# Polynomial parameter estimation of exponential power distribution data

## Authors

• S. V. Zabolotnii Cherkasy State Technological University, Ukraine
• A. V. Chepynoha Cherkasy State Technological University, Ukraine
• Yu. Yu. Bondarenko Cherkasy State Technological University, Ukraine
• M. P. Rud Cherkasy State Technological University, Ukraine

## Keywords:

exponential power distribution, stochastic polynomials, high-order statistics, parameter estimation

## 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.

## Author Biographies

### S. V. Zabolotnii, Cherkasy State Technological University

Zabolotnii S. V., Doc. Sci (Tech), Assoc. Prof

### A. V. Chepynoha, Cherkasy State Technological University

Chepynoha A. V., Cand of Sci (Tech), Assoc. Prof

### Yu. Yu. Bondarenko, Cherkasy State Technological University

Bondarenko Yu. Yu., Cand. of Sci (Tech), Assoc. Prof

### M. P. Rud, Cherkasy State Technological University

Rud M. P., Cand. of Sci (Tech), Assoc. Prof.

## References

The International Vocabulary of Metrology, Basic and General Concepts and Associated Terms (VIM), JCGM 200:2012 [ISO/IEC Guide 99].

Novitskii P. V. and Zograf I. A. (1991) Otsenka pogreshnostei rezul'tatov izmerenii [Estimation of errors of measurement results], Moskow, Energoatomizdat Publ., 304 p.

Box G.E. and Tiao G.C. (1992) Bayesian Inference in Statistical Analysis. DOI: 10.1002/9781118033197

Maugey T., Gauthier J., Pesquet-Popescu B. and Guillemot C. (2010) Using an exponential power model forwyner ziv video coding. 2010 IEEE International Conference on Acoustics, Speech and Signal Processing. DOI: 10.1109/icassp.2010.5496065

Subbotin M.T. (1923) On the law of frequency of error, Mat. Sb., Vil. 31, No 2, pp. 296-301.

Taguchi T. (1978) On a generalization of Gaussian distribution. Annals of the Institute of Statistical Mathematics, Vol. 30, Iss. 1, pp. 211-242. DOI: 10.1007/bf02480215

Varanasi M.K. and Aazhang B. (1989) Parametric generalized Gaussian density estimation. The Journal of the Acoustical Society of America, Vol. 86, Iss. 4, pp. 1404-1415. DOI: 10.1121/1.398700

Nadarajah S. (2005) A generalized normal distribution. Journal of Applied Statistics, Vol. 32, Iss. 7, pp. 685-694. DOI: 10.1080/02664760500079464

Crowder G.E. and Moore A.H. (1983) Adaptive Robust Estimation Based on a Family of Generalized Exponential Power Distributions. IEEE Transactions on Reliability, Vol. R-32, Iss. 5, pp. 488-495. DOI: 10.1109/tr.1983.5221739

Lindsey J.K. (1999) Multivariate Elliptically Contoured Distributions for Repeated Measurements. Biometrics, Vol. 55, Iss. 4, pp. 1277-1280. DOI: 10.1111/j.0006-341x.1999.01277.x

Hassan, M. Y. and Hijazi, R. H. (2010) А bimodal exponential power distribution, Pak. J. Statist, Vol. 26, No 2, pp. 379–396.

Fernandez C., Osiewalski J. and Steel M.F.J. (1995) Modeling and Inference with \$nu\$-Spherical Distributions. Journal of the American Statistical Association, Vol. 90, Iss. 432, pp. 1331-1340. DOI: 10.1080/01621459.1995.10476637

Komunjer I. (2007) Asymmetric power distribution: Theory and applications to risk measurement. Journal of Applied Econometrics, Vol. 22, Iss. 5, pp. 891-921. DOI: 10.1002/jae.961

Zhu D. and Zinde-Walsh V. (2009) Properties and estimation of asymmetric exponential power distribution. Journal of Econometrics, Vol. 148, Iss. 1, pp. 86-99. DOI: 10.1016/j.jeconom.2008.09.038

Zakharov I. P. and Shtefan N. V. (2002) Definition of effective distribution center value at statistical processing of measurement observations, Radioelektronika ta informatyka, No. 3 (20), pp. 97-99

Warsza, Z. L., Galovska, M. (2009) About the best measurand estimators of trapezoidal probability distributions. Przegląd Elektrotechniczny, Vol. 85, No. 5, pp.86–91.

Kunchenko Yu. P. (2002) Polynomial parameter estimations of close to Gaussian random variables, Aachen: Shaker Verlag. DOI:10.1007/978-3-319-77174-3

Warsza Z.L. and Zabolotnii S.W. (2017) A Polynomial Estimation of Measurand Parameters for Samples of Non-Gaussian Symmetrically Distributed Data. Advances in Intelligent Systems and Computing, pp. 468-480. DOI: 10.1007/978-3-319-54042-9_45

Warsza Z. and Zabolotnii S. (2017) Evaluation of the Uncertainty of Trapeze Distributed Measurements by the Polynomial Maximization Method. Pomiary Automatyka Robotyka, Vol. 21, Iss. 4, pp. 59-65. DOI: 10.14313/par_226/59

Zabolotnii S. V., Kucheruk V. Yu., Warsza Z. L. and Khassenov А. K. (2018) Polynomial Estimates of Measurand Parameters for Data from Bimodal Mixtures of Exponential Distributions, Vestnik Karagandinskogo universiteta, No. 2 (90), pp. 71-80.

Mathews J. H. and Fink K. D. (2004) Numerical methods using MATLAB, London, Pearson.

Cramér H. (1946) Mathematical Methods of Statistics (PMS-9). DOI: 10.1515/9781400883868