It is possible to think a model function has three parameters, where in fact there are effectively only two (or just one). This exercise investigates such a situation. (a) Suppose ( ) is a linear...

It is possible to think a model function has three parameters, where in fact there are effectively only two (or just one). This exercise investigates such a situation.

(a) Suppose
() is a linear function of the variable
. In trying to decide if two or three parameters are needed to fit a particular data set, the following possibilities are considered:

Suppose the minimum error using

₂() occurs when

₁
= 1,

₂
= 3, and

₃
= −2. Explain why the minimum error using

₁() occurs when

₁
= −2 and

₂
= 3. What values for
₁,

₂, and

₃
produce the minimum error using

₃()? Finally, explain why (i) corresponds to linear regression, while (ii) and (iii) correspond to nonlinear regression.

(b) For the model functions in part (a), suppose the minimum error using

₁() occurs only when

₁
= 6 and

₂
= −1. Explain why if one uses either of the other two model functions that the minimum error does not occur at unique values for

₁,

₂, and

₃.