  Profile likelihood provides a general framework to infer on a scalar
parameter of a statistical model. A confidence interval is obtained by
numerically finding the two abscissas where the profile log-likelihood curve
intersects an horizontal line. An alternative derivation for this interval can
be obtained by solving a constrained optimisation problem which can broadly be
described as: maximise or minimise the parameter of interest under the
constraint that the log-likelihood is high enough. This formulation allows nice
geometrical interpretations; It can be used to infer on a parameter or on a
known scalar function of the parameter, such as a quantile. Widely available
routines for constrained optimisation can be used for this task, as well as
Markov Chain Monte Carlo samplers. When the interest is on a smooth function
depending on an extra continuous variable, the constrained optimisation
framework can be used to derive Ordinary Differential Equation (ODE) for the
confidence limits. This is illustrated with the return levels of Extreme Value
models based on the Generalised Extreme Value distribution. Moreover the same
ODE-based technique applies as well to the derivation of profile likelihood
contours for couples of parameters. The initial value of the ODE used in the
determination of the interval or the contour can itself be obtained by another
auxiliary ODE with known initial value obtained by using the confidence level
as the extra continuous variable.
