illustrate both the fact that the distribution depends on ¯ θ, and also that despite the distribution not being pivotal, the usual χ 2 1 distribution provides here a pivotal upper bound. Example 3.2. Consider the restriction given by g ( θ ) = θ$_{1}$θ$_{2}$. In this case, G ( θ ) = [ θ$_{2}$, θ$_{1}$ ] , and the Wald statistic for testing H$_{0}$ : θ$_{1}$θ$_{2}$ = 0 takes the form: If either θ$_{1}$ or θ$_{2}$ is non-zero, under H$_{0}$ the limiting distribution is χ 2 $_{1}$. If, however, θ$_{1}$ = θ$_{2}$ = 0 , we have: Writing this expression as Z 2 − Z 4 2 Z 2 $_{1}$+ Z 2 we see that the limit distribution in this case under Assumption 2.3a is strictly below χ 2 $_{1}$, thus it is not pivotal. However, χ 2 1 provides a conservative bound. A more precise bound can be obtained. Write the vector ( Z$_{1}$, Z$_{2}$ ) in polar coordinates: ( r sin φ, r cos φ ) , with r 2 = Z 2 1 + Z 2 $_{2}$, r ≥ 0 and φ = arcsin Z$_{1}$ r . Then the limit ratio in (8) becomes Thus the distribution of 1 $_{4}$r 2 provides an upper bound on the limit distribution of W$_{T}$ under the most general assumptions. 10 1 4 r $^{2}$(sin 2 φ ) 2 . W$_{T}$ p −→ T →∞ Z 2 $_{1}$Z 2 Z 2 1 + Z 2 . (8) W$_{T}$ = T ˆ θ 2 $_{1}$ˆ θ 2 2 ˆ θ 2 1 + ˆ θ 2 2 .