where ¯ a$_{p}$ = ( 1 − √ c$_{p}$ ) 2 and ¯ b$_{p}$ = ( 1 + √ c$_{p}$ ) $^{2}$. Further, let Finally, for any permutation ρ of the sequence (1 , 2 , ..., r ) and any vector z = ( z$_{1}$, ..., z$_{r}$ ), let where S = λ$_{1}$ + ... + λ$_{p}$. Proposition 2 Let the contour K that encircles λ$_{1}$, ..., λ$_{p}$ counter-clockwise be chosen so that for any z ∈ K , Re z < ( ∑ r j =1 h$_{j}$ 1+ h$_{j}$ ) − 1 S . Then where i denotes the imaginary unit, the summation in (19) is over all permutations ρ of the sequence (1 , 2 , ..., r ) , In the next section, we perform the asymptotic analysis of L ( h ; λ ) and L ( h ; µ ) that relies on the Laplace approximations of the contour integrals in (18) and (19) after the contours are suitably deformed without changing the value of the inte- grals. 14 k$_{1}$ = ( − 1) r ( r − 1) / 2 n − pr + r ( r +1) / 2 r ∏ i>j ( h$_{i}$ − h$_{j}$ ) − 1 r ∏ t =1 [ h r − p t (1 + h$_{t}$ ) p − n − $^{1}$( p − t )! ] , and k$_{2}$ = k$_{1}$ ( nS ) pr − r ( r +1) / $^{2}$Γ ( p ( n − r ) + r ( r + 1) / 2) [Γ ( np )] − 1 . L ( h ; µ ) = k$_{2}$ ∑ ρ sgn ρ (2 π i) r ∮ K ... ∮ K q$_{ρ}$ ( z ) ∏ j =1 { e − nf$_{ρ}$$_{(}$$_{j}$$_{)}$ ( z$_{j}$ $^{)}$g$_{j}$ ( z$_{j}$ ) } d z$_{r}$... d z$_{1}$, (19) L ( h ; λ ) = k$_{1}$ det 1 ≤ i,j ≤ r ( 1 2 π i ∮ K e − nf$_{i}$ ( z $^{)}$g$_{j}$ ( z ) d z ) and (18) q$_{ρ}$ ( z ) = ( 1 − r ∑ j =1 h$_{ρ}$$_{(}$$_{j}$$_{)}$ 1 + h$_{ρ}$$_{(}$$_{j}$$_{)}$ z$_{j}$ S ) − p ( n − r ) − r ( r +1) / 2 exp { − r ∑ j =1 nh$_{ρ}$$_{(}$$_{j}$$_{)}$z$_{j}$ 1 + h$_{ρ}$$_{(}$$_{j}$$_{)}$ } , (17) f$_{i}$ ( z ) = − ( h$_{i}$ 1 + h$_{i}$ z − c$_{p}$ ∫ ln ( z − λ ) d F $_{p}$( λ ) ) and (15) g$_{j}$ ( z ) = z j − $^{1}$exp {− Δ$_{p}$ ( z ) } . (16)