Proper confidence regions can be computed for constrained models through the inversion of the chi-squared Wald or likelihood ratio statistics. The inversion of the Wald statistic is discussed in the context of the restricted least squares model by Rust and Burrus (1972) and O'Leary and Rust (1986). Discussion of the inversion of the likelihood ratio statistic in computing confidence regions is found in Cox (1974), Cook and Weisberg (1990), Meeker and Escobar (1995).
Inversion of the Likelihood Ratio Statistic.
Partition a k-vector of parameters, 
, and
let 
be a maximum likelihood estimate of 
, where
is fixed to some value. A 
confidence
region for the parameters in is defined by
Let
where 
is a vector with a one in the i-th position and zeros elsewhere,
and 
is a function describing the constraints.
The lower limit of the ( 
) interval for 
is the value of
such that
A modified secant method is used to find the value of that satisfies
(1).
The upper limit is found by defining 
as a maximum.
Inversion of the Wald statistic.
A ( ) joint Wald-type confidence region for is the hyper-ellipsoid
where V is the covariance matrix of the parameters.
The lower limit of the confidence limit is the solution to
where can be an arbitrary vector of constants
and 
, and
where again we have assumed that the linear constraints and bounds have been
folded in among nonlinear constraints. The upper limit
is the maximum of this same function.
In this form, the function to be minimized is not convex and cannot be solved by the usual methods. However, the problem can be re-stated as a parametric nonlinear programming problem (Rust and Burrus, 1972). Let
The upper and lower limits of the interval are the values of
such that
A modified secant method is used to find the value of that satisfies
(2) (O'Leary and Rust, 1986).
The upper limit is found by defining 
as a maximum.
