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Unformatted text preview: Estimating Estimable Functions of We now shift attention from E ( y ) to the parameter vector . Remember our ttest questions about 1 2 and y 1 y 2 ? Those are questions about or linear combinations of . Weve seen some models where there is a unique solution for so the elements of can be interpreted. Weve seen other models where there are an infinite number of solutions for . There are other models where some components of have unique solutions and other components have an infinite number of solutions. How can we tell when a component of is unique? How can we tell when a linear combination of is unique? What can we say about the statistical properties of solutions for ? Copyright c 2011 Dept. of Statistics (Iowa State University) Statistics 511 1 / 19 The Response Depends on Only through X In the GaussMarkov or Normal Theory GaussMarkov Linear Model, the distribution of y depends on only through X , i.e., y ( X , 2 I ) or y N ( X , 2 I ) If X is not of full column rank, there are infinitely many vectors in the set { b : Xb = X } for any fixed value of . Thus, no matter what the value of E ( y ) , there will be infinitely many vectors b such that Xb = E ( y ) when X is not of full column rank. The response vector y can help us learn about E ( y ) = X , but when X is not of full column rank, there is no hope of learning about alone unless additional information about is available. Copyright c 2011 Dept. of Statistics (Iowa State University) Statistics 511 2 / 19 Treatment Effects Model Remember the dairy cow experiment: six experimental units randomly assigned to two treatments. One possible model for the response is called an effects model . y ij = + i + ij , i = 1 , 2 ; j = 1 , 2 , 3 y 11 y 12 y 13 y 21 y 22 y 23 = + 1 + 1 + 1 + 2 + 2 + 2 + 11 12 13 21 22 23 y 11 y 12 y 13 y 21 y 22 y 23 = 1 1 1 1 1 1 1 1 1 1 1 1 1 2 + 11 12 13 21 22 23 Copyright c 2011 Dept. of Statistics (Iowa State University) Statistics 511 3 / 19 Treatment Effects Model (continued) In this case, it makes no sense to estimate = [ , 1 , 2 ] because there are multiple (infinitely many, in fact) choices of that define the same mean for y ....
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This note was uploaded on 02/11/2012 for the course STAT 511 taught by Professor Staff during the Spring '08 term at Iowa State.
 Spring '08
 Staff

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