Inference for high-dimensional split-plot designs with different dimensions between groups

TL;DR

This paper introduces a novel high-dimensional inference approach for split-plot designs with varying group dimensions, allowing for flexible asymptotic analysis and robust testing across diverse settings.

Contribution

It develops a new statistical framework and estimators for high-dimensional split-plot designs with group-dependent dimensions, including semi-high-dimensional cases, without traditional size-dimension restrictions.

Findings

The proposed test statistic has an asymptotic distribution suitable for high-dimensional settings.

Simulation results demonstrate the method's robustness across different group dimensions.

The approach accommodates both fixed and increasing group dimensions in a unified framework.

Abstract

In repeated Measure Designs with multiple groups, the primary purpose is to compare different groups in various aspects. For several reasons, the number of measurements and therefore the dimension of the observation vectors can depend on the group, making the usage of existing approaches impossible. We develop an approach which can be used not only for a possibly increasing number of groups $a$, but also for group-depending dimension $d_i$, which is allowed to go to infinity. This is a unique high-dimensional asymptotic framework impressing through its variety and do without usual conditions on the relation between sample size and dimension. It especially includes settings with fixed dimensions in some groups and increasing dimensions in other ones, which can be seen as semi-high-dimensional. To find a appropriate statistic test new and innovative estimators are developed, which can be used under these diverse settings on $a,d_i$ and $n_i$ without any adjustments. We investigated the asymptotic distribution of a quadratic-form-based test statistic and developed an asymptotic correct test. Finally, an extensive simulation study is conducted to investigate the role of the single group's dimension.