420Hw02ans - STAT 420 Homework #2 Fall 2007 1. Sometimes it...

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STAT 420 Fall 2007 Homework #2 1. Sometimes it is known in advance that the least-squares regression line must go through the origin, i.e., the regression model is of the form Y i = β x i + ε i , i = 1, 2, … , n , where ε i ’s are i.i.d. N ( 0, σ 2 ), and the equation of the regression line is y ˆ = ˆ x . In this case, finding the least-squares line reduces to finding the value ˆ that minimizes the expression ( ) [ ] = - = n i i i x y f 1 2 . Use the derivative of f with respect to β to derive the formula for the slope of the least-squares regression line in this case. We want to minimize ( ) [ ] = - = n i i i x y b f 1 2 . ( ) [ ] ( ) = = = - = - - = n i i i n i i n i i i i y x x x x y b f 1 1 2 1 2 2 2 ' . To find the extremum points: ( ) 0 ' = f . Therefore, = = = n i i n i i i x y x 1 2 1 ˆ . ( ) = = n i i x f 1 2 2 " > 0. ± ( ) f has minimum at ˆ . OR ( ) [ ] [ ] = = + - = - = n i i i i i n i i i x y x y x y b f 1 2 2 2 1 2 2 = 2 2 1 2 1 1 2 = = ² ² ³ ´ µ µ · + ² ² ³ ´ µ µ · - ² ² ³ ´ µ µ · = n i i x n i i y i x y n i i – parabola. The vertex (minimum) is at = = = = - n i i n i i i x y x a b 1 2 1 2 ˆ .
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2. It has been proposed that the brightness measured in some unit of color or a commercial product is proportional to the time it is in a certain chemical reaction during the production process, or Y i = β x i + ε i , i = 1, 2, … , n , where ε i ’s are i.i.d. N ( 0, σ 2 ), where Y i measures brightness, x i measures time, and β is a parameter. The following data on x and Y are available:
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This note was uploaded on 04/29/2010 for the course STAT stat 420 taught by Professor Stepanov during the Spring '07 term at University of Illinois at Urbana–Champaign.

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420Hw02ans - STAT 420 Homework #2 Fall 2007 1. Sometimes it...

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