Polynomial parameter estimation of exponential power distribution data

Abstract

The paper proposes an original approach to obtain the results of multiple measurements at random errors, which are described by the exponential power (generalized Gaussian) distribution model. The approach is based on the polynomial maximization method (PMM), which is based on Kunchchenko’s mathematical apparatus using stochastic polynomials and a partial description of random variables of high-order statistics (moments or cumulants). The theoretical foundations of PMM are presented in relation to finding the estimates of the informative parameter from an equally distributed random variables sample. There are analytical expressions for finding polynomial estimations. It is shown that r \leqslant 2, then polynomial estimates degenerate in linear arithmetic mean estimates . If the polynomial degree r = 3 then the relative accuracy of polynomial estimations increases. The features of numerical procedures (Newton-Raphson method) for finding the stochastic equation roots are considered. Obtained analytical that describe the dispersion of the PMM estimates for an asymptotic case (for n → ∞). It is shown that the theoretical value of reduction coefficient variance of PMM estimates (in comparison with the linear mean estimates) depends on the magnitude of the random error cumulative coefficients of the 4th and 6th order. Through multiple statistical tests (Monte Carlo method) carried out a comparative accuracy analysis of polynomial estimates with known non-parametric estimates (median, mid-range and mean). It is shown that with increasing size of sample the difference between theoretical and experimental data decreases. The efficiency areas for each method are constructed, depending on the exponential power distribution parameter and sample size. It is shown that the accuracy of the proposed approach can significantly (more than twofold) exceed the classical nonparametric estimation.