BHEP Class of Tests for Multivariate Normality: An Empirical Comparison

This paper compares the empirical power performances of eight tests for multivariate normality classified under Baringhaus-Henze-Epps-Pulley (BHEP) class of tests. The tests are compared under eight different alternative distributions. The result shows that the eight statistics have good control over type-I-error. Also, some tests are more sensitive to distributional differences with respect to their power performances than others. Also, some tests are generally more powerful than others. The generally most powerful ones are therefore recommended.

Significant Subgraph Mining for Neural Network Inference with Multiple Comparisons Correction

We describe how the recently introduced method of significant subgraph mining can be employed as a useful tool in network comparison. It is applicable whenever the goal is to compare two sets of unweighted graphs and to determine differences in the processes that generate them. We provide an extension of the method to dependent graph generating processes as the occur for example in within-subject experimental designs. Furthermore, we present an extensive investigation of error-statistical properties of the method in simulation using Erdos-Renyi models and in empirical data. In particular, we perform an empirical power analysis for transfer entropy networks inferred from resting state MEG data comparing autism spectrum patients with neurotypical controls. From this analysis one may estimate that the appropriate sample size for similar studies should be chosen in the order of n=60 per group or larger.

Overcoming the problems caused by collinearity in mixed-effects logistic model: determining the contribution of various types of violence on depression in pregnant women

Background Collinearity is a common and problematic phenomenon in studies on public health. It leads to inflation in variance of estimator and reduces test power. This phenomenon can occur in any model. In this study, a new ridge mixed-effects logistic model (RMELM) is proposed to overcome consequences of collinearity in correlated binary responses. Methods Parameters were estimated through penalized log-likelihood with combining expectation maximization (EM) algorithm, gradient ascent, and Fisher-scoring methods. A simulation study was performed to compare new model with mixed-effects logistic model(MELM). Mean square error, relative bias, empirical power, and variance of random effects were used to evaluate RMELM. Also, contribution of various types of violence, and intervention on depression among pregnant women experiencing intimate partner violence(IPV) were analyzed by new and previous models. Results Simulation study showed that mean square errors of fixed effects were decreased for RMELM than MELM and empirical power were increased. Inflation in variance of estimators due to collinearity was clearly shown in the MELM in data on IPV and RMELM adjusted the variances. Conclusions According to simulation results and analyzing IPV data, this new estimator is appropriate to deal with collinearity problems in the modelling of correlated binary responses.

Normality Testing of High-Dimensional Data Based on Principle Component and Jarque–Bera Statistics

The testing of high-dimensional normality is an important issue and has been intensively studied in the literature, it depends on the variance–covariance matrix of the sample and numerous methods have been proposed to reduce its complexity. Principle component analysis (PCA) has been widely used in high dimensions, since it can project high-dimensional data into a lower-dimensional orthogonal space. The normality of the reduced data can then be evaluated by Jarque–Bera (JB) statistics in each principle direction. We propose a combined test statistic—the summation of one-way JB statistics upon the independence of the principle directions—to test the multivariate normality of data in high dimensions. The performance of the proposed method is illustrated by the empirical power of the simulated normal and non-normal data. Two real data examples show the validity of our proposed method.

Normality Testing of High-Dimensional Data Based on Principle Component and Jarque-Bera Statistics

The testing of high-dimensional normality has been an important issue and has been intensively studied in literatures, it depends on the Variance-Covariance matrix of the sample, numerous methods have been proposed to reduce the complex of the Variance-Covariance matrix. The principle component analysis(PCA) was widely used since it can project the high-dimensional data into lower dimensional orthogonal space, and the normality of the reduced data can be evaluated by Jarque-Bera(JB) statistic on each principle direction. We propose two combined statistics, the summation and the maximum of one-way JB statistics, upon the independency of each principle direction, to test the multivariate normality of data in high dimensions. The performance of the proposed methods is illustrated by the empirical power of the simulated data of normal data and non-normal data. Two real examples show the validity of our proposed methods.

Normality Testing of High-Dimensional Data Based on Principle Component and Jarque-Bera Statistics

The testing of high-dimensional normality has been an important issue and has been intensively studied in literatures, it depends on the Variance-Covariance matrix of the sample, numerous methods have been proposed to reduce the complex of the Variance-Covariance matrix. The principle component analysis(PCA) was widely used since it can project the high-dimensional data into lower dimensional orthogonal space, and the normality of the reduced data can be evaluated by Jarque-Bera(JB) statistic on each principle direction. We propose two combined statistics, the summation and the maximum of one-way JB statistics, upon the independency of each principle direction, to test the multivariate normality of data in high dimensions. The performance of the proposed methods is illustrated by the empirical power of the simulated data of normal data and non-normal data. Two real examples show the validity of our proposed methods.

Features of the Module «Probability and Statistics» as Part of the Higher Mathematics Course

The article deals with the problem of providing a natural and logically grounded “embeddedness” of the stochastic module in the general course of higher mathematics. Its solution is proposed both by structural means of constructing mathematical theories (primary concepts, axioms, definitions, theorems), and by means of establishing intra-subject connections. In particular, the connections of standard distributions with the theory of summation of series are demonstrated. It is argued that the practice-oriented orientation of stochastic knowledge and methods is achieved by a “model” approach, that is, by bringing the process of solving problems in accordance with the standard stages of mathematical modeling. Using the example of empirical power-law moments, it is proposed to familiarize students with some modern ideas of mathematical analysis. In particular, an algorithm for reconstructing continuous theoretical distributions based on convergence theorems for mean Fourier–Chebyshev expansions is presented.

An Empirical Evaluation of a Test Procedure for the Median of Symmetrical and Asymmetrical Populations Using an Interpolated Nonparametric Confidence Interval

This paper investigates, evaluates, and highlights the performance of a test procedure for the median of a single population using an old nonparametric interpolated confidence interval. Simulation results show that the test procedure under investigation strictly maintains the size at its nominal level and has generally higher empirical power under both symmetrical heavy-tailed and asymmetrical populations.

Discharge coefficient of broad-crested weirs as a function of the relative weir height for different weir lengths

Broad-crested weirs (BCW) are often used in hydraulic engineering and water management. The most complex factor that affects the discharge capacity of BCW is the discharge coefficient. In Ukrainian engineering practice, the flow rate of BCW is defined as a function of the relative height of the spillway wall, while in the most common European methods – as a function of the relative length of the weir. The experimental dependences of the discharge coefficient of rectangular sharp-edged BCW with vertical inlet and outlet walls with the ratio of the weir length and height d/Р = 2; 4 are obtained. A comparison of the obtained results with the values of the discharge coefficient of the same BCW using the methods of Kumin and Hager indicates that this coefficient depends on both the height of the wall and the length of the weir. The corresponding empirical power law dependences are obtained. At the same values of the relative height of the wall, the discharge coefficient for the weir with the ratio d/Р = 4 is significantly lower comparing the weir with d/Р = 2, that can be explained by the more significant effect of friction resistance for the weir with longer threshold.