note02 - Chapter 1 One-Sample Methods 1 Non-Parametric Test...

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Unformatted text preview: Chapter 1: One-Sample Methods 1. Non-Parametric Test of Hypothesis and CI for the Median • Binomial Test (Sign Test) – X 1 ,...,X n ∼ iid F ( x ) where F ( x ) is a continuous cdf – θ . 5 : the population median – Test H : θ . 5 = θ H a : θ . 5 > θ – Define B = n summationdisplay i =1 ψ i where ψ i = braceleftbigg 1 if X i > θ 0 otherwise – Uner H , – Test H : θ . 5 = θ H a : θ . 5 > θ ⇔ H : H a : – Notice that under H , B is a distribution-free statistic over the class of all continuous distributions that have a median equal to θ . – With large sample, by using normal approximation, Z B = B- . 5 × n √ . 25 × n > z 1- α ⇒ reject H at significant level α • Confidence Interval (Sign CI) – 1- α : the desired probability that the interval captures the median 1- α ≈ Pr ( X ( a ) < θ . 5 < X ( b ) ) – For large sample, obtain a and b by a- . 5 × n √ . 25 × n =- z 1- α/ 2 ( b- 1)- . 5 × n √ . 25 × n = z 1- α/ 2 rounding to the nearest integer. 1 2. Estimating the Population CDF and Percentiles • CI for F ( x ) – the empirical cdf: ˆ F ( x ) = 1 n n summationdisplay i =1 I { X i ≤ x } Note that E bracketleftBig ˆ F ( x ) bracketrightBig = F ( x ): unbiased estimator – Y ≡ # of obs for which X i ≤ x = n ˆ F ( x ) = ∑ n i...
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note02 - Chapter 1 One-Sample Methods 1 Non-Parametric Test...

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