Optimal Rank-Based Tests for the Location Parameter of a Rotationally Symmetric Distribution on the Hypersphere

Abstract

Rotationally symmetric distributions on the unit hyperpshere are among the most commonly met in directional statistics. These distributions involve a finite-dimensional parameter ${\pmb {\theta }}$ and an infinite-dimensional parameter $g$, that play the role of “location” and “angular density” parameters, respectively. In this paper, we focus on hypothesis testing on ${\pmb {\theta }}$, under unspecified $g$. We consider (i) the problem of testing that ${\pmb {\theta }}$ is equal to some given ${\pmb {\theta }}_0$, and (ii) the problem of testing that ${\pmb {\theta }}$ belongs to some given great “circle”. Using the uniform local and asymptotic normality result from Ley et al. (Statistica Sinica 23:305–333, 2013), we define parametric tests that achieve Le Cam optimality at a target angular density $f$. To improve on the poor robustness of these parametric procedures, we then introduce a class of rank tests for these problems. Parallel to parametric tests, the proposed rank tests achieve Le Cam optimality under correctly specified angular densities. We derive the asymptotic properties of the various tests and investigate their finite-sample behavior in a Monte Carlo study.

Paindaveine—Research supported by an A.R.C. contract from the Communauté Française de Belgique and by the IAP research network grant P7/06 of the Belgian government (Belgian Science Policy).

Appendix

In this Appendix, we prove Theorems 3 and 4.

Proof of Theorem 3. (i) Recalling that $\hat{{\pmb {\theta }}}$ is an arbitrary consistent estimator of ${\pmb {\theta }}$, we have that under $\mathrm{P}^{(n)}_{{\pmb {\theta }}{,}{g}}$,

$$\begin{aligned}&{\hat{\mathcal{L}}_k(\hat{{\pmb {\theta }}})-\hat{\mathcal{L}}_k({\pmb {\theta }}) = \frac{1}{n} \sum _{i=1}^n \big \{ (\mathbf{{X}}_i^{\prime }{\pmb {\theta }})^2 - (\mathbf{{X}}_i^{\prime }\hat{{\pmb {\theta }}})^2 \big \} }\\&\qquad =\,\frac{1}{n} \sum _{i=1}^n \big \{ \mathbf{{X}}_i^{\prime }({\pmb {\theta }}-\hat{{\pmb {\theta }}}) \mathbf{{X}}_i^{\prime }({\pmb {\theta }}+\hat{{\pmb {\theta }}}) \big \} = ({\pmb {\theta }}-\hat{{\pmb {\theta }}}) \Big \{ \frac{1}{n} \sum _{i=1}^n \mathbf{{X}}_i \mathbf{{X}}_i^{\prime }\Big \} ({\pmb {\theta }}+\hat{{\pmb {\theta }}})=o_\mathrm{P}(1) \end{aligned}$$

as $n\rightarrow \infty$, so that $\hat{\mathcal{L}}_k(\hat{{\pmb {\theta }}})$ is a consistent estimator of $\mathcal{L}_k(g)=1-\mathrm{E}^{(n)}_{{\pmb {\theta }}{,}{g}}[(\mathbf{{X}}_i^{\prime }{\pmb {\theta }})^2]$. Consequently, $Q_{f_{\mathrm{exp}}; \mathrm{Stud}}^{(n)}-Q_{{\pmb {\theta }}{,}g}^{(n)}$ is $o_\mathrm{P}(1)$ as $n\rightarrow \infty$ under $\mathrm{P}^{(n)}_{{\pmb {\theta }}{,}g}$, with

$$\begin{aligned}&{Q_{{\pmb {\theta }}{,}g}^{(n)}:= \frac{n(k-1)}{ \mathcal{L}_k(g)} \, {\bar{\mathbf{{X}}}}^{\prime }(\mathbf{I}_k- {\pmb {\varUpsilon }}_{}({\pmb {\varUpsilon }}_{}^{\prime }{\pmb {\varUpsilon }}_{})^{-1} {\pmb {\varUpsilon }}_{}^{\prime }) {\bar{\mathbf{{X}}}} } \\&\qquad =\,\frac{n(k-1)}{ \mathcal{L}_k(g)} \, {\bar{\mathbf{{X}}}}^{\prime }(\mathbf{I}_k- {\pmb {\varUpsilon }}_{}({\pmb {\varUpsilon }}_{}^{\prime }{\pmb {\varUpsilon }}_{})^{-1} {\pmb {\varUpsilon }}_{}^{\prime }) (\mathbf{I}_k- {\pmb {\theta }}{\pmb {\theta }}^{\prime }) {\bar{\mathbf{{X}}}} = \mathbf {Y}^{\prime }(\mathbf{I}_k- {\pmb {\varUpsilon }}_{}({\pmb {\varUpsilon }}_{}^{\prime }{\pmb {\varUpsilon }}_{})^{-1} {\pmb {\varUpsilon }}_{}^{\prime }) \mathbf {Y}, \end{aligned}$$

where we used the fact that $\left( \mathbf{I}_k- {\pmb {\varUpsilon }}_{}({\pmb {\varUpsilon }}_{}^{\prime }{\pmb {\varUpsilon }}_{})^{-1} {\pmb {\varUpsilon }}_{}^{\prime }\right) (\mathbf{I}- {\pmb {\theta }}{\pmb {\theta }}^{\prime })= \mathbf{I}_k- {\pmb {\varUpsilon }}_{}({\pmb {\varUpsilon }}_{}^{\prime }{\pmb {\varUpsilon }}_{})^{-1} {\pmb {\varUpsilon }}_{}^{\prime }$ and where we let $\mathbf {Y}:=\sqrt{n(k-1)/\mathcal{L}_k(g)}(\mathbf{I}_k- {\pmb {\theta }}{\pmb {\theta }}^{\prime }) {\bar{\mathbf{{X}}}}$. Under $\mathrm{P}^{(n)}_{{\pmb {\theta }}{,}g}$,

$$\begin{aligned} \sqrt{n}(\mathbf{I}- {\pmb {\theta }}{\pmb {\theta }}^{\prime }) {\bar{\mathbf{{X}}}}=\frac{1}{\sqrt{n}} \sum _{i=1}^n \big \{\sqrt{1-(\mathbf{{X}}_i'{\pmb {\theta }})^2} \,{\mathbf {S}}_i({\pmb {\theta }}) \big \} \end{aligned}$$

(21)

is asymptotically normal with mean zero and covariance matrix $\mathcal{L}_k(g)(\mathbf{I}_k- {\pmb {\theta }}{\pmb {\theta }}^{\prime })/$ $(k-1)$, so that $\mathbf {Y}$, under the same, is asymptotically normal with mean zero and covariance matrix $\mathbf{I}_k- {\pmb {\theta }}{\pmb {\theta }}^{\prime }$. By using again the fact that $\left( \mathbf{I}_k- {\pmb {\varUpsilon }}_{}({\pmb {\varUpsilon }}_{}^{\prime }{\pmb {\varUpsilon }}_{})^{-1} {\pmb {\varUpsilon }}_{}^{\prime }\right) (\mathbf{I}- {\pmb {\theta }}{\pmb {\theta }}^{\prime })= \mathbf{I}_k- {\pmb {\varUpsilon }}_{}({\pmb {\varUpsilon }}_{}^{\prime }{\pmb {\varUpsilon }}_{})^{-1} {\pmb {\varUpsilon }}_{}^{\prime }$ and by noting that $\mathrm{tr}[\mathbf{I}_k- {\pmb {\varUpsilon }}_{}({\pmb {\varUpsilon }}_{}^{\prime }{\pmb {\varUpsilon }}_{})^{-1} {\pmb {\varUpsilon }}_{}^{\prime }]=k-s$, it is easy to check that Theorem 9.2.1 in Rao and Mitra (1971) provides the result.

(ii) Le Cam’s third lemma implies that under $\mathrm{P}^{(n)}_{{\pmb {\theta }}+n^{-1/2}{\pmb {\tau }}^{(n)},g}$, the random vector in (21) is asymptotically normal with mean

$$\begin{aligned} \lim _{n\rightarrow \infty } \mathrm{Cov}_{{\pmb {\theta }}{,}g}\Big [ \sqrt{n}(\mathbf{I}- {\pmb {\theta }}{\pmb {\theta }}^{\prime }) {\bar{\mathbf{{X}}}} , {\pmb {\varDelta }}_{{\pmb {\theta }}{,}g}^{(n)}\Big ] {\pmb {\tau }}^{(n)}\end{aligned}$$

(22)

and covariance matrix $\mathcal{L}_k(g)(\mathbf{I}_k- {\pmb {\theta }}{\pmb {\theta }}^{\prime })/(k-1)$. Using (3) and integrating by parts yields

$$ \mathrm{E}_{{\pmb {\theta }}{,}g}\big [ (1-(\mathbf{{X}}_1'{\pmb {\theta }})^2) \varphi _{g}(\mathbf{{X}}_1'{\pmb {\theta }}) \big ] = \int _{-1}^1 (1-t^2)\varphi _{g}(t) \tilde{g}(t)\,dt = k-1, $$

so that (22) can be rewritten as $\mathrm{E}_{{\pmb {\theta }}{,}g}\big [ (1-(\mathbf{{X}}_1'{\pmb {\theta }})^2) \varphi _{g}(\mathbf{{X}}_1'{\pmb {\theta }}) \big ] \mathrm{E}_{{\pmb {\theta }}{,}g}\big [ S_1({\pmb {\theta }})(S_1({\pmb {\theta }}))' \big ] {\pmb {\tau }}= (\mathbf{I}_k- {\pmb {\theta }}{\pmb {\theta }}^{\prime }){\pmb {\tau }}= {\pmb {\tau }}$. Therefore, $\mathbf {Y}$, under $\mathrm{P}^{(n)}_{{\pmb {\theta }}+n^{-1/2}{\pmb {\tau }}^{(n)},g}$, is asymptotically normal with mean ${\pmb \mu }:=\sqrt{(k-1)/\mathcal{L}_k(g)} \,{\pmb {\tau }}$ and covariance matrix $\mathbf{I}_k- {\pmb {\theta }}{\pmb {\theta }}^{\prime }$. From contiguity, we still have that $Q_{f_{\mathrm{exp}}; \mathrm{Stud}}^{(n)}-\mathbf {Y}^{\prime }(\mathbf{I}_k- {\pmb {\varUpsilon }}_{}({\pmb {\varUpsilon }}_{}^{\prime }{\pmb {\varUpsilon }}_{})^{-1} {\pmb {\varUpsilon }}_{}^{\prime }) \mathbf {Y}$ is $o_\mathrm{P}(1)$ under $\mathrm{P}^{(n)}_{{\pmb {\theta }}+n^{-1/2}{\pmb {\tau }}^{(n)},g}$. Theorem 9.2.1 in Rao and Mitra (1971) then shows that under this sequence of probability measures, $Q_{f_{\mathrm{exp}}; \mathrm{Stud}}^{(n)}$ is asymptotically $\chi ^2_{k-s}$ with non-centrality parameter ${\pmb \mu }'(\mathbf{I}_k- {\pmb {\varUpsilon }}_{}({\pmb {\varUpsilon }}_{}^{\prime }{\pmb {\varUpsilon }}_{})^{-1} {\pmb {\varUpsilon }}_{}^{\prime }){\pmb \mu }$, which establishes the result.

(iii) This directly follows from the asymptotic null distribution given in (i) and the classical Helly–Bray theorem.

(iv) Fix $\kappa >0$. Then, it follows from Part (i) of the proof that under $\mathrm{P}^{(n)}_{{\pmb {\theta }}{,}f_{\mathrm{exp},\kappa }}$, with ${\pmb {\theta }}\in \fancyscript{S}^{k-1} \cap \mathcal{M}({\pmb {\varUpsilon }})$, $Q_{f_{\mathrm{exp}}; \mathrm{Stud}}^{(n)}$ is asymptotically equivalent in probability to

$$\begin{aligned} Q_{{\pmb {\theta }};f_{\mathrm{exp},\kappa }}^{(n)}&= \frac{n(k-1)}{ \mathcal{L}_k(f_{\mathrm{exp},\kappa })} \, {\bar{\mathbf{{X}}}}^{\prime }(\mathbf{I}_k- {\pmb {\varUpsilon }}_{}({\pmb {\varUpsilon }}_{}^{\prime }{\pmb {\varUpsilon }}_{})^{-1} {\pmb {\varUpsilon }}_{}^{\prime }) {\bar{\mathbf{{X}}}}\\&= \frac{n\kappa ^2(k-1)}{\fancyscript{J}_k({f_{\mathrm{exp},\kappa }})} \, {\bar{\mathbf{{X}}}}^{\prime }(\mathbf{I}_k- {\pmb {\varUpsilon }}_{}({\pmb {\varUpsilon }}_{}^{\prime }{\pmb {\varUpsilon }}_{})^{-1} {\pmb {\varUpsilon }}_{}^{\prime }) {\bar{\mathbf{{X}}}} , \end{aligned}$$

which is the FvML($\kappa$)-most stringent statistic we derived in (18).   $\square$

In Theorem 3(ii), we assumed that $g\in \fancyscript{F}_\mathrm{ULAN}$ to show, through Le Cam’s third lemma, that $\sqrt{n}(\mathbf{I}- {\pmb {\theta }}{\pmb {\theta }}^{\prime }) {\bar{\mathbf{{X}}}}$, under $\mathrm{P}^{(n)}_{{\pmb {\theta }}+n^{-1/2}{\pmb {\tau }}^{(n)},g}$, is asymptotically normal with mean ${\pmb {\tau }}$ and covariance matrix $\mathcal{L}_k(g)(\mathbf{I}_k- {\pmb {\theta }}{\pmb {\theta }}^{\prime })/(k-1)$. Actually, the result still holds for $g\in \fancyscript{F}$, as it can be shown that as $n\rightarrow \infty$ under $\mathrm{P}^{(n)}_{{\pmb {\theta }}+n^{-1/2}{\pmb {\tau }}^{(n)},g}$,

$$ \sqrt{n}(\mathbf{I}- {\pmb {\theta }}{\pmb {\theta }}^{\prime }) {\bar{\mathbf{{X}}}} = {\mathbf {M}}^{(n)}+(\mathbf {I}_k+{\pmb {\theta }}{\pmb {\theta }}^{\prime }){\pmb {\tau }}+o_\mathrm{P}(1) = {\mathbf {M}}^{(n)}+{\pmb {\tau }}+o_\mathrm{P}(1), $$

where ${\mathbf {M}}^{(n)}:=\sqrt{n}(\mathbf{I}- ({\pmb {\theta }}+n^{-1/2}{\pmb {\tau }}^{(n)})({\pmb {\theta }}+n^{-1/2}{\pmb {\tau }}^{(n)})^{\prime }) {\bar{\mathbf{{X}}}}$, under the same, is clearly asymptotically normal with mean zero and covariance matrix $\mathcal{L}_k(g)(\mathbf{I}_k- {\pmb {\theta }}{\pmb {\theta }}^{\prime })/$ $(k-1)$.

Proof of Theorem 4. (i)–(ii) First note that since ${\pmb {\theta }}'{\pmb {\tau }}^{(n)}=O(n^{-1/2})$, Proposition 2(iv) rewrites

(23)

as $n\rightarrow \infty$ under $\mathrm{P}^{(n)}_{{\pmb {\theta }}{,} g}$. Since Assumption (B) holds, Lemma 4.4 in Kreiss (1987) allows to replace in (23) the deterministic quantity ${\pmb {\tau }}^{(n)}$ with the random one $\sqrt{n}(\hat{{\pmb {\theta }}}-{{\pmb {\theta }}})$, which yields

as $n\rightarrow \infty$, under $\mathrm{P}^{(n)}_{{\pmb {\theta }}{,}g}$. This, jointly with Assumption (B)(iii) (which implies that $(\mathbf{I}_k-{\pmb {\varUpsilon }}_{}({\pmb {\varUpsilon }}_{}^{\prime }{\pmb {\varUpsilon }}_{})^{-1} {\pmb {\varUpsilon }}_{}^{\prime })\hat{{\pmb {\theta }}}=0$ almost surely), entails that under $\mathrm{P}^{(n)}_{{\pmb {\theta }}{,} g}$, with ${\pmb {\theta }}\in \fancyscript{M}({\pmb {\varUpsilon }})$,

as $n\rightarrow \infty$. It follows that $\begin{array}{c}{Q}\\ \widetilde{} \end{array} _{K}^{(n)}= \begin{array}{c}{Q}\\ \widetilde{} \end{array} _{{\pmb {\theta }}{,}K}^{(n)}+o_\mathrm{P}(1)$ as $n\rightarrow \infty$ under $\mathrm{P}^{(n)}_{{\pmb {\theta }}{,} g}$, with ${\pmb {\theta }}\in \fancyscript{M}({\pmb {\varUpsilon }})$, hence also under sequences of local alternatives. The results in (i)–(ii) then follow, as in the proof of Theorem 3(i)–(ii), from Theorem 9.2.1 in Rao and Mitra (1971) and Proposition 2(ii)–(iii) (recall that $(\mathbf{I}_k- {\pmb {\varUpsilon }}_{}({\pmb {\varUpsilon }}_{}^{\prime }{\pmb {\varUpsilon }}_{})^{-1} {\pmb {\varUpsilon }}_{}^{\prime })(\mathbf{I}- {\pmb {\theta }}{\pmb {\theta }}^{\prime })=\mathbf{I}_k- {\pmb {\varUpsilon }}_{}({\pmb {\varUpsilon }}_{}^{\prime }{\pmb {\varUpsilon }}_{})^{-1} {\pmb {\varUpsilon }}_{}^{\prime }$).

(iii) As in the proof of Theorem 3(iii), this is a direct consequence of Part (i) of the result and the classical Helly–Bray theorem.

(iv) Then, under $\mathrm{P}^{(n)}_{{\pmb {\theta }}{,}f}$, with ${\pmb {\theta }}\in \fancyscript{S}^{k-1} \cap \mathcal{M}({\pmb {\varUpsilon }})$, $\begin{array}{c}{Q}\\ \widetilde{} \end{array} _{K_{f}}^{(n)}= \begin{array}{c}{Q}\\ \widetilde{} \end{array} _{{\pmb {\theta }}{,}K_{f}}^{(n)}+o_\mathrm{P}(1)$ as $n\rightarrow \infty$. Now, Proposition 2(i) entails that under the same sequence of hypotheses, $\begin{array}{c}{Q}\\ \widetilde{} \end{array} _{{\pmb {\theta }}{,}K_{f}}^{(n)}$ is asymptotically equivalent in probability to

$$ Q_{{\pmb {\theta }}{,}K_{f}}^{(n)}= \frac{k-1}{\mathcal{J}_k(K_{f})} \big ( {\pmb {\varDelta }}_{{\pmb \theta }, K_{f}}^{(n)}\big )^{\prime }\left( \mathbf{I}_k- {\pmb {\varUpsilon }}_{}({\pmb {\varUpsilon }}_{}^{\prime }{\pmb {\varUpsilon }}_{})^{-1} {\pmb {\varUpsilon }}_{}^{\prime }\right) {\pmb {\varDelta }}_{{\pmb \theta }, K_{f}}^{(n)}\nonumber , $$

which coincides with the $f$-most stringent statistic in (17). The result follows.    $\square$