Test for a Mean Vector with Fixed or Divergent Dimension

Abstract

It has been a long history in testing whether a mean vector with a fixed dimension has a specified value. Some well-known tests include the Hotelling $T^{2}$-test and the empirical likelihood ratio test proposed by Owen [Biometrika 75 (1988) 237–249; Ann. Statist. 18 (1990) 90–120]. Recently, Hotelling $T^{2}$-test has been modified to work for a high-dimensional mean, and the empirical likelihood method for a mean has been shown to be valid when the dimension of the mean vector goes to infinity. However, the asymptotic distributions of these tests depend on whether the dimension of the mean vector is fixed or goes to infinity. In this paper, we propose to split the sample into two parts and then to apply the empirical likelihood method to two equations instead of d equations, where d is the dimension of the underlying random vector. The asymptotic distribution of the new test is independent of the dimension of the mean vector. A simulation study shows that the new test has a very stable size with respect to the dimension of the mean vector, and is much more powerful than the modified Hotelling $T^{2}$-test.