Multivariate saddlepoint tests on the mean direction of the von Mises–Fisher distribution

Abstract

This article provides P values for two new tests on the mean direction of the von Mises–Fisher distribution. The test statistics are obtained from the exponent of the saddlepoint approximation to the density of M-estimators, as suggested by Robinson et al. (Ann Stat 31:1154–1169, 2003). These test statistics are chi-square distributed with asymptotically small relative errors. Despite the high dimensionality of the problem, the proposed P values are accurate and simple to compute. The numerical precision of the P values of the new tests is illustrated by some simulation studies.

Appendix

This appendix provides the detailed proofs of Results 2.1 and 2.2.

Proof

of Result 2.1 We need the specific form of the c.g.f. (8) for the score $\varvec{\psi }$ given in (10). For $\varvec{v} \in \mathbb {R}^p$, let $M(\varvec{v};\varvec{\zeta },\varvec{\zeta }_{0}) = \mathsf{E}[ \exp \{ \langle \varvec{v} , \varvec{\psi }(\varvec{X},\varvec{\eta }(\varvec{\zeta }))\rangle \}]$, then

$$\begin{aligned} M(\varvec{v};\varvec{\zeta },\varvec{\zeta }_{0})&= \int _{\mathbb {S}^{p-1}} \exp \{ \langle \varvec{v} , \varvec{\psi }(\varvec{x},\varvec{\eta }(\varvec{\zeta })) \rangle \} f_p(\varvec{x} | \varvec{\mu }_0, \kappa _0) \mathrm{d}U_p (\varvec{x}) \nonumber \\&= c_p(\kappa _0) \int _{\mathbb {S}^{p-1}} \exp \{ \langle \varvec{v} , \varvec{x} - \varvec{\eta }(\varvec{\zeta })\rangle + \langle \varvec{\zeta }_0 , \varvec{x} \rangle \} \mathrm{d}U_p (\varvec{x}) \nonumber \\&= \frac{c_p(||\varvec{\zeta }_0||)}{c_p (|| \varvec{\zeta }_0 + \varvec{v} ||)} \exp \{ - \langle \varvec{\eta } (\varvec{\zeta }), \varvec{v} \rangle \} \nonumber \\&= \left( \frac{|| \varvec{\zeta }_0 ||}{|| \varvec{\zeta }_0 + \varvec{v} ||} \right) ^{\frac{p}{2}-1} \frac{I_{\frac{p}{2}-1} (|| \varvec{\zeta }_0 + \varvec{v} ||) }{I_{\frac{p}{2}-1} (|| \varvec{\zeta }_0 ||) } \exp \left\{ - \frac{A_p(|| \varvec{\zeta } ||)}{|| \varvec{\zeta } ||} \langle \varvec{\zeta }, \varvec{v} \rangle \right\} , \end{aligned}$$

(16)

from (1) and (4), provided $|| \varvec{\zeta }_0 + \varvec{v} || \ne 0$. The last equality uses (3) in order to obtain $\varvec{\eta } (\varvec{\zeta }) = A_p(|| \varvec{\zeta } ||)/ || \varvec{\zeta } || \, \varvec{\zeta }$. The desired c.g.f. is $K = \log M$.

From the penultimate line of (16) we compute

$$\begin{aligned} \frac{\partial }{\partial \varvec{v}} K(\varvec{v};\varvec{\zeta },\varvec{\zeta }_{0})&= - \frac{c_p'(||\varvec{\zeta }_0 + \varvec{v} ||)}{c_p (|| \varvec{\zeta }_0 + \varvec{v} ||)} \frac{\partial }{\partial \varvec{v}} ||\varvec{\zeta }_0 + \varvec{v} || - \frac{A_p(|| \varvec{\zeta } ||)}{|| \varvec{\zeta } ||} \varvec{\zeta } \\&= A_p(|| \varvec{\zeta }_0 + \varvec{v} ||) \frac{ \varvec{\zeta }_0 + \varvec{v} }{|| \varvec{\zeta }_0 + \varvec{v} ||} - A_p(|| \varvec{\zeta } ||) \frac{ \varvec{\zeta } }{|| \varvec{\zeta } ||}, \end{aligned}$$

where the second equality follows from the re-expression of (7) as

$$\begin{aligned} A_p(u)&= - \frac{c_p'(u)}{c_p(u)}, \; \forall u > 0. \end{aligned}$$

(17)

By equating this gradient to zero and by solving w.r.t. $\varvec{v}$ one finds the saddlepoint $\varvec{v}_0 = \varvec{\zeta } - \varvec{\zeta }_0$.

According to (9), define $h_{\zeta _{0}} ( \varvec{\zeta } ) = \text{ sup }_{v \in \mathbb {R}^p} \{ - K(\varvec{v} ; \varvec{\zeta },\varvec{\zeta }_{0}) \}$. It follows from strict convexity that $h_{\zeta _{0}} ( \varvec{\zeta } ) = - K(\varvec{v}_0 ; \varvec{\zeta },\varvec{\zeta }_{0}) = - K(\varvec{\zeta } - \varvec{\zeta }_0 ; \varvec{\zeta },\varvec{\zeta }_{0})$, which yields (11).

The validity of the chi-square asymptotic approximation with relative error $\mathrm{O}(n^{-1})$ uniformly over the normal deviations region follows from Theorem 1 of Robinson et al. (2003). $\square$

Proof

of Result 2.2 For the score $\varvec{\psi }$ given in (14) and $\varvec{v} \in \mathbb {R}^p$, we define $M(\varvec{v};\varvec{\mu },\varvec{\mu }_{0},\kappa _0) = \mathsf{E}[ \exp \{ \langle \varvec{v} , \varvec{\psi }(\varvec{X},\varvec{\mu }) \rangle \}]$ and obtain

$$\begin{aligned} M(\varvec{v};\varvec{\mu },\varvec{\mu }_{0},\kappa _0)&= \int _{\mathbb {S}^{p-1}} \exp \{ \langle \varvec{v} , \varvec{\psi }(\varvec{x},\varvec{\mu }) \rangle \} f_p(\varvec{x} | \varvec{\mu }_0, \kappa _0) \mathrm{d}U_p (\varvec{x}) \nonumber \\&= c_p(\kappa _0) \int _{\mathbb {S}^{p-1}} \exp \{ \langle \kappa _0 \varvec{\mu }_0 + P_\mu \varvec{v} , \varvec{x} \rangle \} \mathrm{d}U_p (\varvec{x}) \nonumber \\&= \frac{c_p(\kappa _0)}{c_p (|| \kappa _0 \varvec{\mu }_0 + P_\mu \varvec{v} ||)} \nonumber \\&= \left( \frac{\kappa _0}{|| \kappa _0 \varvec{\mu }_0 + P_\mu \varvec{v} ||} \right) ^{\frac{p}{2}-1} \frac{I_{\frac{p}{2}-1} (|| \kappa _0 \varvec{\mu }_0 + P_\mu \varvec{v} ||) }{I_{\frac{p}{2}-1} (\kappa _0) }, \end{aligned}$$

(18)

from (1), provided $|| \kappa _0 \varvec{\mu }_0 + P_\mu \varvec{v} || \ne 0$. The desired c.g.f. is $K = \log M$.

From the penultimate line of (18) we obtain

$$\begin{aligned} \frac{\partial }{\partial \varvec{v}} K(\varvec{v};\varvec{\mu },\varvec{\mu }_{0},\kappa _0)&= - \frac{c_p'(||\kappa _0 \varvec{\mu }_0 + P_\mu \varvec{v} ||)}{c_p (|| \kappa _0 \varvec{\mu }_0 + P_\mu \varvec{v} ||)} P_\mu \frac{\kappa _0 \varvec{\mu }_0 + P_\mu \varvec{v}}{||\kappa _0 \varvec{\mu }_0 + P_\mu \varvec{v}||} \\&= A_p(|| \kappa _0 \varvec{\mu }_0 + P_\mu \varvec{v} ||) P_\mu \frac{ \kappa _0 \varvec{\mu }_0 + P_\mu \varvec{v}}{|| \kappa _0 \varvec{\mu }_0 + P_\mu \varvec{v}||} , \end{aligned}$$

where the second equality follows from (17). By equating this gradient to zero and by solving w.r.t. $\varvec{v}$ one finds $\varvec{v}_0 = - \kappa _0 P_\mu \varvec{\mu }_0$ as possible solution. By noting that $K(\varvec{v};\varvec{\mu },\varvec{\mu }_{0},\kappa _0)$ depends on $\varvec{v}$ only through $P_\mu \varvec{v}$, we deduce that $\varvec{v}_0$ is the unique solution over the $(p-1)$-dimensional hyperplane tangent to $\mathbb {S}^{p-1}$ at $\varvec{\mu }$.

Following (9), define $h_{\mu _{0},\kappa _0} ( \varvec{\mu } ) = \text{ sup }_{v \in \mathbb {R}^p} \{ - K(\varvec{v} ; \varvec{\mu },\varvec{\mu }_{0},\kappa _0) \}$. Strict convexity implies $h_{\mu _{0},\kappa _0} ( \varvec{\mu } ) = - K(\varvec{v}_0 ; \varvec{\mu },\varvec{\mu }_{0},\kappa _0) = - K( - \kappa _0 P_\mu \varvec{\mu }_0; \varvec{\mu },\varvec{\mu }_{0},\kappa _0)$, which yields (15).

The validity of the chi-square approximation with relative error $\mathrm{O}(n^{-1})$ uniformly over the normal deviations region follows from Theorem 1 of Robinson et al. (2003). $\square$