Measures Of Skewness Research Articles

Statistical Inference of the Half-Logistic Inverse Rayleigh Distribution.

The inverse Rayleigh distribution finds applications in many lifetime studies, but has not enough overall flexibility to model lifetime phenomena where moderately right-skewed or near symmetrical data are observed. This paper proposes a solution by introducing a new two-parameter extension of this distribution through the use of the half-logistic transformation. The first contribution is theoretical: we provide a comprehensive account of its mathematical properties, specifically stochastic ordering results, a general linear representation for the exponentiated probability density function, raw/inverted moments, incomplete moments, skewness, kurtosis, and entropy measures. Evidences show that the related model can accommodate the treatment of lifetime data with different right-skewed features, so far beyond the possibility of the former inverse Rayleigh model. We illustrate this aspect by exploring the statistical inference of the new model. Five classical different methods for the estimation of the model parameters are employed, with a simulation study comparing the numerical behavior of the different estimates. The estimation of entropy measures is also discussed numerically. Finally, two practical data sets are used as application to attest of the usefulness of the new model, with favorable goodness-of-fit results in comparison to three recent extended inverse Rayleigh models.

Deciphering the tumor microenvironment through radiomics in non-small cell lung cancer: Correlation with immune profiles.

Growing evidence suggests that the efficacy of immunotherapy in non-small cell lung cancers (NSCLCs) is associated with the immune microenvironment within the tumor. We aimed to explore radiologic phenotyping using a radiomics approach to assess the immune microenvironment in NSCLC. Two independent NSCLC cohorts (training and test sets) were included. Single-sample gene set enrichment analysis was used to determine the tumor microenvironment, where type 1 helper T (Th1) cells, type 2 helper T (Th2) cells, and cytotoxic T cells were the targets for prediction with computed tomographic (CT) radiomic features. Multiple algorithms were in the modeling followed by final model selection. The training dataset comprised 89 NSCLCs and the test set included 60 cases of lung squamous cell carcinoma and adenocarcinoma. A total of 239 CT radiomic features were used. A linear discriminant analysis model was selected for the final model of Th2 cell group prediction. The area under the curve value of the final model on the test set was 0.684. Predictors of the linear discriminant analysis model were skewness (total and outer pixels), kurtosis, variance (subsampled from delta [subtraction inner pixels from outer pixels]), and informational measure of correlation. The performances of radiomics on test set of Th1 and cytotoxic T cell were not accurate enough to be predictable. A radiomics approach can be used to interrogate an entire tumor in a noninvasive manner and provide added diagnostic value to identify the immune microenvironment of NSCLC, in particular, Th2 cell signatures.

A New coefficient of Skewness for grouped data

The primary objective of this paper is to introduce a new measure for detecting skewness for grouped data, which is simpler than the current measures in its application. The new proposed coefficient of skewness based on the cumulative frequency data and hence uses more information from the tails of the distribution and thus will be more appropriate to detect asymmetry in the data. Another advantage of the new statistic is that it is bounded by -1 and +1; hence, the coefficients of skewness can be interpreted easily. Simulation study is employed to assess the performance of the proposed coefficient of skewness with three of the classical measure of skewness appeared in the literature using the mean square error (MSE) and mean absolute error (MAE). The simulation study strongly supports the use of the proposed measure for comparing the degrees of skewness of different frequency distributions.

Issues with integrating carbonate sand texture data generated by different analytical approaches: A comparison of standard sieve and laser-diffraction methods

Palaeoenvironmental reconstructive work relies on sediment-texture information as a proxy for particle-transport dynamics. As the integration of data from multiple sources is often sought for studies at the regional scale, a body of work exists that addresses compatibility levels of different analytical approaches. Past insights are derived predominantly from siliciclastic deposits. This study instead addresses standard sieve and laser-diffraction methods of texture analysis on a suite of carbonate beach sands from San Salvador Island, Bahamas. While the near-equant grain shapes of siliciclastic sands generate comparable results between methods, our findings show that laser diffraction tends to overestimate the coarser fractions in carbonate sands, given indiscriminate measurement of elongate particle axes. Peloidal grains that have width-to-length ratios on the order of 0.5 can pass through much smaller mesh sizes when sieved than diffraction would suggest. The resulting deviation in analytical outputs significantly influences measures of skewness, sorting, and other derivative metrics, which vary by up to several Wentworth classes between procedures. Meaningful integration of texture data obtained by different methods is ill-advised when dealing with carbonate beach sands. Similar problems are documented for muddy deposits, wherein platy clay minerals favor size-overestimation by laser diffraction and a size underestimation by settling tube technique. An improved understanding of methodological biases resulting from application of different particle-size methods is needed to help facilitate data-reconciliation efforts. This study provides a glimpse of issues facing the integration of carbonate sand-texture information, troublesome given prevalence of irregular grain shapes.

Slash Maxwell Distribution: Definition, Modified Maximum Likelihood Estimation and Applications

In this study slash Maxwell (SM) distribution, defined as a ratio of a Maxwell random variate to a power of an independent uniform random variate, is introduced. Its stochastic representation and some distributional properties such as moments, skewness and kurtosis measures are provided. The maximum likelihood (ML) method is used for estimating the unknown parameters. However, closed forms of the ML estimators cannot be obtained since the likelihood equations include nonlinear functions of the unknown parameters. We therefore use Tiku's (1967,1968) modified maximum likelihood (MML) methodology which allows to obtain explicit forms of the estimators. Some asymptotic properties of the MML estimators are derived. A Monte-Carlo simulation study is also carried out to compare the performances of the ML and MML estimators. Two data sets taken from the literature are modelled using the SM distribution in application part of the study.

Asymptotic distributions and performance of empirical skewness measures

A number of skewness measures have been proposed and applied to theoretical distributions. However, the corresponding empirical counterparts have been analyzed only rarely, especially with respect to their asymptotic properties and limit distributions. Six of these empirical measures are considered. After discussing some general properties, the limiting distribution for each measure is derived under weak assumptions. The performance of these estimators is analyzed in simulations using tests and the coverage probabilities of confidence intervals. A particular focus is put on the standardized central third moment as the most popular measure of skewness. Since it turns out to behave poorly, especially when sample sizes are small, the use of alternative and more suitable skewness measures is recommended. A real data application illustrates some of the findings.

The Properties of a Decile-Based Statistic to Measure Symmetry and Asymmetry

This paper studies a simple skewness measure to detect symmetry and asymmetry in samples. The statistic can be obviously applied with only three short central tendencies; i.e., the first and ninth deciles, and the median. The strength of the statistic to find symmetry and asymmetry is studied by employing numerous Monte Carlo simulations and is compared with some alternative measures by applying some simulation studies. The results show that the performance of this statistic is generally good in the simulation.

Measuring skewness: we don’t assume much

Skewness plays a vital role in different engineering phenomena so it is desired to measure this characteristic accurately. Several measures to quantify the extent of skewness in distributions have been developed over the course of history but each measure has some serious limitations. Therefore, in this article, we propose a new skewness measuring functional, based on distribution function evaluated at mean with minimal assumptions and limitations. Four well recognized properties for an appropriate measure of skewness are verified and demonstrated for the new measure. Comparisons with the conventional moment-based measure are carried out by employing both functionals over range of distributions available in literature. Furthermore, the robustness of the proposed measure against unusual data points is explored through the application of influence function. The Mathematical findings are verified through meticulous simulation studies and further verified by real data sets coming from diverse fields of inquiries. It is witnessed that the suggested measure passes all the checks with distinction while comparing to the classical moment-based measure. Based on computational simplicity, applicability in more general environment and preservation of c-ordering of distribution, it may be considered as an attractive addition to the family of skewness measures.

Cross-Asset Skew

We find that realized skewness is a significant indicator of returns across a range of assets from different asset classes, namely commodities, government bonds, equity indices and currencies. Taking on skewness risk is broadly compensated within, but more substantially across asset classes. Portfolios in these four asset classes with long positions on most negatively (or least positively) skewed assets and short positions on least negatively (or most positively) skewed assets generate on average a Sharpe ratio of 0.35 between 1990 and 2017. We find little evidence of a common risk driver among these portfolios, to the extent that their combination benefits substantially from diversification, delivering a Sharpe ratio of 0.72. The patterns are not subsumed by other known factors that drive returns, such as value, momentum or carry factors and, consequently, mean-variance efficient multi-factor portfolios assign a positive weight to skewness. Our results remain robust to different measures of skewness and across sub-samples.

On moments of doubly truncated multivariate normal mean–variance mixture distributions with application to multivariate tail conditional expectation

Multivariate normal mean–variance mixture (NMVM) distributions are alternatives to the multivariate normal distribution when, in practice, we encounter data sets possessing large skewness and/or kurtosis measures. In this paper, we focus on truncated forms of NMVM distributions and derive explicit expressions for the first two moments. Our results are general which can be applied for any NMVM distribution. In particular, we derive explicit expressions for the first two moments of doubly truncated multivariate generalized hyperbolic (GH) distribution. We show that by using the results established here, the multivariate tail conditional expectation (MTCE) can be obtained for any NMVM distribution.

The extended gamma distribution with regression model and applications

<abstract> <p>This paper introduces a new extension of the gamma distribution, named as a <italic>new extended gamma</italic> distribution, via mixture representation of xgamma and gamma distributions. The statistical properties of the proposed distribution are derived such as moment generating and characteristic functions, variance, skewness, and kurtosis measures, Lorenz curve, and mean residual life function. The maximum likelihood, parametric bootstrap, method of moments, least squares, and weighted least squares estimation methods are considered to obtain the unknown model parameters. The finite sample performance of estimation methods is discussed via a simulation study. Using the proposed distribution, we propose a new regression model for the right-skewed response variable as an alternative to the gamma regression model. Two real data sets are analyzed to convince the readers for the usefulness of the proposed model.</p> </abstract>

Discrete Burr-Hatke Distribution With Properties, Estimation Methods and Regression Model

A new one-parameter discrete distribution, namely discrete Burr-Hatke distribution is introduced and its mathematical properties are studied comprehensively. The main properties of the discrete Burr-Hatke distribution such as mean, variance, skewness and kurtosis measures are obtained in explicit forms. Several parameter estimation methods are used to obtain unknown model parameters and these estimation methods are compared via simulation study. The discrete Burr-Hatke distribution is over-dispersed since its variance is greater than its mean. This property of the proposed distribution opens a new opportunity to model over-dispersed data sets. To show the importance of the proposed distribution against the existing discrete probability distributions, three data sets in different fields are analyzed. Additionally, count regression model based on the discrete Burr-Hatke distribution is introduced with its residual analysis.

Modeling Skewness Determinants in Accounting Research

Skewness-based proxies are widely used in accounting and finance research. To study how the skewness of a dependent variable Y varies with conditioning variables X, researchers typically compute firm-specific skewness measures over a short rolling window and regress them on X. However, we show that this standard approach can cause severe biases and produce false findings of both conditional skewness on average and systematic variation in conditional skewness. These biases generalize beyond rolling-window skewness. We develop alternative methods that address these biases by directly modeling the conditional skewness of Y for each observation as a function of X. Simulations confirm that our methods have good type-I errors and test power even in scenarios in which the standard method is severely biased. Our methods are transparent, robust, and can be implemented in a few lines of code. Use of our methods changes a major prior finding.

Stock Return Asymmetry in China

In this paper, we find that the upside asymmetry calculated based on a new distribution-based asymmetry measure proposed by Jiang, Wu, Zhou, and Zhu (2020) is negatively related to average future returns in the crosssection of Chinese stock returns. By contrast, when using a conventional skewness measure, the relationship between asymmetry and average returns is unclear. Furthermore, the asymmetry factor constructed from the new asymmetry measure cannot be explained by the three-factor (CH-3) and four-factor (CH-4) models proposed by Liu, Stambaugh, and Yuan (2019). When augmenting the CH-3 model with our asymmetry factor, the augmented four-factor model is able to explain 32 anomalies out of a universe of 37 significant anomalies in the Chinese stock market, outperforming both the CH-3 and CH-4 models.

Bias, Precision, and Accuracy of Skewness and Kurtosis Estimators for Frequently Used Continuous Distributions

Several measures of skewness and kurtosis were proposed by Hogg (1974) in order to reduce the bias of conventional estimators when the distribution is non-normal. Here we conducted a Monte Carlo simulation study to compare the performance of conventional and Hogg’s estimators, considering the most frequent continuous distributions used in health, education, and social sciences (gamma, lognormal and exponential distributions). In order to determine the bias, precision and accuracy of the skewness and kurtosis estimators for each distribution we calculated the relative bias, the coefficient of variation, and the scaled root mean square error. The effect of sample size on the estimators is also analyzed. In addition, a SAS program for calculating both conventional and Hogg’s estimators is presented. The results indicated that for the non-normal distributions investigated, the estimators of skewness and kurtosis which best reflect the shape of the distribution are Hogg’s estimators. It should also be noted that Hogg’s estimators are not as affected by sample size as are conventional estimators.

Mixtures of traces of Wishart and inverse Wishart matrices

Traces of Wishart matrices appear in many applications, for example in finance, discriminant analysis, Mahalanobis distances and angles, loss functions and many more. These applications typically involve mixtures of traces of Wishart and inverse Wishart matrices that are concerned in this paper. Of particular interest are the sampling moments and their limiting joint distribution. The covariance matrix of the marginal positive and negative spectral moments is derived in closed form (covariance matrix of where ). The results are obtained through convenient recursive formulas for and Moreover, we derive an explicit central limit theorem for the scaled vector Y, when and present a simulation study on the convergence to normality and on a skewness measure.

A note on a measure of asymmetry

A recently proposed measure of asymmetry (Patil et al. in Stat Papers 53: 971–985, 2012) is analyzed in detail. Several examples illustrate the peculiar behavior of this measure $$\eta $$ as a measure of asymmetry or skewness. These findings are supported by theoretical considerations. Specifically, $$\eta $$ is revealed to be a measure of similarity with the exponential distribution rather than an asymmetry measure. To illustrate this, we consider a related goodness of fit test for exponentiality. Moreover, we show that the partly erratic behavior of $$\eta $$ also has a negative impact on the estimation of the measure.

Some remarks on Koziol’s kurtosis

James Koziol, in his 1987 and 1989 papers, proposed to use the sums of either the squared fourth-order cumulants or moments as test statistics for multivariate normality. His proposals are by far less popular than Mardia’s measure of multivariate kurtosis, that is the fourth moment of the Mahalanobis distance of a random vector from its mean. We investigate some properties of Koziol’s measures of multivariate kurtosis which motivate their use in statistical practice. Firstly, we show some of their connections with Mahalanobis angles. Secondly, we use inequalities to highlight their connections with other measures of multivariate skewness and kurtosis. Thirdly, we obtain their analytical formulae for some well-known multivariate statistical models. Simple examples illustrate the interpretation of Koziol’s measures of multivariate kurtosis and detect a wrong statement about them which appeared in the statistical literature. We suggest that Mardia’s and Koziol’s measures of kurtosis should be used together to detect interesting data structures.

Modelling return behaviour of global real estate investment trusts equities

PurposeThis paper aims to contribute to knowledge by investigating the return behaviour of seven global real estate investment trusts (REITs) with respect to the appropriate distributional fit that captures tail and shape characteristics. The study adds to the knowledge of distributional properties of seven global REITs by using the generalised lambda distribution (GLD), which captures fairly well the higher moments of the returns.Design/methodology/approachThis is an empirical study with GLD through three rival methods of fitting tail and shape properties of seven REIT return data from January 2008 to November 2017. A post-Global Financial Crisis (GFC) (from July 2009) period fits from the same methods are juxtaposed for comparison.FindingsThe maximum likelihood estimates outperform the methods of moment matching and quantile matching in terms of goodness-of-fit in line with extant literature; for the post-GFC period as against the full-sample period. All three methods fit better in full-sample period than post-GFC period for all seven countries for the Region 4 support dynamics. Further, USA and Singapore possess the strongest and stronger infinite supports for both time regimes.Research limitations/implicationsThe REITs markets, however, developed, are of wide varied sizes. This makes comparison less than ideal. This is mitigated by a univariate analysis rather than multivariate one.Practical implicationsThis paper is a reminder of the inadequacy of the normal distribution, as well as the mean, variance, skewness and kurtosis measures, in describing distributions of asset returns. Investors and policymakers may look at the location and scale of GLD for decision-making about REITs.Originality/valueThe novelty of this work lies with the data used and the detailed analysis and for the post-GFC sample.

Darna Distribution: Properties and Application

In this paper, Darna distribution (DD) is suggested as a new continuousprobability density function. The statistical properties of the DD as the moments,shapes of the distribution, measures of skewness, kurtosis, coefficientof variation are presented as well as some calculations are provided. Also,the maximum likelihood estimators, the Bonferroni and Lorenz curves, andGini Index are obtained. The Stress - Strength Reliability, R´enyi entropy,mean and median deviations are derived and proved. The distribution oforder statistics are presented. The reliability analysis including hazard, reliability,odds, and reverse hazard functions are presented. An application ofWheaton River data is considered.