engleski

Statistical inference for reciprocal gamma diffusion process

It is now generally accepted that heavy-tailed distributions occur commonly in practice. Their use is now widespread in communication networks, risky asset and insurance modeling. However, the study of stationary processes having these heavy-tailed distributions as their one-dimensional distributions have received rather little attention. In this paper we focus on such process with reciprocal gamma marginals. An important property of reciprocal gamma ergodic diffusion, such as the spectral representation of the transition density function obtained as the principal solution of the corresponding Fokker-Planck equation, is clarified. This principal solution is written in form of the finite sum of discrete eigenfunctions (Bessel polynomials) and the integral which is taken over the essential spectrum of the corresponding Sturm-Liouville operator. The statistical part of the paper contains parametric and semiparametric estimation of parameters of stationary reciprocal gamma diffusion. We use the method of moments and the martingale estimation equation approach. The method of moments yields estimators for parameters of reciprocal gamma diffusion that correspond to scale and shape parameters of marginal distribution in an explicit form. Since the reciprocal gamma diffusion process satisfies mixing conditions with the exponentially decaying rate, using the proper functional central limit theorem and the functional delta method, we are able to prove consistency and asymptotical normality of these moment estimators. Using the finite system of orthonormal Bessel polynomials we developed a method for calculation of moments of the form E[X_{; ; ; s + t}; ; ; ^{; ; ; m}; ; ; X_{; ; ; s}; ; ; ^{; ; ; n}; ; ; ], where m and n are at most equal to the finite number of Bessel polynomials. This method made it possible for us to calculate the explicit form of the limiting covariance matrix of a bivariate estimator of scale and shape parameter. This makes an important problem of constructing asymptotic confidence intervals for unknown parameters operational. The martingale estimation equation method (due to Sorensen et al.) that provides a P-consistent and asymptotically normal estimator is used here for estimation of an unknown autocorrelation parameter. The statistical part also deals with testing reciprocal gamma distributional assumptions and it is based on the Stein equation for reciprocal gamma diffusion. The orthonormal Bessel polynomials are underlying basis of this approach. However, we were not able to prove their robustness as test functions in the proposed procedure.

Asymptotical normality; Bessel polynomials; Consistency; Heavy-tailed distribution; Martingale estimation equation; Moment estimation; Pearson equation; Reciprocal gamma distribution; Reciprocal gamma diffusion; Stationary distribution; Stein equation; Stochastic differential equation.

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