869 research outputs found
Large Parameter Cases of the Gauss Hypergeometric Function
We consider the asymptotic behaviour of the Gauss hypergeometric function
when several of the parameters a, b, c are large. We indicate which cases are
of interest for orthogonal polynomials (Jacobi, but also Krawtchouk, Meixner,
etc.), which results are already available and which cases need more attention.
We also consider a few examples of 3F2-functions of unit argument, to explain
which difficulties arise in these cases, when standard integrals or
differential equations are not available.Comment: 21 pages, 4 figure
Large Degree Asymptotics of Generalized Bessel Polynomials
Asymptotic expansions are given for large values of of the generalized
Bessel polynomials . The analysis is based on integrals that follow
from the generating functions of the polynomials. A new simple expansion is
given that is valid outside a compact neighborhood of the origin in the
plane. New forms of expansions in terms of elementary functions valid in
sectors not containing the turning points are derived, and a new
expansion in terms of modified Bessel functions is given. Earlier asymptotic
expansions of the generalized Bessel polynomials by Wong and Zhang (1997) and
Dunster (2001) are discussed.Comment: 22 pages, 1 figur
Computation of the reverse generalized Bessel polynomials and their zeros
ABSTRACT:It is well known that one of the most relevant applications of the reverse Bessel polynomials \u1d703�n(z) is filter design. In particular, the poles of the transfer function of a Bessel filter are basically the zeros of \u1d703�n(z). In this article we discuss an algorithm to compute the zeros of reverse generalized Bessel polynomials \u1d703�n(z;a). A key ingredient in the algorithm will be a method to compute the polynomials. For this purpose, we analyze the use of recurrence relations and asymptotic expansions in terms of elementary functions to obtain accurate approximations to the polynomials. The performance of all the numerical approximations will be illustrated with examples.The authors acknowledge financial support from Ministerio de Ciencia e Innovación, Spain, project PGC2018-098279-B-I00 (MCIU/AEI/FEDER, UE)
Asymptotic approximations to the nodes and weights of Gauss-Hermite and Gauss-Laguerre quadratures
Asymptotic approximations to the zeros of Hermite and Laguerre polynomials
are given, together with methods for obtaining the coefficients in the
expansions. These approximations can be used as a standalone method of
computation of Gaussian quadratures for high enough degrees, with Gaussian
weights computed from asymptotic approximations for the orthogonal polynomials.
We provide numerical evidence showing that for degrees greater than the
asymptotic methods are enough for a double precision accuracy computation
(- digits) of the nodes and weights of the Gauss--Hermite and
Gauss--Laguerre quadratures.Comment: Submitted to Studies in Applied Mathematic
Asymptotic Approximations to the Nodes and Weights of Gauss-Hermite and Gauss-Laguerre Quadratures
Asymptotic approximations to the zeros of Hermite and Laguerre polynomials are given, together with methods for obtaining the coefficients in the expansions. These approximations can be used as a stand-alone method of computation of Gaussian quadratures for high enough degrees, with Gaussian weights computed from asymptotic approximations for the orthogonal polynomials. We provide numerical evidence showing that for degrees greater than 100, the asymptotic methods are enough for a double precision accuracy computation (15-16 digits) of the nodes and weights of the Gauss-Hermite and Gauss-Laguerre quadratures.The authors acknowledge financial support from Ministerio de Economía y Competitividad, project MTM2015-67142-P (MINECO/FEDER, UE)
Maximum-likelihood estimation for diffusion processes via closed-form density expansions
This paper proposes a widely applicable method of approximate
maximum-likelihood estimation for multivariate diffusion process from
discretely sampled data. A closed-form asymptotic expansion for transition
density is proposed and accompanied by an algorithm containing only basic and
explicit calculations for delivering any arbitrary order of the expansion. The
likelihood function is thus approximated explicitly and employed in statistical
estimation. The performance of our method is demonstrated by Monte Carlo
simulations from implementing several examples, which represent a wide range of
commonly used diffusion models. The convergence related to the expansion and
the estimation method are theoretically justified using the theory of Watanabe
[Ann. Probab. 15 (1987) 1-39] and Yoshida [J. Japan Statist. Soc. 22 (1992)
139-159] on analysis of the generalized random variables under some standard
sufficient conditions.Comment: Published in at http://dx.doi.org/10.1214/13-AOS1118 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
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