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Dr. Hackney STA Solutions pg 124

Dr. Hackney STA Solutions pg 124 - Second Edition 8-3...

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Unformatted text preview: Second Edition 8-3 Differentiation will show that in the numerator ˆ θ = ( ∑ i x i + ∑ j y j ) / ( n + m ), while in the denominator ˆ θ = ¯ x and ˆ μ = ¯ y . Therefore, λ ( x , y ) = n + m ∑ i x i + ∑ j y j n + m exp- n + m ∑ i x i + ∑ j y j ∑ i x i + ∑ j y j n ∑ i x i n exp- n ∑ i x i ∑ i x i m ∑ j y j m exp- m ∑ j y j ∑ j y j = ( n + m ) n + m n n m m ( ∑ i x i ) n ∑ j y j m ∑ i x i + ∑ j y j n + m . And the LRT is to reject H if λ ( x , y ) ≤ c . b. λ = ( n + m ) n + m n n m m ∑ i x i ∑ i x i + ∑ j y j ! n ∑ j y j ∑ i x i + ∑ j y j ! m = ( n + m ) n + m n n m m T n (1- T ) m . Therefore λ is a function of T . λ is a unimodal function of T which is maximized when T = n m + n . Rejection for λ ≤ c is equivalent to rejection for T ≤ a or T ≥ b , where a and b are constants that satisfy a n (1- a ) m = b n (1- b ) m . c. When H is true, ∑ i X i ∼ gamma( n,θ ) and ∑ j Y j ∼ gamma( m,θ ) and they are indepen-...
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