as095 - STAT 854/454 Sampling Theory and Practice...

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Unformatted text preview: STAT 854/454 Sampling Theory and Practice Assignment 5 (For Practice Only, Do Not Hand-In!) 1. Suppose the finite population follows the model ( ξ ): y i = βx i + ε i , i = 1 , 2 , ··· ,N , where x i > 0 and the ε i ’s are independent random variables with E ξ ( ε i ) = 0 and V ξ ( ε i ) = x i σ 2 . The finite population mean μ x for the auxiliary variable x is known. Let { ( y i ,x i ) ,i ∈ s } be a random sample obtained using a probability sampling design. (a) Show that the weighted least square estimator of β under the model ξ is given by ˆ β = ¯ y/ ¯ x . (b) Show that ˆ μ y = ˆ βμ x is a model-unbiased prediction estimator for the finite population mean μ y . (c) Find the model-based variance V ξ (ˆ μ y- μ y ). (d) Note that μ y = N- 1 ( n ¯ y + ∑ i/ ∈ s y i ), another prediction estimator of μ y is given by ˆ μ * y = N- 1 ( n ¯ y + ˆ β ∑ i/ ∈ s x i ), where for i / ∈ s , y i is predicted as ˆ βx i . Show that ˆ μ * y is also model-unbiased for...
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This note was uploaded on 11/23/2010 for the course STAT 454 taught by Professor Da during the Fall '09 term at Waterloo.

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