Abstract:
Many-multi parameter distributions have limit cases containing fewer parameters. This paper demonstrates that,
when fitting distributions to data realized from a distribution resembling one of these limit cases, the parameter estimates
obtained vary wildly between estimators. Special attention is paid to the modelling of financial log-returns. Two classes
of estimators are used in order to illustrate the behaviour of the parameter estimates; the maximum likelihood estimator
and the empirical characteristic function estimator. This paper discusses numerical problems associated with the maximum
likelihood estimator for certain distributions and proposes a solution using Fourier inversion. In addition to simulation results,
parameter estimates are obtained by fitting the normal inverse Gaussian and Meixner distributions to smooth bootstrap
samples from the log-returns of the Dow Jones Industrial Average index are included as examples.
