  This paper develops a smooth test of goodness-of-fit for elliptical
distributions. The test is adaptively omnibus, invariant to affine-linear
transformations and has a convenient expression that can be broken into
components. These components have diagnostic capabilities and can be used to
identify specific departures. This helps in correcting the null model when the
test rejects. As an example, the results are applied to the multivariate normal
distribution for which the R package ECGofTestDx is available. It is shown that
the proposed test strategy encompasses and generalizes a number of existing
approaches. Some other cases are studied, such as the bivariate Laplace,
logistic and Pearson type II distribution. A simulation experiment shows the
usefulness of the diagnostic tools.
