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Publication date: 11/10/2021

Nonlinear Models

Denote the vector of n responses by Y = (Y1, Y2, ..., Yn)‘. A nonlinear model is defined by the following properties:

The Yi are independent and identically distributed with an exponential family distribution.

The expected value of each Yi given a vector of predictor values xi is a nonlinear function of parameters, θ. Denote this function as follows:

Equation shown here

Each Yi is expressed as follows:

Equation shown here

The vector of errors, ε = (ε1, ε2, ..., εn)‘ has mean 0 and covariance matrix σ2I, where I is the n x n identity matrix.

Denote the matrix of first partial derivatives of the function f with respect to the parameters θ by X. Under general conditions, the least squares estimator of θ is asymptotically unbiased, with asymptotic covariance matrix given as follows:

Equation shown here

For the proof of this result, see Wu (1981) and Jennrich (1969).

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