that success in research is the result of several prerequisites that are multiplicatively related. Great success results from prerequisites that must be all given. If at least one prerequisite is not given, failure follows, whereby the failure is specific to the set of given and missing prerequisites. High productivity is given for the few scientists who fulfill all prerequisites (e.g., high motivation, pronounced creativity, reputational professional position, early important papers in high-impact journals), and low productivity is connected to a specific combination of missing and fulfilled prerequisites for many scientists. Besides the AKP as theoretical principle, the program for studying productivity differences includes a mathematical concept explaining skewed distributions and statistical methods for empirical productivity analyses.

Skew-symmetric families of distributions such as the skew-normal and skew-t represent supersets of the normal and t distributions, and they exhibit richer classes of extremal behaviour. By defining a non-stationary skew-normal process, which allows the easy handling of positive definite, non-stationary covariance functions, we derive a new family of max-stable processes – the extremal skew-t process. This process is a superset of non-stationary processes that include the stationary extremal-t processes. We provide the spectral representation and the resulting angular densities of the extremal skew-t process and illustrate its practical implementation.

A distribution that maximizes an entropy can be found by applying two different principles. On the one hand, Jaynes (1957a,b) formulated the maximum entropy principle (MaxEnt) as the search for a distribution maximizing a given entropy under some given constraints. On the other hand, Kapur (1994) and Kesavan and Kapur (1989) introduced the generalized maximum entropy principle (GMaxEnt) as the derivation of an entropy for which a given distribution has the maximum entropy property under some given constraints. In this paper, both principles were considered for cumulative entropies. Such entropies depend either on the distribution function (direct), on the survival function (residual) or on both (paired). We incorporate cumulative direct, residual, and paired entropies in one approach called cumulative Φ entropies. Maximizing this entropy without any constraints produces an extremely U-shaped (=bipolar) distribution. Maximizing the cumulative entropy under the constraints of fixed mean and variance tries to transform a distribution in the direction of a bipolar distribution, as far as it is allowed by the constraints. A bipolar distribution represents so-called contradictory information, which is in contrast to minimum or no information. In the literature, to date, only a few maximum entropy distributions for cumulative entropies have been derived. In this paper, we extended the results to well known flexible distributions (like the generalized logistic distribution) and derived some special distributions (like the skewed logistic, the skewed Tukey λ and the extended Burr XII distribution). The generalized maximum entropy principle was applied to the generalized Tukey λ distribution and the Fechner family of skewed distributions. Finally, cumulative entropies were estimated such that the data was drawn from a maximum entropy distribution. This estimator will be applied to the daily S&P500 returns and time durations between mine explosions.

The performance of ectotherms integrated over time depends in part on the position and shape of the distribution of body temperatures (Tb) experienced during activity. For several complementary reasons, physiological ecologists have long expected that Tb distributions during activity should have a long left tail (left‐skewed), but only infrequently have they quantified the magnitude and direction of Tb skewness in nature. To evaluate whether left‐skewed Tb distributions are general for diurnal desert lizards, we compiled and analysed Tb (∑ = 9,023 temperatures) from our own prior studies of active desert lizards in three continents (25 species in Western Australia, 10 in the Kalahari Desert of Africa and 10 species in western North America). We gathered these data over several decades, using standardized techniques. Many species showed significantly left‐skewed Tb distributions, even when records were restricted to summer months. However, magnitudes of skewness were always small, such that mean Tb were never more than 1°C lower than median Tb. The significance of Tb skewness was sensitive to sample size, and power tests reinforced this sensitivity. The magnitude of skewness was not obviously related to phylogeny, desert, body size or median body temperature. Moreover, a formal phylogenetic analysis is inappropriate because geography and phylogeny are confounded (i.e. are highly collinear). Skewness might be limited if lizards pre‐warm inside retreats before emerging in the morning, emerge only when operative temperatures are high enough to speed warming to activity Tb, or if cold lizards are especially wary and difficult to spot or catch. Telemetry studies may help evaluate these possibilities. A plain language summary is available for this article. Plain Language Summary

Background Image processing techniques have been widely used in the analysis of leaf characteristics. Earlier techniques for processing digital RGB color images of plant leaves had several drawbacks, such as inadequate de-noising, and adopting normal-probability statistical estimation models which have few parameters and limited applicability. Results We confirmed the skewness distribution characteristics of the red, green, blue and grayscale channels of the images of tobacco leaves. Twenty skewed-distribution parameters were computed including the mean, median, mode, skewness, and kurtosis. We used the mean parameter to establish a stepwise regression model that is similar to earlier models. Other models based on the median and the skewness parameters led to accurate RGB-based description and prediction, as well as better fitting of the SPAD value. More parameters improved the accuracy of RGB model description and prediction, and extended its application range. Indeed, the skewed-distribution parameters can describe changes of the leaf color depth and homogeneity. Conclusions The color histogram of the blade images follows a skewed distribution, whose parameters greatly enrich the RGB model and can describe changes in leaf color depth and homogeneity.

Regional Convolutional Neural Network (RCNN)−based detectors have played a crucial role in object detection in remote sensing images due to their exceptional detection capabilities. Some studies have shown that different stages should have different Intersections of Union (IoU) thresholds to distinguish positive and negative samples because each stage has different IoU distributions. However, these studies have overlooked the fact that the IoU distribution at each stage changes continuously during the training process. Therefore, the IoU threshold at each stage should also be adjusted continuously to adapt to the changes in the IoU distribution. We realized that the IoU distribution at each stage is very similar to a Gaussian skewed distribution. In this paper, we introduce a novel dynamic IoU threshold method based on the Cascade RCNN architecture, called the Dynamic Cascade detector, with reference to the Gaussian skewed distribution. We tested the effectiveness of this method by detecting horizontal storage tanks and rotated ships in optical remote sensing images. Our experiments demonstrated that this technique can significantly improve detection results, as evaluated based on the COCO metric. In addition, the threshold range of the last stage impacts other stages, so the threshold range of one stage may change significantly when the number of stages changes. Furthermore, the threshold may not always increase during the training process and may decrease when the IoU distribution resembles a negatively skewed distribution.

We investigate the statistical mechanism of clicks and replies to posts on the BBS forum. Empirical analysis indicates that the heavy-tailed distribution of replies is correlated and determined by similarly skewed distribution of clicks. The investigation suggests that the highly skewed distribution of one could result from the skewed distribution of the other. Power law distribution of both replies and clicks can be interpreted as the macro behaviours of the systems that consist of micro units. The empirical results of this paper provide another perspective for us to understand collective online behaviour.