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

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online