Math 558 Lecture #10
Estimation
Proposition: Orthogonal projection of the response vector Y on the treatment subspace is an unbiased estimator of τ, the vector of treatment effects.
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Proof: The vector Y ∈ V. Project the vector Y onto the treatment space VT. The orthogonal projection of Y on VT is PVT Y.
E(PVT Y) = PVT E(Y) = PVT τ = τ
Estimation
We also know that the orthogonal projection of Y minimizes the sum of squares ||Y − PvT Y||2
Estimation
Furthermore, if {u1,u2,…..ut} is an orthogonal basis for VT, we have t Y.ui
PVT Y = u .u u ∑i
Y.ui =y1ui1+y2ui2+….+yNuiN =SUMT=i
This is the sum of observation for treatment i on all the plots treatment i appears.
ui.ui = ri. Therefore
t SUMT =i PVT Y = r u
Estimation
From table in lectures 7 and 8
y.uAuA = y6 +y8 +y9 +y11uA
Therefore, or
y.uBuB = y4 +y5 +y7 +y10uB
y.uAuC = y1 +y2 +y3 +y12uB
PVTY = y.uAuA + y.uBuB + y.uCuC rA rB rC
Estimation
Discussion
Discussion
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