# Projection Test with Sparse Optimal Direction for High-Dimensional One Sample Mean Problem

• Wanjun Liu
• Runze Li
Chapter

## Abstract

Testing whether the mean vector from some population is zero or not is a fundamental problem in statistics. In the high-dimensional regime, where the dimension of data p is greater than the sample size n, traditional methods such as Hotelling’s $$T^2$$ test cannot be directly applied. One can project the high-dimensional vector onto a space of low dimension and then traditional methods can be applied. In this paper, we propose a projection test based on a new estimation of the optimal projection direction $$\varSigma ^{-1}\mu$$. Under the assumption that the optimal projection $$\varSigma ^{-1}\mu$$ is sparse, we use a regularized quadratic programming with nonconvex penalty and linear constraint to estimate it. Simulation studies and real data analysis are conducted to examine the finite sample performance of different tests in terms of type I error and power.

## Notes

### Acknowledgments

This work was supported by a NSF grant DMS 1820702 and a NIDA, NIH grant P50 DA039838. The content is solely the responsibility of the authors and does not necessarily represent the official views of NSF, NIH or NIDA.

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