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Abstract
We consider the problem of parameter estimation for an ergodic diffusion with the symmetric scaled Student invariant distribution, where the spectral representation of the transition density is given in terms of the finite number of polynomial eigenfunctions (Routh–Romanovski polynomials) and absolutely continuous spectrum of the negative infinitesimal generator of observed diffusion. We prove the consistency and asymptotic normality of the proposed estimators and, based on the Stein equation for Student diffusion, consider the statistical test for the Student distributional assumptions.
The research is partly supported by the EPSRC grant EP/D057361 (RCMT 119) and Marie Curie grant of the European Communities PIRSES-GA-2008-230804 (RCMT 152), the Wales Institute of Mathematics and Computer Science and the grant of the Croatian Foundation for Science, Higher Education and Technological Development. The authors are grateful to Professors F. Avram, M. Marletta, A. Veretennikov, and Doctor K.-M. Schmid for helpful discussions, to the anonymous referee for constructive remarks, and specially to Professor C. C. Heyde (1939–2008) who was one of the initiators of this research.