Dr. Hackney STA Solutions pg 145

Dr. Hackney STA Solutions pg 145 - Second Edition 9-3 The...

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Unformatted text preview: Second Edition 9-3 The sign of the derivative is given by the expression in square brackets, a parabola. It is easy to see that for λ ≥ 0, the parabola changes sign from positive to negative. Since this is the sign change of the derivative, the function must increase then decrease. Hence, the function is an upside-down bowl, and the set is an interval. 9.5 a. Analogous to Example 9.2.5, the test here will reject H if T < k ( p ). Thus the confidence set is C = { p : T ≥ k ( p ) } . Since k ( p ) is nondecreasing, this gives an upper bound on p . b. k ( p ) is the integer that simultaneously satisfies n X y = k ( p ) n y p y (1- p ) n- y ≥ 1- α and n X y = k ( p )+1 n y p y (1- p ) n- y < 1- α. 9.6 a. For Y = ∑ X i ∼ binomial( n,p ), the LRT statistic is λ ( y ) = ( n y ) p y (1- p ) n- y ( n y ) ˆ p y (1- ˆ p ) n- y = p (1- ˆ p ) ˆ p (1- p ) y 1- p 1- ˆ p n where ˆ p = y/n is the MLE of p . The acceptance region is A ( p ) = y : p ˆ p y 1- p 1- ˆ p n- y ≥ k * where...
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This note was uploaded on 02/03/2012 for the course STA 1014 taught by Professor Dr.hackney during the Spring '12 term at UNF.

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