Genotypes AA, Aa, and as occur with probabilities [θ 2 ,2θ (1 — θ), (1 - θ) 2 ]. A multinomial sample of size n has frequencies (n 1 , n 2 , n 3 ) of these three genotypes. a. Form the log likelihood....

Genotypes AA, Aa, and as occur with probabilities [θ²,2θ (1 — θ), (1 - θ)²]. A multinomial sample of size n has frequencies (n₁, n₂, n₃) of these three genotypes.

a. Form the log likelihood. Show that θ = (2n₁ + n₂)/(n₁ + n₂ + 2n₃).

b. Show that —??²L(θ)/ ?? θ² = [(2n₁ + n₂)/θ²] + [n₂ + 2n₃)/ (1 - 0)²] and that its expectation is 2n/ θ (1 — θ). Use this to obtain an asymptotic standard error of θ.

c. Explain how to test whether the probabilities truly have this pattern.