Genotypes AA, Aa, and aa occur with probabilities 2 , 2(1 ), (1 ) 2 A multinomial sample of size n has frequencies (n 1 , n 2 , n 3 ) of these three genotypes a Form the log likelihood Show that (2n 1 n 2 ) (2n 1 2n 2 2n 3 ) b Show that 2 L() 2 (2n 1 n 2 ) 2 (n 2 2n 3 ) (1 ) 2 and that its expectat...

a The kernel of the log likelihood is L n 1 log 2 n 2 log 21 n 3 log 1 2 Take L 2n 1 n ...

Genotypes AA, Aa, and aa occur with probabilities [ 2 , 2(1 ), (1 )

Question:

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.

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