Applied Statistics and Probability for Engineers

where x

[1, x 1 , x 2 ,p, xk]. Since the expected value of the model errors is zero, the ex-

pected value of the response variable is

We usually refer to f(x,) as the expectation functionfor the model. Obviously, the expec-

tation function here is just a linear function of the unknown parameters.

Any model that is not linear in the unknown parameters is a nonlinear regression model.

For example, the model

(S12-5)

is not linear in the unknown parameters  1 and  2. We will use the symbol to represent a pa-

rameter in a nonlinear model to emphasize the difference between the linear and the nonlinear

case. Nonlinear models often arise in cases where the relationship between the response and

the regressors is a differential equation or the solution to a differential equation.

In general, we will write the nonlinear regression model as

(S12-6)

where is a p 1 vector of unknown parameters, and is an uncorrelated random error term

with E()0 and Var()^2. We also typically assume that the errors are normally distrib-

uted, as in linear regression. Since

(S12-7)

we call f(x,) the expectation functionfor the nonlinear regression model. This is very sim-

ilar to the linear regression case, except that now the expectation function is a nonlinearfunc-

tion of the parameters.

In a nonlinear regression model, at least one of the derivatives of the expectation function

with respect to the parameters depends on at least one of the parameters. In linear regression,

these derivatives are notfunctions of the unknown parameters. To illustrate these points, con-

sider a linear regression model

with expectation function Now

where. Notice that in the linear case the derivatives are notfunctions of the ’s.

Now consider the nonlinear model

 1 e^2 x

yf 1 x,  2 

x 0  1

f 1 x,  2

j

xj, j0, 1,p, k

f 1 x,  2  0 gkj 1 jxj.

Y 0  1 x 1  2 x 2 pkxk

f 1 x,  2

E 1 Y 2 E 3 f 1 x,  2  4

Yf 1 x,  2 

Y 1 e^2 x

f 1 x,  2

E 1 Y 2 E 3 f 1 x,  2  4

12-6

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