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Title: Saddlepoint adjusted inversion of characteristic functions Authors:  Berent Aanund Stroemnes Lunde - University of Stavanger (Norway) [presenting]
Tore Selland Kleppe - University of Stavanger (Norway)
Hans Skaug - University of Bergen (Norway)
Abstract: For certain types of statistical models, the characteristic function is available in closed form, whereas the probability density function will have an intractable form, typically as an infinite sum of probability weighted densities. Important examples include solutions of stochastic differential equations with jumps, the Tweedie model, and certain mixture models. Likelihood optimisation, using inversion of the characteristic function, is made difficult by possible multi-modality of the density, which it is shown renders the unimodal saddlepoint approximation (SPA) useless. Direct numerical integration techniques will only work up-until a constant, which creates numerical problems of taking logarithms of the approximation to the density in the tails, something very intractable for optimisation routines. As a solution, the integrand of the problem is optimized for ``well behaviour'' under numerical inversion, much like the original SPA, creating a SPA weighted numerical inversion technique that is exact over the whole domain. The routine is computationally stable under optimisation, while also being much faster than ordinary SPA renormalisation routines, along with the tractable property of being exact. The method is applied to likelihood estimation of jump diffusion models, the Tweedie model, and mixture models and is also empirically seen to be stable, efficient, and accurate.