Let X1,...,Xn be independently normally distributed with common variance 2 and means i = + ti

Let X1,...,Xn be independently normally distributed with common variance σ2 and means ξi = α + βti + γt2 i , where the ti are known. If the coefficient vectors (t k 1 ,...,tk n), k = 0, 1, 2, are linearly independent, the parameter space ΠΩ has dimension s = 3, and the least-squares estimates ˆα, β, ˆ γˆ are the unique solutions of the system of equations α
t k i + β
t k+1 i + γ
t k+2 i = t k i Xi (k = 0, 1, 2).
The solutions are linear functions of the X’s, and if ˆγ = ciXi, the hypothesis γ = 0 is rejected when |γˆ|/
c2 1 i 
Xi − αˆ − βtˆ i − γt ˆ 2 i 2 /(n − 3)
> C0.
Section 7.7