The quantification of β-glucan in oats is of immense importance for plant breeders and food scientists to develop plant varieties and food products with a high quantity of β-glucan. However, the chemical analysis of β-glucan is time consuming, destructive, and laborious. In this study, near-infrared (NIR) spectroscopy in conjunction with Chemometrics was employed for rapid and non-destructive prediction of β-glucan content in oats. The interval Partial Least Square (iPLS) along with correlation matrix plots were employed to analyze the NIR spectrum from 700–1300 nm, 1300–1900 nm, and 1900–2500 nm for the selection of important wavelengths for the prediction of β-glucan. The NIR spectral data were pre-treated using Savitzky Golay smoothening and normalization before employing partial least square regression (PLSR) analysis. The PLSR models were established based on the selection of wavelengths from PLS loading plots that present a high correlation with β-glucan content. It was observed that wavelength region 700–1300 nm is sufficient for the satisfactory prediction of β-glucan of hulled and naked oats with R²c of 0.789 and 0.677, respectively, and RMSE < 0.229.

In recent years, the mixture of two-component normal distributions (MixN) has attracted considerable interest due to its flexibility in capturing a variety of density shapes. In this paper, we investigate the problem of discretizing a MixN by a fixed number of points under the minimum mean squared error (MSE-RPs). Motivated by the Fang-He algorithm, we provide an effective computational procedure with high precision for generating numerical approximations of MSE-RPs from a MixN. We have explored the properties of the nonlinear system used to generate MSE-RPs and demonstrated the convergence of the procedure. In numerical studies, the proposed computation procedure is compared with the k-means algorithm. From an application perspective, MSE-RPs have potential advantages in statistical inference.Our numerical studies show that MSE-RPs can significantly improve Kernel density estimation.

Testing multivariate normality is an ever-lasting interest in the goodness-of-fit area since the classical Pearson’s chi-squared test. Among the numerous approaches in the construction of tests for multivariate normality, normal characterization is one of the common approaches, which can be divided into the necessary and sufficient characterization and necessary-only characterization. We construct a test for multivariate normality by combining the necessary-only characterization and the idea of statistical representative points in this paper. The main idea is to transform a high-dimensional sample into a one-dimensional one through the necessary normal characterization and then employ the representative-point-based Pearson’s chi-squared test. A limited Monte Carlo study shows a considerable power improvement of the representative-point-based chi-square test over the traditional one. An illustrative example is given to show the supplemental function of the new test when used together with existing ones in the literature.

As a classic parameter from the binomial distribution, the binomial proportion has been well studied in the literature owing to its wide range of applications. In contrast, the reciprocal of the binomial proportion, also known as the inverse proportion, is often overlooked, even though it also plays an important role in various fields including clinical studies and random sampling. The maximum likelihood estimator of the inverse proportion suffers from the zero-event problem, and to overcome it, alternative methods have been developed in the literature. Nevertheless, there is little work addressing the optimality of the existing estimators, as well as their practical performance comparison. Inspired by this, we propose to further advance the literature by developing an optimal estimator for the inverse proportion in a family of shrinkage estimators. We further derive the explicit and approximate formulas for the optimal shrinkage parameter under different settings. Simulation studies show that the performance of our new estimator performs better than, or as well as, the existing competitors in most practical settings. Finally, to illustrate the usefulness of our new method, we also revisit a recent meta-analysis on COVID-19 data for assessing the relative risks of physical distancing on the infection of coronavirus, in which six out of seven studies encounter the zero-event problem.