Regularized t$$ t $$ distribution: definition, properties, and applications

For gene expression data analysis, an important task is to identify genes that are differentially expressed between two or more groups. Nevertheless, as biological experiments are often measured with a relatively small number of samples, how to accurately estimate the variances of gene expression becomes a challenging issue. To tackle this problem, we introduce a regularized t$$ t $$ distribution and derive its statistical properties including the probability density function and the moment generating function. The noncentral regularized t$$ t $$ distribution is also introduced for computing the statistical power of hypothesis testing. For practical applications, we apply the regularized t$$ t $$ distribution to establish the null distribution of the regularized t$$ t $$ statistic, and then formulate it as a regularized t$$ t $$‐test for detecting the differentially expressed genes. Simulation studies and real data analysis show that our regularized t$$ t $$‐test performs much better than the Bayesian t$$ t $$‐test in the “limma” package, in particular when the sample sizes are small.