Testing for central symmetry and inference of the unknown center

In this paper, we consider testing for central symmetry and inference of the unknown center with multivariate data. Our proposed test statistics are based on weighted integrals of empirical characteristic functions. With two special weight functions, we obtain test statistics with simple and closed forms. The test statistics are easy to implement. In fact, they are based merely on pairwise distances between points in the sample. The asymptotic results are developed. It is proven that our proposed tests can converge to finite limit at the rate of n−1 under the null hypothesis and can detect any fixed alternatives. For the unknown center, we also propose two classes of minimum distance estimators based on the previously introduced test statistics. The asymptotic normalities are derived. Efficient algorithms are also developed to compute the estimators in practice. We further consider checking whether the unknown center is equal to a specified value μ0. Extensive simulation studies and one medical data analysis are conducted to illustrate the merits of the proposed methods.