On optimal tests for circular reflective symmetry about an unknown central direction

Abstract

Symmetry is one of the most fundamental of dividing hypotheses, its rejection, or not, heavily influencing subsequent modeling strategies. In this paper, the authors construct tests for circular reflective symmetry about an unknown central direction that are asymptotically valid within a semi-parametric class of distributions and maintain certain parametric local and asymptotic optimality properties. The asymptotic distributions of the test statistics under the null hypothesis and under local alternatives are established, and a pre-existing omnibus test is identified as a special case of the proposed construction. The finite-sample properties of the semi-parametric tests are compared with those of other testing approaches in a simulation experiment, and recommendations made regarding testing for reflective symmetry in practice. Analyses of data on the directions of cracks in hip replacements illustrate the proposed methodology.

Appendices

Proof of Lemma 1

We show that ${\hat{{\varGamma }}}_{f_0,g_0;11}-{{\varGamma }}_{f_0,g_0;11}=o_{{\mathrm{P}}}(1)$ as $n\rightarrow \infty$ under ${\mathrm{P}}^{(n)}_{(\mu ,0)';g_0}$. Showing that ${\hat{{\varGamma }}}_{g_0,p;12}-{{\varGamma }}_{g_0,p;12}=o_{{\mathrm{P}}}(1)$ as $n\rightarrow \infty$ under ${\mathrm{P}}^{(n)}_{(\mu ,0)';g_0}$ proceeds along the same lines. In this proof, we set ${\mu }^{(n)}:= \mu +n^{-1/2} \tau _1^{(n)}$ for some bounded sequence $\tau _1^{(n)}$ as in Theorem 1. Because of the local discreteness of ${\hat{\mu }}^{(n)}$ (Assumption B), it is sufficient to show that

$$\begin{aligned} n^{-1} \sum _{i=1}^n {\dot{\varphi }}_{f_0}\left( {\varTheta }_i-{\mu }^{(n)}\right) -\mathrm{E}_{g_0}[{\dot{\varphi }}_{f_0}\left( {\varTheta }_i-{\mu }\right) ]=o_\mathrm{P}(1) \end{aligned}$$

as $n\rightarrow \infty$ under ${\mathrm{P}}^{(n)}_{(\mu ,0)';g_0}$. The law of large numbers leads to

$$\begin{aligned} n^{-1} \sum _{i=1}^n {\dot{\varphi }}_{f_0}\left( {\varTheta }_i-{\mu }\right) -\mathrm{E}_{g_0}[{\dot{\varphi }}_{f_0}\left( {\varTheta }_i-{\mu }\right) ]=o_{\mathrm{P}}(1) \end{aligned}$$

as $n\rightarrow \infty$ under ${\mathrm{P}}^{(n)}_{(\mu ,0)';g_0}$ so that it only remains to show that

$$\begin{aligned} S_n:=n^{-1} \sum _{i=1}^n \left( {\dot{\varphi }}_{f_0}\left( {\varTheta }_i-{\mu }^{(n)}\right) -{\dot{\varphi }}_{f_0}\left( {\varTheta }_i-{\mu }\right) \right) \end{aligned}$$

is $o_{{\mathrm{P}}}(1)$ as $n\rightarrow \infty$ under ${\mathrm{P}}^{(n)}_{(\mu ,0)';g_0}$. As the ${\varTheta }_i$ are i.i.d., we find

$$\begin{aligned} {{\mathrm{E}}}[| S_n|]\le & {} n^{-1} \sum _{i=1}^n {{\mathrm{E}}}[|{\dot{\varphi }}_{f_0}\left( {\varTheta }_i-{\mu }^{(n)}\right) -{\dot{\varphi }}_{f_0}\left( {\varTheta }_i-{\mu }\right) |] \\= & {} {{\mathrm{E}}}[|{\dot{\varphi }}_{f_0}\left( {\varTheta }_1-{\mu }^{(n)}\right) -{\dot{\varphi }}_{f_0}\left( {\varTheta }_1-{\mu }\right) |]. \end{aligned}$$

Since ${\dot{\varphi }}_{f_0}$ is continuous on a compact support, it is bounded. The result then follows by applying Lebesgue’s dominated convergence theorem. $\square$

Proof of Lemma 2

To prove (i), we start by showing that ${\varDelta }_{f_0,j}^{(n)\star }({{\hat{\mu }}^{(n)}})-{\varDelta }_{f_0,j}^{(n)\star }(\mu )=o_P(1)$ under ${\mathrm{P}}^{(n)}_{(\mu ,0)';g_0}$ as $n\rightarrow \infty$. First note that, due to Assumption B, we have under ${\mathrm{P}}^{(n)}_{(\mu ,0)';g_0}$ as $n\rightarrow \infty$ that

$$\begin{aligned} n^{-1/2} \sum _{i=1}^n \varphi _{f_0}({\varTheta }_i- {\hat{\mu }}^{(n)})= & {} {\varDelta }_{f_0}^{(n)}(\mu ) -{\mathrm{E}}_{g_0} [{\dot{\varphi }}_{f_0}({\varTheta }_i- {\mu })] n^{1/2} ({\hat{\mu }}^{(n)}- \mu )+ o_{{\mathrm{P}}}(1) \nonumber \\= & {} {\varDelta }_{f_0}^{(n)}(\mu )-{\varGamma }_{f_0,g_0;11} n^{1/2} ({\hat{\mu }}^{(n)}- \mu )+ o_{{\mathrm{P}}}(1). \end{aligned}$$

(12)

Therefore, using (4) with (12), it follows that

$$\begin{aligned} {\varDelta }_{f_0,j}^{(n)\star }({{\hat{\mu }}^{(n)}})= & {} {\varDelta }_{j}^{(n)}(\mu )- \eta {\varDelta }_{f_0}^{(n)}(\mu ) - ({\varGamma }_{g_0,j;12}- \eta {\varGamma }_{f_0,g_0;11}) n^{1/2} ({\hat{\mu }}^{(n)}- \mu )+ o_{{\mathrm{P}}}(1) \\= & {} {\varDelta }_{f_0,j}^{(n)\star }(\mu )+ o_{{\mathrm{P}}}(1) \end{aligned}$$

under ${\mathrm{P}}^{(n)}_{(\mu ,0)';g_0}$ as $n\rightarrow \infty$ since ${\varGamma }_{g_0,j;12}- \eta {\varGamma }_{f_0,g_0;11}=0$. It remains to show that

$$\begin{aligned} {{\hat{{\varDelta }}}}_{f_0,j}^{(n)\star }({{\hat{\mu }}^{(n)}})-{\varDelta }_{f_0,j}^{(n)\star }({{\hat{\mu }}^{(n)}})=-\left( \frac{{\hat{{\varGamma }}}_{g_0,j;12}}{{\hat{{\varGamma }}}_{f_0,g_0;11}}-\frac{{\varGamma }_{g_0,j;12}}{{\varGamma }_{f_0,g_0;11}}\right) {\varDelta }_{f_0}^{(n)}\left( {\hat{\mu }}^{(n)}\right) =o_{{\mathrm{P}}}(1)\nonumber \\ \end{aligned}$$

(13)

under ${\mathrm{P}}^{(n)}_{(\mu ,0)';g_0}$ as $n\rightarrow \infty$. To prove (13), first note that (12) and the central limit theorem (CLT) imply that ${\varDelta }_{f_0}^{(n)}\left( {\hat{\mu }}^{(n)}\right)$ is $O_{\mathrm{P}}(1)$ under ${\mathrm{P}}^{(n)}_{(\mu ,0)';g_0}$ as $n\rightarrow \infty$. Therefore, we only need to show that

$$\begin{aligned} \frac{{\hat{{\varGamma }}}_{g_0,j;12}}{{\hat{{\varGamma }}}_{f_0,g_0;11}}-\frac{{\varGamma }_{g_0,j;12}}{{\varGamma }_{f_0,g_0;11}}=o_{\mathrm{P}}(1) \end{aligned}$$

(14)

as $n\rightarrow \infty$ under ${\mathrm{P}}^{(n)}_{(\mu ,0)';g_0}$ as $n\rightarrow \infty$. Since

$$\begin{aligned} \frac{{\hat{{\varGamma }}}_{g_0,j;12}}{{\hat{{\varGamma }}}_{f_0,g_0;11}}-\frac{{\varGamma }_{g_0,j;12}}{{\varGamma }_{f_0,g_0;11}}= & {} \frac{{\hat{{\varGamma }}}_{g_0,j;12}-{\varGamma }_{g_0,j;12}}{{\hat{{\varGamma }}}_{f_0,g_0;11}}-\frac{{\varGamma }_{g_0,j;12}\left( {\hat{{\varGamma }}}_{f_0,g_0;11} - {\varGamma }_{f_0,g_0;11} \right) }{{\hat{{\varGamma }}}_{f_0,g_0;11} {\varGamma }_{f_0,g_0;11}}, \end{aligned}$$

the result follows directly from Lemma 1.

Turning to the proof of (ii), and working along the same lines as those at the end of the proof of Lemma 1, it is easily shown that

$$\begin{aligned}&n^{-1} \sum _{i=1}^n \varphi _{f_0}^2({\varTheta }_i- {\hat{\mu }}^{(n)})-\mathrm{E}_{g_0}[\varphi _{f_0}^2({\varTheta }_i- {\mu })], \nonumber \\&n^{-1} \sum _{i=1}^n \sin ^2(j({\varTheta }_i- {\hat{\mu }}^{(n)}))-\mathrm{E}_{g_0}[\sin ^2(j({\varTheta }_i- {\mu }))] \end{aligned}$$

(15)

and

$$\begin{aligned} n^{-1} \sum _{i=1}^n \sin (j({\varTheta }_i- {\hat{\mu }}^{(n)})) \varphi _{f_0}({\varTheta }_i- {\hat{\mu }}^{(n)})-{\mathrm{E}}_{g_0}[\sin (j({\varTheta }_i- {\mu })) \varphi _{f_0}({\varTheta }_i- {\mu })]\nonumber \\ \end{aligned}$$

(16)

are $o_{{\mathrm{P}}}(1)$ under ${\mathrm{P}}^{(n)}_{(\mu ,0)';g_0}$ as $n\rightarrow \infty$. Using the law of large numbers, $C_{f_0,j}({\mu })$ converges to ${\mathrm{E}}_{g_0}\left[ \left( \sin (j({\varTheta }_i-\mu )) -\frac{{\varGamma }_{g_0,j;12}}{{\varGamma }_{f_0,g_0;11}} \varphi _{f_0}({\varTheta }_i-\mu )\right) ^2\right]$, in probability, under ${\mathrm{P}}^{(n)}_{(\mu ,0)';g_0}$. From this last result, it follows that $C_{f_0,j}({\hat{\mu }}^{(n)})-C_{f_0,j}({\mu }) =o_{{\mathrm{P}}}(1)$ under ${\mathrm{P}}^{(n)}_{(\mu ,0)';g_0}$ as $n\rightarrow \infty$. Therefore it remains to show that ${\hat{C}}_{f_0,j}({\hat{\mu }}^{(n)}) -C_{f_0,j}({\hat{\mu }}^{(n)})$ is $o_{\mathrm{P}}(1)$ under $\mathrm{P}^{(n)}_{(\mu ,0)';g_0}$ as $n\rightarrow \infty$ . We readily obtain that

$$\begin{aligned}&{\hat{C}}_{f_0,j}\left( {\hat{\mu }}^{(n)}\right) -C_{f_0,j}\left( {\hat{\mu }}^{(n)}\right) = \left( \frac{{\hat{{\varGamma }}}_{g_0,j;12}^2}{{\hat{{\varGamma }}}_{f_0,g_0;11}^2}-\frac{{\varGamma }_{g_0,j;12}^2}{{\varGamma }_{f_0,g_0;11}^2}\right) n^{-1} \sum _{i=1}^n \varphi _{f_0}^2({\varTheta }_i- {\hat{\mu }}^{(n)}) \\&\quad -2 \left( \frac{{\hat{{\varGamma }}}_{g_0,j;12}}{{\hat{{\varGamma }}}_{f_0,g_0;11}}-\frac{{\varGamma }_{g_0,j;12}}{{\varGamma }_{f_0,g_0;11}}\right) n^{-1} \sum _{i=1}^n \sin (j({\varTheta }_i- {\hat{\mu }}^{(n)})) \varphi _{f_0}({\varTheta }_i- {\hat{\mu }}^{(n)}) \end{aligned}$$

so (15) and (16) together with (14) and the continuous mapping theorem imply that ${\hat{C}}_{f_0,j}\left( {\hat{\mu }}^{(n)}\right) -C_{f_0,j}\left( {\hat{\mu }}^{(n)}\right)$ is $o_{\mathrm{P}}(1)$ under ${\mathrm{P}}^{(n)}_{(\mu ,0)';g_0}$ as $n\rightarrow \infty$. The result follows.

$\square$

Proof of Theorem 2

Fix $g_0\in {\mathcal {G}}$ and $\mu \in [-\pi ,\pi )$. Lemma 2 combined with Slutsky’s lemma leads to

$$\begin{aligned} Q^{(n)}_{f_0,j}= \frac{{{\hat{{\varDelta }}}}_{f_0,j}^{(n)\star }({{\hat{\mu }}^{(n)}})}{\sqrt{{{\hat{C}}}_{f_0,j}({\hat{\mu }}^{(n)})}}=\frac{{\varDelta }_{f_0,j}^{(n)\star }( \mu )}{\sqrt{ C_{f_0,j}({\mu })}}+o_P(1)= \frac{{\varDelta }_{f_0,j}^{(n)\star }( \mu )}{V^{g_0}_{f_0}(j)}+o_P(1) \end{aligned}$$

(17)

as $n\rightarrow \infty$ under ${\mathrm{P}}^{(n)}_{(\mu ,0)';g_0}$. Part (i) then follows from the CLT.

Part (ii) is obtained via Le Cam’s third lemma. First, it is necessary to calculate the joint distribution of ${{\hat{{\varDelta }}}}_{f_0,j}^{(n)\star }({{\hat{\mu }}^{(n)}})$ and $\log (d{\mathrm{P}}^{(n)}_{(\mu ,n^{-1/2}\tau _2^{(n)})';g_0,k}/d\mathrm{P}^{(n)}_{(\mu ,0)';g_0})$ under ${\mathrm{P}}^{(n)}_{(\mu ,0)';g_0}$. We use Lemma 2 and the fact that

$$\begin{aligned}&n^{-1/2} \sum _{i=1}^n \left( \begin{array}{c} \sin (j( {\varTheta }_i- \mu )) -\eta \varphi _{f_0}({\varTheta }_i- \mu )\\ \tau _2^{(n)} \sin (k ({\varTheta }_i- \mu )) \end{array} \right) -\left( \begin{array}{c} 0\\ \frac{1}{2} (\tau _2^{(n)})^2 {\varGamma }_{g_0,k;22} \end{array} \right) \overset{{\mathcal {D}}}{\rightarrow } \\&{\mathcal {N}}_2\left( \left( \begin{array}{c} 0\\ -\frac{1}{2} (\tau _2)^2 {\varGamma }_{g_0,k;22} \end{array} \right) , \left( \begin{array}{cc} V^{g_0}_{f_0}(j)&{} \tau _2 C^{g_0}_{f_0}(j,k) \\ \tau _2 C^{g_0}_{f_0}(j,k) &{}(\tau _2)^2 {\varGamma }_{g_0,k;22} \end{array} \right) \right) \\ \end{aligned}$$

as $n\rightarrow \infty$ under ${\mathrm{P}}^{(n)}_{(\mu ,0)';g_0}$, obtained using the multivariate CLT. Now, since $\mathrm{P}^{(n)}_{(\mu ,0)';g_0}$ and $\mathrm{P}^{(n)}_{(\mu ,n^{-1/2}\tau _2^{(n)})';g_0,k}$ are mutually contiguous, applying Le Cam’s third lemma, we obtain that ${{\hat{{\varDelta }}}}_{f_0,j}^{(n)\star }({{\hat{\mu }}^{(n)}}) \overset{{\mathcal {D}}}{\rightarrow }{\mathcal {N}} (\tau _2 C^{g_0}_{f_0}(j,k), V^{g_0}_{f_0}(j))$ under $\mathrm{P}^{(n)}_{(\mu ,n^{-1/2}\tau _2^{(n)})';g_0,k}$, as $n\rightarrow \infty$.

We now turn to Part (iii). From (17), it is easily seen that

$$\begin{aligned} n^{-1/2} \sum _{i=1}^n\left[ \sin (j({\varTheta }_i-\mu )) - \frac{{\varGamma }_{f_0,j;12}}{{\varGamma }_{f_0,f_0;11}} \varphi _{f_0}({\varTheta }_i-\mu )\right] = {\varDelta }_{f_0,j}^{(n)\star }({{\hat{\mu }}^{(n)}}) + o_{{\mathrm{P}}}(1),\nonumber \\ \end{aligned}$$

(18)

as $n\rightarrow \infty$ under ${\mathrm{P}}^{(n)}_{(\mu ,0)';f_0}$, and therefore under contiguous alternatives. Now, consider the following parametric testing problem

$$\begin{aligned} \left\{ \begin{array}{l} {\mathcal {H}}_{0;f_0}:=\cup _{\mu \in [-\pi ,\pi )}\mathrm{P}^{(n)}_{(\mu ,0)^{\prime };f_0}, \\ {\mathcal {H}}_{1;f_0,j}:=\cup _{\lambda \ne 0\in [-1,1]}\cup _{\mu \in [-\pi ,\pi )}\mathrm{P}^{(n)}_{(\mu ,\lambda )^{\prime };f_0,j}, \end{array}\right. \end{aligned}$$

and consider the parametric test that rejects ${\mathcal {H}}_{0;f_0}$ when the absolute value of the parametric test statistic, that can be derived by taking as numerator the first part of the equality in (18), is greater than $z_{1-\alpha /2}$. It follows from Le Cam theory that this test is locally and asymptotically maximin for testing ${\mathcal {H}}_{0;f_0}$ against ${\mathcal {H}}_{1;f_0,j}$. The result in (iii) then follows from (18) and the optimality features of the parametric test for all $f_0\in {\mathcal {G}}$ satisfying Assumption A. $\square$