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Optimal Sign Test for High Dimensional Location Parameters

This article concerns tests for location parameters in cases where the data dimension is larger than the sample size. We propose a family of tests based on the optimality arguments in Le Cam (1986) under elliptical symmetric. The asymptotic normality of these tests are established. By maximizing the asymptotic power function, we propose an uniformly optimal test for all elliptical symmetric distributions. The optimality is also confirmed by a Monte Carlo investigation.


Introduction
Testing the population mean vector is a fundamental problem in statistics. A classical method to deal with this problem is the famous Hotelling's T 2 test. However, it can not work in high dimensional settings because the sample covariance matrix is not invertible. With the rapid development of technology, various types of high-dimensional data have been generated in many areas, such as internet portals, microarray analysis. By replacing the Mahalanobis distance by the Euclidean distance, many modified Hotelling's T 2 tests for high dimensional data are proposed in many literatures, such as Bai and Saranadasa (1996), Chen and Qin (2010), Srivastava (2009) , Feng, et al. (2015b). However, the statistical performance of the moment-based tests mentioned above would be degraded when the non-normality is severe, especially for heavy-tailed distributions.
Many nonparametric methods have been developed, as a reaction to the Gaussian approach of Hotelling's test, with the objective of extending to the multivariate context the classical univariate rank and signed-rank techniques. There are three main groups. One relies on componentwise rankings (Puri and Sen , 1971), but is not affine invariant. The second group is based on spatial signs and ranks with the so called Oja median (Oja, 2010).
Some efforts have been devoted to extending this type of method to the high dimensional data. Wang, et al. (2015) propose a high dimensional spatial sign test by replacing the scatter matrix with identity matrix. Feng, et al. (2015a) also propose a scalar-invariant high dimensional sign test for the two sample location problem. They demonstrate that the multivariate sign and rank are still very efficient methods in constructing robust test in high dimension settings. The last group use the concept of interdirections (Randles , 1992). In an important work, Hallin and Paindaveine (2002) propose a class of tests based on interdirections and pseudo-Mahalanobis ranks. Depending on the score function considered, they allow for locally asymptotically maximin test at selected densities. However, to the best of our knowledge, there are no optimal tests for high dimensional location parameters.
In this article, we propose an uniformly optimal test for high dimensional data. Based on the optimality arguments in Le Cam (1986), we introduce a high dimensional form of the locally and asymptotically optimal testing procedure. The asymptotic normality of this class of tests are established. By maximizing the power function of these tests, we propose an uniformly optimal test for high dimensional location problem. In the multivariate case, the optimal score function deeply depends on the underlying distributions. However, the optimal weighted function for our high dimensional test is unique. So our proposed test procedure is uniformly optimal for the elliptical symmetric distributions. We also derive the asymptotic relative efficiency of our test with respect to Chen and Qin (2010)'s test 's test. It is not surprised that they are all no less than one for the elliptical symmetric distributions. And for the heavy tailed distributions, such as multivariate t-distributions or mixture multivariate normal distributions, our test would perform eventually better than these two tests. Simulation studies also demonstrate these results.
2 Uniformly Optimal Test 2.1 High Dimensional Weighted Sign tests random sample from p-variate elliptically symmetric distribution with density function det(Σ) −1/2 g(||Σ −1/2 (x − θ)||) where θ is the symmetry centers and Σ is the positive definite symmetric p × p scatter matrices. We consider the following one sample testing problem (1) When the dimension p is fixed, according to the local asymptotic normality theory (Le Cam , 1986), the form of locally and asymptotically optimal testing procedures for (1) under specified Σ and g is where U(x) = x/||x||I(x = 0), ψ g = −g ′ /g and c p,g is a scaled parameter. Hallin and Paindaveine (2002) proposed a class of tests based on interdirections and pseudo-Mahalanobis ranks which are of the asymptotic form is a continuous weighted function. However, the scatter matrix Σ is not available in high dimensional settings. Motivated by Bai and Saranadasa (1996) and Chen and Qin (2010), we simply replace Σ by I p and exclude the same term in R n . We propose the following generally weighted sign test statistic: where r i = ||X i ||. Let K(t) = t, W n would be the one-sample high dimensional t-test statistic proposed in Chen and Qin (2010). Similarly, we can obtain the high dimensional sign test  with K(t) = 1. We will determine the optimal weighted function K(t) in the next section. First, we propose an asymptotic analysis for W n .
Recently, there are many high dimensional scalar-invariant tests in literature (Park and Ayyala , 2013;Srivastava , 2009;Feng, et al. , 2015a,b). The idea is replacing Σ by its diagonal matrix. And then all the variables have the same scale. Here we also standardize each variables first by the estimated diagonal matrix in Feng, et al. (2015a), which make W n invariant under the scale transformation. Details about the scalar-invariant test are given in the appendix. To expedite our discussion, we assume the diagonal matrix of Σ are known and equal to one without loss of generality.
The following conditions are needed.
The first condition in (C1) is similar to condition (3.8) in Chen and Qin (2010). Obviously, (C1) will hold if all the eigenvalues of Σ are bounded. The second condition in Condition (C1) is used to reduce the difference between the module ||ε|| and ||Σ 1/2 ε||. Then, we can get an explicit relationship between the variance of W n and Σ. Condition (C2) is similar to Theorem 1 Under Conditions (C1)-(C2) and H 0 , as (p, n) → ∞, Similar to Wang, et al. (2015), we propose the following ratio-consistent estimator of . And then we reject the null hypothesis if W n /σ n > z α where z α is the upper α quantile of N(0, 1).
Next, we consider the asymptotic distribution of W n under the alternative hypothesis Condition (C3) require the difference between µ and 0 is not large so that the variance of W n is still asymptotic σ 2 n . It can be viewed as a high-dimensional version of the local alternative hypotheses.

High Dimensional Optimal Sign test
According to Theorem 1 and 2, the asymptotic power of our weighted sign test becomes .
The power function of W n is an increasing function of E{K 2 (r i )} . By the Cauchy inequality, we have Consequently, we propose the following high dimensional optimal sign test Chen and Qin (2010) and Wang, et al. (2015) show that the asymptotic power of their proposed tests are Thus, the asymptotic relative efficiency of our proposed test with respect to these two tests In this case, ψ g (t) = (p + v)t/(v + t 2 ) → pt −1 as t → ∞. So, our uniformly optimal weighted function K(t) would be consistent with the "optimal" weighted function ψ g (t).
Formulas of asymptotic relative efficiency with these two distributions are given in the Supplementary Material.

Simulation
Here we report a simulation study designed to evaluate the performance of the proposed test.
All the simulation results are based on 2,500 replications. We consider the following five elliptical distributions: (I) N(θ, Σ); (II) t p (θ, Σ, 3); (III) t p (θ, Σ, 4); (IV) MN(0.2, 10, Σ); (V) MN(0.8, 10, Σ) and two independent component model  all the information of the module of the observations. All these results suggest that our proposed test is very efficient and robust in a wide range of distributions.

Discussion
In this paper, we propose a weighted sign test and determine the "optimal" weight function by maximizing the power function. Our asymptotic and numerical results together suggest that the proposed optimal sign test is quite robust and efficient in testing the population mean vector. This article concerns the one sample location problem. Testing the equality of two sample locations are also a very important problem (Srivastava and Du, 2008;Cai, Liu and Xia, 2014;Chen et al., 2011;Gregory et al., 2015). In the two sample problem, the common mean vector is not specified and need to be estimated. How to extend our method deserves further study. Furthermore, the proposed test procedure is essentially developed under the framework of L 2 -norm-based tests. In another direction, Cai, Liu and Xia (2014) and Zhong, Chen and Xu (2013) used the max-norm or thresholding approach to construct tests rather than the L 2 -norm. Generally speaking, the max-norm test is for more sparse and stronger signals whereas the L 2 -norm test is for denser but fainter signals. Fan, Liao and Yao (2015) also proposed a power-enhancement test based on a screening technique. Developing a spatial-sign-based test for sparse signals is of interest in the future study.

Appendix A: Scalar-invariant test
Here we replace Σ in R n with its diagonal matrix and define the following test statistic whereD ij is the corresponding diagonal matrix estimator using leave-two-out sample {X k } k =i,j in Feng, et al. (2015a). Now, T n is invariant under scalar transformations X i → BX i , Furthermore, we need another technical condition for the consistency ofD ij .

Theorem 3 Under Conditions (C1
Correspondingly, the ratio-consistent estimator ofσ 2 n is So the asymptotic power function of T n is By the Cauchy inequality, the optimal weighted function is also K(t) = t −1 .
Thus, we only need to proof the normality of W n . Define Let F n,i = σ{V 1 , · · · , V i } be the σ-field generated by {V j , j ≤ i}. Obviously, E(Z ni | F n,i−1 ) = 0 and it follows that {W nk , F n,k ; 2 ≤ k ≤ n} is a zero mean martingale. The central limit theorem (Hall and Hyde, 1980) will hold if we can show n j=2 E(Z 2 nj | F n,j−1 ) σ 2 n p → 1.