Optimal inference about the tail weight in multivariate Student $t$ distributions

The multivariate Student $t$ distribution is at the core of classical statistical inference and is also a well-known model for empirical financial data. In the present paper, we propose a simple yet efficient testing procedure about its tail weight parameter $\nu$ by applying the Le Cam methodology to the multivariate Student $t$ setting. After establishing the uniform local asymptotic normality property of the multivariate location-scatter-tail weight Student $t$ model, we derive the locally and asymptotically optimal (in the maximin sense) test for tail weight under unspecified location and scatter and show that our test performs asymptotically as well as the classical approaches (likelihood ratio test, Wald test, Rao score test). In fact, our test happens to have the same form as score tests, but the Le Cam approach permits to replace the unknown location and scatter quantities by any root-$n$ consistent estimators, hence avoiding numerical maximum likelihood estimation under the null. The resulting test thus improves on the classical tests by its flexibility and simplicity; moreover, we can write out explicitly the power of our test against sequences of contiguous local alternatives. The finite-sample properties of our test are analyzed in a Monte Carlo simulation study, and we apply our method on a financial data set. We conclude the paper by indicating how to use the framework developed in the present paper for efficient point estimation.