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On parameter estimation in multi-parameter distributions

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.