engleski

Parameter Estimation for Fisher-Snedecor Diffusion

Ergodic diffusion with an invariant Fisher-Snedecor distribution is studied. In particular, the problem of parameter estimation is treated and the quality of estimators is illustrated by the simulation results. Classical approach to the asymptotic analysis of estimators is implied by some properties of the Fisher-Snedecor diffusion, such as the ergodicity, stationarity and exponentially decaying alpha-mixing property. The unknown parameters which are treated here are the autocorrelation parameter and parameters which coincide with the shape parameters of the invariant Fisher-Snedecor distribution. Estimation procedure for all parameters is based on the discrete observations form the process of interest. Estimation of the autocorrelation parameter and estimation of shape parameters of invariant distribution are treated separately. In particular, the autocorrelation parameter is estimated under the assumption that the shape parameters of the invariant distribution are known. The corresponding estimator is determined by the generalized method of moments based on the consistent estimator for the autocorrelation function - Pearson's sample autocorrelation function. Furthermore, the shape parameters of the invariant distribution are estimated under the assumption that the autocorrelation parameter is known. The bivariate estimator of the shape parameters is determined by the method of moments based on the empirical counterparts of the first and the second moment of the invariant distribution. Ergodicity, stationarity and exponentially decaying alpha-mixing property of the Fisher-Snedecor diffusion are crucial for the analysis of asymptotic properties of these estimators. Consistency of estimators is implied by the ergodic theorem for stationary sequences and the continuous mapping theorem. Asymptotic normality of the bivariate estimator of shape parameters of invariant distribution follows from the central limit theorem for alpha-mixing sequences and the standard delta method. Furthermore, the explicit form of the limiting covariance matrix of the bivariate estimator of shape parameters is calculated according to the method based on the closed form expression for the spectral representation of transition density and important properties of the finite system of orthogonal polynomial eigenfunctions of the infinitesimal generator of Fisher-Snedecor diffusion. The estimation procedure based on the martingale estimation equations due to Forman and Sorensen (2008) is briefly discussed. The discrete observations from the Fisher-Snedecor diffusion are simulated in statistical software R and used for further analysis of the quality of presented estimators.

Asymptotic normality; Consistency; Method of moments; Orthogonal polynomials; Transition density

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