The Jarque–Bera test is commonly used in statistics and econometrics to test the hypothesis that sample elements adhere to a normal distribution with an unknown mean and variance. This paper proposes several modifications to this test, allowing for testing hypotheses that the considered sample comes from: a normal distribution with a known mean (variance unknown); a normal distribution with a known variance (mean unknown); a normal distribution with a known mean and variance. For given significance levels, $\alpha =0.05$ and $\alpha =0.01$, we compare the power of our normality test with the most well-known and popular tests using the Monte Carlo method: Kolmogorov–Smirnov (KS), Anderson–Darling (AD), Cramér–von Mises (CVM), Lilliefors (LF), and Shapiro–Wilk (SW) tests. Under the specific distributions, 1000 datasets were generated with the sample sizes $n=25,50,75,100,150,200,250,500,$ and 1000. The simulation study showed that the suggested tests often have the best power properties. Our study also has a methodological nature, providing detailed proofs accessible to undergraduate students in statistics and probability, unlike the works of Jarque and Bera.

The presented article is aimed at solving the problem of implementing effective quality management of the regional ecosystem and preventing environmental threats. Trends in the reduction of species biological diversity and climate change are observed. Forecast and prevention of large-scale emergencies of various nature should become the basis for a new environmental and economic policy, at the center of which should be an updated system of environmental regulation. The problem arises of standardizing the quality of the environment, which requires the development of a methodology for its assessment based on a certain tool that determines the permissible or unacceptable anthropogenic load and thereby warns of the emergence of a threat. The scientific novelty of the research consists in the development of a methodology for constructing an environmental monitoring system (IAS “Environmental Barometer”), aimed at measuring negative pressure on the territorial ecosystem and based on a digital information and analytical system used by federal and regional authorities for effective management of environmental quality and sustainable development of the territory on the basis of formed strategies. The scientific novelty of the research consists in the development of a methodology for the implementation of environmental monitoring.

We investigate the asymptotic properties of the plug-in estimator for the Jeffreys divergence, the symmetric variant of the Kullback–Leibler (KL) divergence. This study focuses specifically on the divergence between discrete distributions. Traditionally, estimators rely on two independent samples corresponding to two distinct conditions. However, we propose a one-sample estimator where the condition results from a random event. We establish the estimator’s asymptotic unbiasedness (law of large numbers) and asymptotic normality (central limit theorem). Although the results are expected, the proofs require additional technical work due to the randomness of the conditions.