An inverse norm weight spatial sign test for high-dimensional directional data

Abstract: In this paper, we focus on the high-dimensional location testing problem of directional data under the assumption of rotationally symmetric distributions, where the data dimension is potentially much larger than the sample size. We study the family of directional weighted spatial sign tests for this testing problem and establish the asymptotic null distributions and local power properties of this family. In particular, we find that the test based on the inverse norm weight, named as the inverse norm weight spatial sign test, has the maximum asymptotic power in this family. As demonstrated by extensive numerical results, the inverse norm weight spatial sign test has advantages in empirical power compared with some other members in the family as well as some existing tests.


Introduction
Directional data have been widely studied in meteorology [7], astronomy [2], earth science [22] and biology [10]. These fields naturally yield a large number of directional data, such as the wind direction data and the earth scale spatial 3250 H. Wang et al. data, which are commonly considered as an implementation of a random vector on a hypersphere. Besides, some data from other fields are essential only for their relative size, hence being projected onto a unit hypersphere, where the projection removes the overall scale factor associated with the size. For analyzing directional data, the family of rotationally symmetric distributions is one of the most commonly used distributions, which has played a central role in the application of directional statistics [15]. One prominent member of the family of rotationally symmetric distributions is the Fisher-von Mises-Langevin (FvML) distribution, which plays a critical role in directional statistics, on par with that of the Gaussian distribution in the classic multivariate setting.
The literature on the inference of location parameters under the assumption of rotationally symmetric distributions is abundant. For example, [23], [3], [11], [13] and [19] considered the location inference problems in low-dimensional situations. [21] and [5] tackled the problem for spherical regression, while [1] considered the location testing problem under axial frames. [8] considered the problem of testing rotational symmetry on hyperspheres, and introduced two locally asymptotically maximin tests against two classes of directional distributions in the Le Cam sense.
In this paper, we focus on the high-dimensional location testing problem of directional data under the assumption of rotationally symmetric distributions, where the null hypothesis is that the location parameter vector is equal to a given vector on the unit hypersphere. To deal with such a testing problem, [23] proposed a traditional Watson test based on the mean of the spherical sample. [17] proposed a class of rank tests and discussed the Le Cam optimality of the tests. Then, [12] proposed a high-dimensional Watson test by standardizing the traditional Watson statistic.
To develop some alternative high-dimensional tools for testing location under rotationally symmetric distributions, in this paper we study the family of directional weighted spatial sign tests for testing location of directional data under rotationally symmetric distributions. In particular, we find that the test based on the inverse norm weight, named as the inverse norm weight spatial sign test, has the maximum asymptotic power in this family. Indeed, the proposed inverse norm weight spatial sign test is an extension of the inverse norm sign test devised for general data. We then present its asymptotic properties under the unified framework of directional weighted spatial sign tests, and demonstrate its empirical power advantages via extensive numerical results.
The rest of the paper is organized as follows. In Section 2, we introduce the family of directional weighted spatial sign tests for high-dimensional directional data under rotationally symmetric distributions. Besides, we establish the corresponding theoretical results, including the limiting null distributions, the asymptotic power under the local alternative and the asymptotic relative efficiency results in Section 3. Then, we investigate the numerical performance of the proposed test in comparison with its main competitors in Section 4. Finally, we conclude the paper with some discussions in Section 5, and relegate some additional numerical results as well as the technical proofs to the appendix.
for some given location θ 0 ∈ S p−1 . To test (2.1), we study the family of directional weighted spatial sign tests, where the test statistics are constructed as Here, ω(·) is a nonnegative and continuous weight function on R + , I p denotes the p × p identity matrix and U(·) denotes the spatial sign function with U(a) . =a/ a if a = 0 and U(a) . =0 if a = 0. Define σ 2 n (ω) .
According to Theorems 3.1 and 3.2 presented in the next section, for each ω, H 0 will be rejected when T n (ω)/ σ 2 n (ω) > z α , where z α is the upper α-quantile of the standard normal distribution N (0, 1) with significance level α. We will call this test the T n (ω)-based test.
In fact, some members of this family are closely related to existing tests for (2.1). For example, taking ω(t) = ω N (t) . =t for t ≥ 0, i.e. using the norm (N) weight function ω N , the test statistic is The main part of T n (ω N ) is the same as that of the high-dimensional Watson test statistic proposed by [12]: The relationship between the two statistics is discussed in the following proposition. To present this relationship, we need to impose the following condition.
It can be seen that the distributions of u i and v i do not depend on θ. Proposition 2.1. As n, p → ∞, under condition (C0) and H 0 , This proposition indicates that the T n (ω N )-based and W n -based tests are asymptotically equivalent, where the W n -based test rejects H 0 when W n > z α . We recall that all proofs are discussed in the appendix.
In fact, W n is a standardized version of the traditional Watson test statistic [23,20] with asymptotic mean p − 1 and asymptotic variance 2(p − 1) under the null hypothesis, whereX n .
. =1 for t ≥ 0, i.e. using the constant (C) weight function ω C , the test statistic is which is very similar to the test that was mentioned in [18]. In addition to these existing ones, we also consider some other members of this family. Taking . =t −1 for t ≥ 0, i.e. using the inverse norm (IN) weight function ω IN , the test statistic is

6)
Inverse norm weight spatial sign test 3253 which can be regarded as an extension of the inverse norm sign test (INST) proposed by [6] on directional data. In this paper, we name the T n (ω IN )-based test as the inverse norm weight spatial sign test. Taking . =t 1/2 respectively, the test statistics are and Now, we have introduced the family of directional weighted spatial sign tests. In the following section, we will establish the theoretical results of the whole family, which imply that the T n (ω IN )-based test has the maximum asymptotic power among this family. Then, extensive numerical results will demonstrate the empirical power advantages of the T n (ω IN )-based test compared with some other members of this family as well as some existing tests.

Theoretical results
In this section, we will establish the theoretical results of the whole family of directional weighted spatial sign tests with the general form of weight function ω(·).

Asymptotic power
Then, we investigate the asymptotic distribution of T n (ω) under the following local alternative. [20] due to var(u i ) ≤ E(u 2 i ), which was used to derive the local power of W n . Condition (C2) ensures that the difference between θ and θ 0 is not too large, so that the variance of T n (ω) can be asymptotically described by σ 2 n (ω).
According to Theorems 3.1 and 3.3, the local power of the T n (ω)-based test against an alternative θ that satisfies condition (C2) is Then, we obtain the asymptotic power of the T n (ω)-based test: To find a weight function ω that can make β(ω) reach the maximum value, we only need to find the maximum value of , we will obtain that taking ω = ω IN makes β(ω) reach the maximum value.
Inverse norm weight spatial sign test

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In contrast, the asymptotic power of the T n (ω C )-based, T n (ω N )-based, T n (ω S )-based and T n (ω R )-based tests are respectively. Next, we consider the asymptotic power of the T n (ω)-based test in situation of the FvML distribution.
which does not depend on the choice of the weight function ω.
This corollary indicates that if X i follows a FvML distribution, all the directional weighted spatial sign tests have the same asymptotic power.

Asymptotic relative efficiency
Then, we derive the asymptotic relative efficiencies (AREs) between the T n (ω IN )based test and the T n (ω C )-based, T n (ω N )-based, T n (ω S )-based, T n (ω R )-based tests, respectively. Specifically, For rotationally symmetric distributions, there are many choices of angular functions. Below, we present the formulas for calculating the AREs under some commonly used angular functions. First, when X i follows a rotationally symmetric distribution, the density function of Next, when X i follows the mixture of two rotationally symmetric distributions with the same location parameter, the density function of X i can be denoted as In the following, we investigate the ARE results numerically under the nonmixed and mixed rotationally symmetric distributions respectively. Here, we consider the following three types of angular functions.  Table 1 presents the corresponding ARE results of the non-mixed rotationally symmetric distributions, where some different choices of κ and p are considered. Then, Tables 2, 3 and 4 present the corresponding ARE results of the mixed distributions for (F1)-(F3) respectively, where two settings are considered: (1) λ = 0.9 with κ 1 = κ/10 and κ 2 = 10κ; (2) λ = 0.6 with κ 1 = κ/5 and κ 2 = 5κ. Although the T n (ω IN )-based test does not show obvious advantages in Table 1, it has obvious advantages as shown in Tables 2, 3 and 4, which indicates that the proposed T n (ω IN )-based test performs better than the other four directional weighted spatial sign tests in situation of mixed distributions. Table 1 The ARE results of the non-mixed rotationally symmetric distributions for the three types of angular functions.

Two rank-based tests
To make a more extensive comparison in the following simulation studies, in this subsection we consider two rank-based weights mentioned in [16]. The statistics are    and The R W -based and R S -based tests rejects H 0 when R W / σ 2 n (R W ) > z α and R S / σ 2 n (R S ) > z α , respectively.

Simulation results
In this section, we present some simulation results to investigate the performance of the W n -based,  We consider two settings of mixed distributions in (3.2), where for setting I, (λ, κ 1 , κ 2 , δ) = (0.9, κ/10, 10κ, 2Δ), and for setting II, (λ, κ 1 , κ 2 , δ) = (0.6, κ/5, 5κ, Δ/2). κ and Δ will be set according to the angular function type in the following text. For angular function (F1), we let n = 50, p ∈ {100, 200, 300} and consider two cases:  Figure 1 suggest that all these nine tests can   Figure 4 presents the power curves for angular function (F2), while Figure 6 presents the power curves for angular function (F3). From Figures 4  and 6, we see that T IN and T C can outperform the remaining tests in power comparison, while T IN performs much better than T C . In addition, we place some additional simulation results for the non-mixed distributions in Appendix.

Discussion
We have studied the general framework of directional weighted spatial sign tests for testing location of directional data under the commonly used rotationally symmetric distributions, which includes many members closely related to the existing tests. The asymptotic properties of the family of tests have been established under the rotationally symmetric distributions. Within this framework of asymptotic properties, we find that the T n (ω IN )-based test has the maximum asymptotic power in this family. Then, the power advantages of the T n (ω IN )based test have been fully demonstrated by the theoretical and numerical results.

Some additional simulation results
In this subsection, we present some additional simulation results of the nine tests, as X i follows the non-mixed rotationally symmetric distributions.
For angular function (F1), we perform simulations under the following three settings: which are the same as settings (i)-(iii) used in [20], with concentration κ and location Let (n, p) ∈ {(100, 400), (200, 800)}. Table 5 summarizes the empirical size and power results of the nine tests by using the angular function (F1). The size results in Table 5 suggest that all these nine tests can properly control the empirical size. For power performance, the power results in Table 5 suggest that the power of the T n (ω)-based weighted spatial sign tests are quite similar to that of the W n -based and W n -based tests. The power of the R W -based and R S -based tests are significantly smaller than those of the remaining tests.
Next  Table 6 summarizes the empirical size of the nine tests for angular functions (F2) and (F3), which suggests that all these tests generally control the size. In particular, we find that the empirical size of the T n (ω)-based weighted spatial sign tests are slightly larger than that of the W n -based and W n -based tests. These may indicate a slight anti-conservatism of the proposed tests.
For power comparison, Figure 7 shows the power curves of κ = 2p and 3p respectively for angular function (F2), while Figure 8 shows the power curves of κ = 2p and 3p respectively for angular function (F3). From Figures 7 and 8, it can be seen that some of the proposed directional weighted spatial sign tests are more powerful, but not significantly more powerful, than the W n -based and W n -based tests, considering the slight anti-conservatism of the proposed tests. In addition, the empirical power of the R S -based and R W -based tests are much lower than that of the remaining tests. In short, in the case of non-mixed rotationally symmetric distributions, the power of all tests except the R S -based and R W -based tests are similar. Table 5 The empirical size and power results of the nine tests under the non-mixed rotationally symmetric distributions at 5% level with angular function (F1).  Table 6 The empirical size results of the nine tests under the non-mixed rotationally symmetric distributions at 5% level with angular functions (F2) and (F3).

Technical details
If X i are rotationally symmetric on S p−1 about θ, then we have the tangentnormal decomposition of X i , In addition, S i are independent with u i , and S i is uniformly distributed on S p−1 (θ ⊥ ) : {S ∈ R p : S = 1, S T θ = 0}. Before starting to prove, we recall Lemma B.1. in [4].  Lemma A.1. Let X 1 , · · · , X n be a sequence of independent and identically distributed observations with a rotationally symmetric distribution, due to

Proof of Theorem 3.1
First, it is noteworthy that under H 0 , X i are rotationally symmetric with θ 0 , hence S i and u i are independent. We have E{T n (ω)} = 0 under H 0 due to E(S i ) = 0. Then, the variance of T n (ω) is The normality of T n (ω) has yet to be proven.
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. According to the Martingale central limit theorem in [9], we only need to show It can be shown that We first consider C n1 . Due to Thus, In addition, where we used Condition (C1) in the last equality. Then, we have Similarly, where i = j in the fourth equality and we used A = (p − 1) −1 b 2 (ω)(I p − θ 0 θ T 0 ) and Lemma A.1 (ii) in the seventh equality. Then (A.2) holds.
Next, we only need to show Note that can be decomposed as Q + P , where Then, we complete the proof.

Proof of Theorem 3.2
We have First of all, where the last equality is due to Condition (C1), which leads to

Proof of Theorem 3.3
We have H. Wang et al. Thus, Because of the tangent-normal decomposition, we have then, Due to Condition (C2), we have var(α i ) = o(1) and then α i = o p (1). Thus, n (ω)}, where the second last equality is due to α i → 0 and Lemma A.1 (iii). This leads to A 1 = o p {σ n (ω)}. Next, for the first term in A 2 , we have n (ω)}, where the second last equality is due to Lemma A.1 (i) and Condition (C2).
Then, due to i 1 = i 2 < j, we have that where i = j in the first equality, and the second last equality is due to Condition (C1). In addition,

hence the last equation holds and
Note that β 2 0 ≤ E(β 2 i ). We let Next, we will show where the last equality is due to Condition (C2). Hence we have where i < j = l in the second inequality. At first, we focus on D 1 . Due to i < j = l, where the first inequality is because of the Cauchy inequality and |u i | ≤ 1. The Condition (C2) brings about the last equality. Next, similar to the above process, we have Therefore, we can conclude that where the last equality is due to Condition (C2), which leads to Combining the above results, we get using the same procedure as in the proof of Theorem 1. Then, we obtain the conclusion.

Proof of theorem 3.4
Under conditions (C1)-(C2), given θ and θ 0 , we find that the right side of (3.1) is an increasing function of Interestingly, according to the Cauchy inequality, it can be seen that The above inequality holds for any nonnegative continuous weight function satisfying the condition (C1). So, the maximum value of the asymptotic power in the whole family is Besides, when taking ω = ω IN , the asymptotic power is which equals to the maximum value of the asymptotic power in the whole family. In addition, we will verify that under the FvML and mixed FvML distributions, the weight ω IN satisfies condition (C1).
When X i follows a FvML distribution with density function c p,κ exp(κx T θ). Then, the density function of So, we have Then, according to (S.2.7) in the supplement of [4], we have .
Then, we have Then, we obtain the conclusion that under the FvML and mixed FvML distributions, the weight ω IN satisfies condition (C1). Similarly, it is easy to verify that under the FvML and mixed FvML distributions, the weight ω N , ω C , ω R , ω S satisfies condition (C1).
Hence, the asymptotic power of all the T n (ω)-based tests is

Proof of Proposition 3.1
Under H 0 , since S i = U{(I p − θ 0 θ T 0 )X i } and v i are independent, so S i and R i are also independent. We have where Z ni = 2/{n(n − 1)} Next, we consider C n1 and note that A = E(R 2 i )(I p − θ 0 θ T 0 )/(p − 1) due to Lemma A.1. Furthermore, we have where S i = (I p − θ 0 θ T 0 )X i /v i and the last equality holds due to E(R 2 i R 2 j ) = E 2 (R 2 i ){1 − o(1)}. We also obtain . Thus, Then, we have C n1 /σ 2 n (R W ) → p 1. Similarly, where i = j in the fourth equality and we used A = (p − 1) −1 E(R 2 i )(I p − θ 0 θ T 0 ) and Lemma A.1 (ii) in the seventh equality. Then (A.11) holds.
Next, we only need to show can be decomposed as Q + P , where j )E 2 (R 2 i ) = o{σ 4 n (R W )} due to (A.13). Similarly, due to Lemma A.1 (iii), we can show that So, we have complete the proof of R W / σ 2 n (R W ). Similarly, R S / σ 2 n (R S ) is also asymptotically standard normal. Finally, we obtain the conclusion.