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.

Statistical Inference for Student Diffusion Process

Statistical inference for reciprocal gamma diffusion process

This report gives a rather arbitrary choice of formulas for (q-)hypergeometric orthogonal polynomials which the author missed while consulting Chapters 9 and 14 in the book \Hypergeometric orthogonal polynomials and their q-analogues" by Koekoek, Lesky and Swarttouw. The systematics of these chapters will be followed here, in particular for the numbering of subsections and of references.

Additions to the formula lists in \Hypergeometric orthogonal polynomials and their q-analogues" by Koekoek, Lesky and Swarttouw

The integral is calculated and the system of orthogonal polynomials with weight equal to the corresponding integrand is constructed. This weight decreases polynomially, therefore only finitely many of its moments converge. As a?result the system of orthogonal polynomials is finite.Systems of orthogonal polynomials related to -Dougall's formula and the Askey integral is also constructed. All the three systems consist of Wilson polynomials outside the domain of positiveness of the usual weight.

http://iopscience.iop.org/article/10.1070/SM2002v193n07ABEH000670/pdf

Beta-integrals and finite orthogonal systems of Wilson polynomials

/pdf/beta-integraly-i-konechnye-ortogonalnye-sistemy-mnogochlenov-2srtuaabfc.pdf

P.A. Lesky

Papers

Einordnung der Polynome von Romanovski-Bessel in das Askey-Tableau