In this work, first we revisit the estimation of the model parameters of a Weibull distribution based on i.i.d. observations using the maximum likelihood estimation (MLE) method, which does not yield closed expressions of the estimators. Among other results, it has been shown analytically that the MLEs obtained by solving the highly nonlinear equations do exist (i.e., finite), and are unique. (Our method is completely different from that of Balakrishnan and Kateri [6].) We then proceed to study the sampling distributions of the MLEs through both theoretical as well as computational means. It has been shown that the sampling distributions of the two model parameters’ MLEs can be approximated fairly well by suitable Weibull distributions too. Heuristic justifications have been given behind this phenomenon. Results of our comprehensive simulation study on the proposed sampling distributions corroborate some recent analytical results on the first-order bias and first-order mean squared error (MSE) expressions of the MLEs. Last but not least, real-life datasets have been used to demonstrate how the proposed Weibull sampling distributions of the MLEs can help in subsequent inferences, especially from a reliability and life testing standpoint.
Financial crises often amplify the limitations of standard return models, which often underestimate the probability of extreme price movements. This paper investigates the distributional properties of daily log-returns for gold and Bitcoin across two major crises: the COVID-19 pandemic and the Russia–Ukraine war, using pre-crisis, crisis, and post-crisis sub-periods to capture regime-specific behaviour. Ten candidate distributions are considered, including normal and Laplace families, their skewed extensions, and contaminated variants to accommodate heavy tails and asymmetry. The evaluated distributional fit across assets and regimes, provides insights into tail thickness and skewness under varying market conditions. Results reveal pronounced differences: gold demonstrates a comparatively stable distributional structure across regimes, while Bitcoin exhibits heightened sensitivity to market stress, with high kurtosis and regime-dependent skewness. These findings underscore the importance of flexible heavy-tailed models for accurate risk quantification and portfolio construction during periods of extreme volatility. By identifying which models best capture return behaviour under crisis conditions, this paper contributes to improving risk management strategies and reconciling mixed evidence on the distributional behaviour of these assets under market stress.
The two-sample testing problem is a fundamental topic in various scientific fields, with numerous nonparametric test statistics proposed over the years. This study derives the exact expected value and variance of a two-sample rank statistic with arbitrary score functions. Utilizing the exact first two moments, an approximation to the distribution of various test statistics is considered for finite sample sizes. Furthermore, the exact expected value and variance are applied to a test statistic designed for testing of location and scale parameters. Monte Carlo simulation studies are conducted to evaluate the validity of the proposed approximation and to compare the power of the suggested test against existing methods. The simulation results demonstrate that the proposed statistic is a competitive alternative to established simultaneous test statistics. The effectiveness of our approach is demonstrated through applications to real dataset, and we conclude with a discussion of our findings and potential directions for future research.
The generative adversarial network (GAN) is an important model developed for high-dimensional distribution learning in recent years. However, there is a need for a comprehensive understanding of its generalization error and convergence behavior. In this research, we study the generalization error of GANs based on classes of functions encompassing discriminator and generator neural networks with piecewise linear activation functions. Under our assumptions, the resulting function classes are VC type with bounded envelope functions, enabling the application of Talagrand’s inequality. By employing Talagrand’s inequality and the Borel–Cantelli lemma, we establish almost sure convergence rates for the generalization error of GANs under fixed discriminator and generator architectures. We further apply the same empirical-process technique to several existing GAN error formulations and obtain corresponding almost sure convergence bounds. The resulting sample-size dependence is compared with existing results under the same fixed-architecture setting, while network-complexity quantities are treated consistently as fixed constants with respect to the sample sizes. In particular, under suitable range and symmetry conditions on the discriminator class, the error defined by the neural network distance is a special case of our general measuring-function objective.
In reliability and actuarial studies, assessing the similarity between the probability distributions of residual or past lifetimes of two populations is a fundamental concern. Matusita’s measure is a widely used similarity metric that quantifies the degree of overlap between such distributions. This paper focuses on the nonparametric estimation of Matusita’s measure when data from both populations represent residual or past lifetimes. We propose nonparametric estimators for residual and past Matusita’s measures and study their asymptotic properties, such as consistency and asymptotic normality. Finally, we conduct a simulation study to evaluate the performance of the proposed estimators, and we apply our methodology to a real dataset to illustrate its practical utility.
