The asymmetry measure proposed in Patil et al. (2012) is based on the correlation between the density and the distribution function evaluated at a continuous random variable. This note studies this measure for the class of two-piece distributions generated from symmetric distributions. It is shown that, within this class, the measure is linear with respect to the two-piece asymmetry parameter, whereas the effect of the symmetric baseline distribution is summarized by a proportionality constant. Explicit expressions are obtained for benchmark models, including the normal, logistic, Student-t, and contaminated normal distributions. Finally, a Monte Carlo study compares the finite-sample variability of a nonparametric estimator of this measure with that of Bowley’s quartile skewness coefficient, and a brief empirical application illustrates the measure using real data.
Quantile estimation is a fundamental problem in statistical inference. This paper investigates the asymptotic properties of sample quantiles for linearly extended negative quadrant dependent (LENQD) observations. We establish the rate of strong consistency, derive a Bahadur representation, and obtain a Berry–Esseen-type bound for LENQD samples. Finally, a numerical study and a real-data analysis are presented to illustrate the accuracy of our theoretical results for finite sample sizes.
This paper provides a general framework to characterize the limit behavior of extreme order statistics of sequences of independent non-identically distributed random variables. Under the natural uniformity assumptions, which serve as the heterogeneous analogue of the classical independent and identically distributed random variables setting, we establish convergence in distribution for both maxima and minima. Our results encompass the standard Gumbel, Fréchet, and Weibull types, as well as novel non-classical limit laws that arise within this framework. We also briefly examine, as a cautionary digression in Subsect. 1.2, the richer landscape of possible limits when the uniformity assumptions are relaxed; however, the main contribution of this work is the comprehensive treatment under the uniformity assumptions. We further investigate the asymptotic independence of maxima and minima, the asymptotic behavior of the normalizing constants, and the limiting distribution of the sample range, with direct applications in financial risk management.
In the Bayesian framework, statistical inference for structural equation models (SEMs) relies on Markov chain Monte Carlo (MCMC) methods. However, MCMC approaches are constrained by low computational efficiency, slow convergence, and high memory requirements. Gaussian variational Bayes (GVB) has emerged as an effective alternative to (MCMC), providing closed-form analytical solutions. GVB not only facilitates the construction of credible intervals for the joint posterior distribution of all model parameters, but also more accurately captures posterior dependencies than mean-field variational Bayes (MFVB). As a result, it alleviates variance underestimation and overly narrow credible intervals, improving the reliability of inference. Despite these advantages, the application of GVB in SEMs remains limited. To address the quadratic growth of the approximate posterior covariance matrix as the number of latent variables increases, we propose a novel approach combining Gaussian approximations of the joint posterior of parameters with exact conditional posteriors of latent variables, and develop a fast GVB (FGVB) algorithm for SEMs with covariance matrix factorization and reparameterization. A series of simulation studies on SEMs demonstrates that FGVB balances computational efficiency with coverage of credible intervals, providing more accurate credible intervals than MFVB and outperforming MCMC in computation time. It is further applied to investigate functional connectivity differences between younger and older adults using functional magnetic resonance imaging (fMRI) data.
This paper studies the reliability implications of a quadratic functional variable transformation applied to a normalised exponential-gamma baseline distribution. The resulting Quadratic-Transformed Exponential-Gamma distribution is derived and its statistical properties are investigated. Closed-form expressions are obtained for the density, distribution function, hazard rate, survival function, entropy, order statistics, and Fisher information matrix. The normalising constant of the exponential-gamma construction is established, and the corrected baseline is shown to occupy the q=12 position in the generalised Gamma family, with formal proof of statistical distinctness from the standard Gamma. Maximum likelihood estimation is developed with a semi-closed starting value, and the existence and uniqueness of the estimator are established. A Monte Carlo simulation study across three distinct hazard behaviours and four sample sizes confirms convergence and well-calibrated coverage properties. The practical usefulness of the distribution is illustrated using three benchmark lifetime datasets compared with several competing distributions using standard goodness-of-fit criteria. The results indicate that the proposed distribution provides a flexible two-parameter model for modelling heavily right-skewed lifetime data.
