Estimating confidence regions over bounded domains

Estimating a density function over a bounded domain can be very complicated and resulting in an unsatisfactory or unrealistic density estimate. In many cases, a one-to-one transformation can be applied to the considered data set, but there are also situations where such a unique transformation may not exist. A method to estimate confidence regions over bounded domains, when a one-to-one transformation either does not exist or its existence is difficult to verify, is proposed. By taking into account parameter restrictions of a underlying model, a nonlinear grid can be constructed, over which the density function can be estimated. The method is illustrated by applying it to the kurtosis/first-order autocorrelation of squared observations of the GARCH(1,1) model.