Detecting the direction of a signal on high-dimensional spheres: Non-null and Le Cam optimality results

We consider one of the most important problems in directional statistics, namely the problem of testing the null hypothesis that the spike direction $\theta$ of a Fisher-von Mises-Langevin distribution on the $p$-dimensional unit hypersphere is equal to a given direction $\theta_0$. After a reduction through invariance arguments, we derive local asymptotic normality (LAN) results in a general high-dimensional framework where the dimension $p_n$ goes to infinity at an arbitrary rate with the sample size $n$, and where the concentration $\kappa_n$ behaves in a completely free way with $n$, which offers a spectrum of problems ranging from arbitrarily easy to arbitrarily challenging ones. We identify various asymptotic regimes, depending on the convergence/divergence properties of $(\kappa_n)$, that yield different contiguity rates and different limiting experiments. In each regime, we derive Le Cam optimal tests under specified $\kappa_n$ and we compute, from the Le Cam third lemma, asymptotic powers of the classical Watson test under contiguous alternatives. We further establish LAN results with respect to both spike direction and concentration, which allows us to discuss optimality also under unspecified $\kappa_n$. To investigate the non-null behavior of the Watson test outside the parametric framework above, we derive its local asymptotic powers through martingale CLTs in the broader, semiparametric, model of rotationally symmetric distributions. A Monte Carlo study shows that the finite-sample behaviors of the various tests remarkably agree with our asymptotic results.Comment: 47 pages, 4 figure

Disability and satisfaction after Rotator Cuff decompression or repair: a sex and gender analysis

<p>Abstract</p> <p>Background</p> <p>Rotator-cuff pathology is the most common cause of pain and disability in the shoulder. Examining the combined effect of biological and societal factors on disability would potentially identify existing differences between men and women with rotator cuff pathology which would help to provide suggestions for better models of care. Purpose of this study was to determine the overall differences in disability between men and women and to examine the relationship between factors that represent sex (biological factors) and gender (non-biological factors) with disability and satisfaction with surgical outcome 6 months after rotator cuff surgery.</p> <p>Methods</p> <p>Patients with impingement syndrome and/or rotator cuff tear who underwent rotator cuff surgery completed the Western Ontario Rotator Cuff (WORC) index, the American Shoulder & Elbow Surgeons (ASES) assessment form, and the Quick Disabilities of the Arm, Shoulder and Hand (QuickDASH) outcome measures prior to surgery and 6 months post-operatively. They also rated their satisfaction with surgery at their follow-up appointment.</p> <p>Results and Discussion</p> <p>One hundred and seventy patients entered into the study (85 men and 85 women). One hundred and sixty patients (94%) completed the 6-month assessment. Women reported more disability both prior to and after surgery. Disability at 6 months was associated with pain-limited range of motion, participation limitation, age and strength. Satisfaction with surgery was associated with level of reported disability, expectations for improved pain, pain-limited range of motion and strength.</p> <p>Conclusions</p> <p>The results of this study indicate that women with rotator cuff pathology suffer from higher levels of pre- and post-operative disability and sex and gender qualities contribute to these differences. Gender-sensitive approach will help to identify existing differences between men and women which will help to promote more effective and tailored care by health professionals.</p