Under the polyploid model for two loci, show that the expected information for is J =

Under the polyploid model for two loci, show that the expected information for θ is Jθθ = c2r2(1 − θr)c−2(1 − r)c[1 − 2(1 − r)c + (1 − θr)c]

[1 − (1 − θr)c][1 − 2(1 − r)c + (1 − r)c(1 − θr)c]

.

Argue that Jθθ has a maximum as a function of r near r = 1 c+1 when

θ is near 0.

(Hint: Be careful because limθ→0 Jθθ = ∞. The singularity at θ = 0 is removable in the function ∂

∂r ln Jθθ.)

This result suggests that the value r = 1 c+1 is nearly optimal for small θ in the sense of providing the smallest standard error of the maximum likelihood estimate ˆθ of θ. In this regard note that Jθr and Jrr have finite limits as θ → 0.

Thus for small θ, the approximate standard error of ˆθ is proportional to √

1 Jθθ

even when r is jointly estimated with θ.