100BHWS1 - STAT 100B HW1 Solution Problem1 Linear Let = E =...

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STAT 100B HW1 Solution Problem1 Linear Let ˆ β = i w i y i E ( ˆ β ) = E ( X i w i y i ) = X i w i E ( y i ) = X i w i βx i = β < w, x > Unbiased < w, x > = 1 V ar ( ˆ β ) = V ar ( X i w i y i ) = X i w 2 i V ar ( y i ) = σ 2 X i w 2 i = σ 2 || w || 2 Best want to min || w || < w, x > = || w |||| x || cos θ = 1 ⇒ || w || = 1 || x || cos θ 1 || x || Smallest value happens when θ = 0 o . Let w = ax || w || = || ax || = a || x || = 1 || x || a = 1 || x || 2 w = ax = x || x || 2 ˆ β = < w, y > = < x, y > || x || 2 = x i y i x 2 i Problem2 ∂β n X i =1 ( y i - βx i ) 2 = n X i =1 ( y i - βx i )( - x i ) = 0 ˆ β = x i y i x 2 i Problem3 x i y i = 774 . 1159, x 2 i = 385, ˆ β = 2 . 0107 Problem4 E ( n i =1 ( y i /x i ) /n ) = 1 /n n i =1 E ( y i ) /x i = 1 /n n i =1 β = β , ˆ β = 2 . 1157 E ( n i =1 y i / n i =1 x i ) = n i =1 E ( y i ) / n i =1 x i = β n i =1 x i / n i =1 x i = β , ˆ β = 2 . 0408 Problem5 ˆ β 1 ˆ β 2 ˆ β 3 ˆ β 4 ˆ β 5 5 j =1 ( ˆ β j - 2) 2 / 5 x i y i / x 2 i 2.0107 1.9519 1.9695 2.0039 1.9423 0.0013 n i =1 ( y i /x i ) /n 2.1157 1.9368 1.9736 1.9960 1.9578 0.0040 n i =1 y i / n i =1 x i 2.0408 1.9596 1.9749
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