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.

This paper discusses new approaches to parameter estimation of gamma distribution based on representative points. In the first part, the existence and uniqueness of gamma mean squared error representative points (MSE-RPs) are discussed theoretically. In the second part, by comparing three types of representative points, we show that gamma MSE-RPs perform well in parameter estimation and simulation. The last part proposes a new Harrel–Davis sample standardization technique. Simulation studies reveal that the standardized samples can be used to improve estimation performance or generate MSE-RPs. In addition, a real data analysis illustrates that the proposed technique yields efficient estimates for gamma parameters.