Let X ∼ Binomial(n, p1) and Y ∼ Binomial(m, p2) and suppose X and Y are independent. The hypotheses to be tested are: H0 : p1 = p2 HA : p1 < p2 or p1>p2

Annie French 2022-11-20 Answered
Generalized likelihood ratio statistic for two binomial distributions
This question develops hypothesis tests for the difference between two population proportions.Let X ∼ Binomial(n, p1) and Y ∼ Binomial(m, p2) and suppose X and Y are independent. The hypotheses to be tested are: H 0 : p 1 = p 2 , H A : p 1 < p 2   o r   p 1 > p 2
(a) Find the generalized likelihood ratio statistic Λ for testing H0 vs. HA based on the data X and Y.
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Answers (1)

Stella Andrade
Answered 2022-11-21 Author has 19 answers
Instead of
L 1 ( s 2 ( s 2 + a 2 ) 2 ) ( t ) = 0 t sin t cos ( a t a τ ) d τ
you should have
L 1 ( s 2 ( s 2 + a 2 ) 2 ) ( t ) = 1 a 0 t sin ( a τ ) cos ( a t a τ ) d τ
because the convolution of functions f,g is defined as
( f g ) ( t ) = R g ( τ ) f ( t τ ) d τ
where f ( t ) := cos ( a t ) 1 [ 0 , ) ( t ), g ( t ) := 1 a sin ( a t ) 1 [ 0 , ) ( t )
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