International Journal of Statistics and Probability, Issue: Vol.15, No.2

Multicollinearity in Regression Models and Its Geometric Solutions

Tue, 30 Jun 2026 00:15:23 +0000

Multicollinearity is a pervasive challenge in regression analysis that inflates variance estimates and obscures the interpretation of predictor effects. This study develops intuitive geometric approaches for addressing multicollinearity within an observation-axes framework. We examine the structure of multicollinearity and introduce three complementary geometric approaches. First, we present a simple and effective method for resolving multicollinearity when the causal ordering among predictors is known. Second, we refine and extend the geometric interpretation of Principal Component Analysis (PCA) proposed by Wickens (2014) by incorporating angular information between predictor variables. Third, motivated by Partial Least Squares (PLS) regression, we develop a geometry-based method that identifies directions jointly determined by predictor variance and alignment with the response. Together, these solutions demonstrate that geometric reasoning provides richer and more intuitive insights into variable relationships, offering substantial pedagogical and methodological value as a resource for beginners and as a foundation for future research in regression analysis.

Large and Small Sample Problems for Normal or t-Test

Sat, 20 Jun 2026 08:53:20 +0000

Sample size plays a fundamental role in statistical inference using normal and t-tests, yet inappropriate sample sizes can lead to serious inferential distortions. Large samples tend to produce artificially small p-values and overly narrow confidence intervals, resulting in test bias and potentially misleading conclusions. In contrast, small samples often yield unstable variance estimates, reducing the reliability and power of statistical tests.

This paper addresses both extremes within a unified framework. For large samples, we develop the Random Group Method (RGM) and a correction factor approach to mitigate bias by stabilizing variability across subgroups. For small samples, we introduce pseudo-sample expansion techniques, emphasizing a mean-based method that preserves expectation while reducing variance inflation caused by duplication.

A key contribution of this work is the formulation of a relative variance (RV) criterion for determining a “good” sample size. We show that the optimal sample size is not a single value but lies within an interval over which statistical inference remains stable. Theoretical results are supported by numerical examples illustrating the relationship between sample size, variance control, and hypothesis testing behavior.

The proposed methods provide practical tools for improving statistical inference in both traditional settings and modern large-scale data applications, including artificial intelligence and survey analysis.

An Investigation of Bivariate Pareto Distribution

Fri, 26 Jun 2026 09:34:31 +0000

In this paper, some representations of the Bivariate Pareto Distributions were studied for their mathematical properties. A numerical study was undertaken to compare the performance of some Archimedean Copulas for these bivariate pareto representations. Details are provided in the main text of the paper.

Recent Advances in XGamma Extensions

Thu, 25 Jun 2026 17:31:58 +0000

The formulation of generalized class of distributions for modeling and analyzing data in diverse domains is of enormous practical significance. Captivated by the need for greater flexibility and relevance when modeling data in practice, academics in probability and distribution theory have proposed, studied and implemented novel techniques of generating new distributions from existing models. The XGamma distribution has serious shortfalls when it comes to applications in real world data set. This has attracted researchers to develop more generalized XGamma distributions to provide the desired outcomes in applications. The focus of this curated compilation constitute freshly conceived illuminating extensions, generalizations and modifications of the XGamma distribution. This is expected to serve as a vibrant platform and future research direction for researchers in probability and distribution theory.

Two-sample Tests of Sub-mean Vectors Under Two-step Monotone Missing Data

Tue, 30 Jun 2026 00:50:32 +0000

This study proposes a novel test statistic for the two-sample problem involving sub-mean vectors under a two-step monotone missing data structure. The proposed procedure is constructed based on the structure of Rao's U-statistic by combining a Hotelling's T^2-type statistic for monotone missing data with the standard Hotelling's T^2 statistic, thereby efficiently utilizing the available information in incomplete observations. We consider the problem of testing the equality of sub-mean vectors between two populations under the assumption that a subset of the mean components is common. The asymptotic expansion of the null distribution of the proposed statistic is derived, and its distribution function and approximate upper percentiles are obtained. To improve the accuracy of the chi-squared approximation in finite samples, Bartlett and Bartlett-type correction methods are also developed. The performance of the proposed approximations and correction procedures is investigated through extensive Monte Carlo simulations under various dimensional and sample size settings. A numerical example based on real data is presented to illustrate the applicability and practical usefulness of the proposed methodology.

Reviewer Acknowledgements for International Journal of Statistics and Probability, Vol. 15, No. 2

Tue, 30 Jun 2026 02:18:14 +0000

Reviewer Acknowledgements for International Journal of Statistics and Probability, Vol. 15, No. 2