Show that, in the case of the binomial distribution with index m and parameter , the Legendre

Show that, in the case of the binomial distribution with index m and parameter π, the Legendre transformation is y log (

y

μ ) + (m − y)log (

m−y m−μ )

where μ = mπ. Hence show that the saddlepoint approximation is

ρ

2 13 (θˆ) = ρ

∗2 13

(x) ρ

2 23 (θˆ) = ρ

∗2 23

(x)

ρ4 (ˆθ) = −ρ

∗

4 (x) + ρ

∗2 13

(x) + 2ρ

∗2 23

(x).

n

−1/2Y r = Z r − κ

r,s,tκs,iκt,jZ iZ j/6

+ {8κ

r,s,tκ

u,v,wκs,iκt,uκv,jκw,k − 3κ

r,s,t,uκs,iκt,jκu,k}Z iZ jZ k/72

π

y(1−π)

m−ymm+1/2

(2π)

1/2 y

y+1/2(m−y)

m−y+1/2

.

In what circumstances is the saddlepoint approximation accurate? Derive the above as a double saddlepoint approximation to the conditional distribution of Y1 given Y1+Y2 = m, where the Ys are independent Poisson random variables.