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Unformatted text preview: Section Notes for Week 11 In this section, we are going to discuss ”Pivot Method” for confidence intervals, which is one of the most important methods to find confidence intervals, and is convenient to implement. Definition : If X 1 , ··· ,X n ∼ f ( x,θ ), and θ is unknown. A pivot is a function of the data X = ( X 1 , ··· ,X n ) and of θ whose distribution does not depend on θ . Note: A pivot is not a statistics, but a statistics is a pivot. Exp1: If X 1 , ··· ,X n ∼ N ( θ, 1) and are i.i.d, and θ is unknown: ¯ X ∼ N ( θ, 1 /n ), and ¯ X- θ ∼ N (0 , 1 /n ). Thus, ¯ X- θ is a pivot. There are two important types of pivots: t-pivot and F-pivot: Definition :A t-pivot has the following form: T ( x )- q ( θ ) S ( x ) , where S ( x ) > Recall how we introduced t distribution to get some idea of why this is called ”t-type”. Definition :A F-pivot has the following form: T ( x ) q ( θ ) , where T ( x ) > ,q ( θ ) > ....
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- Summer '08
- Normal Distribution, Student's t-distribution