Genotypes AA, Aa, and aa occur with probabilities [ 2 , 2(1 ), (1 ) 2 ]. A multinomial sample of size n

Genotypes AA, Aa, and aa 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 θ̂ = (2n₁ + n₂)/(2n₁ + 2n₂ + 2n₃).

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

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