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This paper proposes a bias-corrected expression for maximum likelihood estimators using the sequential number-theoretic method for optimization (SNTO) to improve the efficiency and accuracy of the estimators in three extreme value distributions (EVDs). It is well known that the widely used maximum likelihood estimation (MLE) could be often biased for small-size samples in EVDs. Meanwhile, numerical simulation results reveal that maximum likelihood estimators are suffered from high variance when the sample size is small and the impact is non-negligible. A comprehensive comparison study which includes classical bias-corrected methods and more recent ones is presented. Based on the simulation studies, the bias-correction estimator via SNTO is highly recommended to reduce the bias and variance of estimators. In addition, a real data set is illustrated to employ different techniques. [ABSTRACT FROM AUTHOR]

The multivariate normal is a common assumption in many statistical models and methodologies for high-dimensional data analysis. The exploration of approaches to testing multivariate normality never stops. Due to the characteristics of the multivariate normal distribution, most approaches to testing multivariate normality show more or less advantages in their power performance. These approaches can be classified into two types: multivariate and univariate. Using the multivariate normal characteristic by the Mahalanobis distance, we propose an approach to testing multivariate normality based on representative points of the simple univariate F-distribution and the traditional chi-square statistic. This approach provides a new way of improving the traditional chi-square test for goodness-of-fit. A limited Monte Carlo study shows a considerable power improvement of the representative-point-based chi-square test over the traditional one. An illustration of testing goodness-of-fit for three well-known datasets gives consistent results with those from classical methods. [ABSTRACT FROM AUTHOR]