On completeness of the general linear model with spherically symmetric errors

We consider the canonical form of the general linear model, with spherically symmetric errors, which may be viewed as a random vector in RnRn partitioned as (XtUt)t with a spherically symmetric density σ−ng({‖x−θ‖2+‖u‖2}σ−2) around a mean vector, partitioned as (θt0t)t, where dimX=dimθ=p and dimU=dim0=k with p+k=np+k=n. When the location parameter θθ and the scale parameter σσ are unknown and the generating function g(⋅)g(⋅) is known, we show that the statistic (X,‖U‖2)(X,‖U‖2) is minimal sufficient and we investigate whether it is a complete statistic or not. In particular, when g(t)g(t) has support contained in a compact interval not containing zero, we show non-completeness of the minimal sufficient statistic. Of course if the distribution is normal, well known results for exponential families imply its completeness. We also show that (X,‖U‖2)(X,‖U‖2) is complete for the generalized multivariate tt distribution.