butions with means and covariance matrices characterized by (27-30). To complete the proof of Theorem 1, we need to establish the tightness of ln L ( h ; λ ) and ln L ( h ; µ ), viewed as random elements of the space C [ 0 , h ] $^{r}$, as n, p → ∞ so that p/n → c. Formulae (25-26) and the facts that S − p = O$_{p}$ (1) , and that Δ$_{p}$ ( z$_{i}$$_{0}$ ) = O$_{p}$ (1) for i = 1 , ..., r, where O$_{p}$ (1) are uniform in h ∈ ( 0 , ¯ h ] $^{r}$, imply that for an arbitrarily small positive ε, there must exist B > 0 such that Pr ( sup$_{h}$$_{∈}$ ( 0 , ¯ h ] r | ln L ( h ; λ ) | > B ) < ε and Pr ( sup$_{h}$$_{∈}$ ( 0 , ¯ h ] r | ln L ( h ; µ ) | > B ) < ε for sufficiently large n and p. Since, as implied by Proposition 1, ln L ( h ; λ ) and ln L ( h ; µ ) are continuous functions on h ∈ [ 0 , h ] $^{r}$, sup$_{h}$$_{∈}$ ( 0 , ¯ h ] r | ln L ( h ; λ ) | = sup$_{h}$$_{∈}$ [ 0 , ¯ h ] r | ln L ( h ; λ ) | , and sup$_{h}$$_{∈}$ ( 0 , ¯ h ] r | ln L ( h ; µ ) | = sup$_{h}$$_{∈}$ [ 0 , ¯ h ] r | ln L ( h ; µ ) | , so that the tightness of ln L ( h ; λ ) and ln L ( h ; µ ) follows. □ To save space, we only derive the asymptotic power envelope for the relatively more difficult case of real-valued data and µ -based tests. According to the Neyman- Pearson lemma, the most powerful test of the null h = 0 against a point alternative h = ( h$_{1}$, ..., h$_{r}$ ) is the test which rejects the null when ln L ( h ; µ ) is larger than a critical value C. It follows from Theorem 1 that, for such a test to have asymptotic size α , C must be where Now, according to Le Cam’s third lemma and Theorem 1, under h = ( h$_{1}$, ..., h$_{r}$ ) , ln L ( h ; µ ) d → N ( m ( h ) + W ( h ) , W ( h )) . Therefore, the asymptotic power β$_{µ}$ ( h ) m ( h ) = 1 2 r ∑ i,j =1 ( ln ( 1 − h$_{i}$h$_{j}$ c ) + h$_{i}$h$_{j}$ c ) and W ( h ) = − r ∑ i,j =1 ( ln ( 1 − h$_{i}$h$_{j}$ c ) + h$_{i}$h$_{j}$ c ) . C = √ W ( h )Φ − $^{1}$(1 − α ) + m ( h ) , (63) Proof of theorem 2 37