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

Let X1,..., Xn be independently normally distributed with common variance σ2 and means ξi = α + βti + γ t 2

i , where the ti are known. If the coefficient vectors (t k

1 ,..., t k n ), k = 0, 1, 2, are linearly independent, the parameter space 

has dimension s = 3, and the least squares estimates α,ˆ 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 γˆ = ci Xi , the hypothesis

γ = 0 is rejected when

| ˆγ |/

c2 i



Xi − ˆα − βˆti − ˆγ t 2 i

2

/(n − 3)

> C0.

Section 7.7