<p>In various fields of study, events are represented by axial data that lacks information on the direction of propagation. The axial data are employed in human activity recognition that uses tri-axial acceleration and in biomedical signals such as electrocardiogram and electromyography signal classification. Although the classifiers are available for directional data, it is pertinent to investigate the classifiers for axial data. In this paper, we study the classification problem for axial data based on the angular central Gaussian (ACG) distribution. Firstly, inference of the parameter matrix for the ACG distribution is studied. Bayes estimator and a class of equivariant estimators of the parameter matrix for the distribution are derived. We propose a shrinkage estimator of the parameter matrix using its maximum likelihood estimator (MLE) and an appropriate target matrix. The likelihood ratio test and an improved test are derived to test the equality of the parameter matrices of several ACG distributions. For axial data, we derive MLE, Bayes estimator, shrinkage estimator-based rules, and likelihood ratio-based classification rule. The kernel density classifier is proposed utilizing a directional kernel and a bandwidth parameter. The classification rules are extended to <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(k (&gt; 2)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo stretchy="false">(</mo> <mo>&gt;</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> populations. We derive the estimators of the parameter matrices of mixture densities associated with the ACG distribution and propose the classification rule for mixture distributions. Finally, intensive simulations are carried out to compute the expected probability of misclassification and compare the proposed rules on the unit hypersphere.</p>