If the distribution of the vector Z is spherical (that is depends on r only) , then the distribution of φ is uniform and independent of r ; it follows that r sin 2 φ has then the same distribution as r sin φ. Indeed, conditionally on r (denoting by F$_{·}$$_{|}$$_{·}$ ( · ) the conditional distribution) Then the limit of W$_{T}$ is given by the distribution of 1 $_{4}$Z 2 1 (the same as 1 $_{4}$Z 2 $_{2}$) . Under normality this is distributed as 1 $_{4}$χ 2 $_{1}$. If the distribution of Z is such that each marginal is normal but the joint is not, then 1 $_{4}$χ 2 1 provides an upper bound but not necessarily the distribution. The limit 1 $_{4}$χ 2 1 distribution under normality was obtained by Glonek (1993) who also demonstrated that this asymptotic distribution does not depend on the covariance matrix V. Thus the limit distribution for test of this hypoth- esis for a normal Z is either χ 2 1 or 1 $_{4}$χ 2 $_{1}$, therefore is not pivotal. However, χ 2 1 provides a conservative bound, so that there is a pivotal upper bound. In the above examples, standard critical values are conservative in non- regular cases. So here if we do not know whether we are in a regular case or not, usual critical values are the appropriate ones: the test never over-rejects (asymptotically) under the null hypothesis when using critical values entailed by usual regularity assumptions. However, it is also possible that the standard limit distribution does not 11 F$_{sin 2}$$_{φ}$$_{|}$$_{r}$ ( α r ) = F$_{2}$$_{φ}$$_{|}$$_{r}$ ( arcsin α r ) = 2 F$_{φ}$$_{|}$$_{r}$ ( 1 2 arcsin α r ) = 2 ∫ 1 2 arcsin( α/r ) 0 I (0 ≤ φ ≤ 2 π ) 1 2 π dφ = F$_{φ}$$_{|}$$_{r}$ ( arcsin α r ) .