  We propose a new testing procedure about the tail weight parameter of
multivariate Student $t$ distributions by having recourse to the Le Cam
methodology. Our test is asymptotically as efficient as the classical
likelihood ratio test, but outperforms the latter by its flexibility and
simplicity: indeed, our approach allows to estimate the location and scatter
nuisance parameters by any root-$n$ consistent estimators, hereby avoiding
numerically complex maximum likelihood estimation. 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
this framework for efficient point estimation.
