# Limiting cases of criterion of minimum-extent

### Abstract

The minimum-extent criterion is appropriate to use for the problem formulation of processing of data obtained with a complicated noise environment and represented by parametric or nonparametric models. In this paper, the various limiting cases of minimum-extent criterion, which are carried out by tending the values of its three free parameters to their limiting or special values, are summarized. The goal of this paper is to obtain the relations between the minimum-extent criterion and other data processing criteria.

By calculating the corresponding limits, there are obtained the relations between the minimum-extent criterion and the following criteria. Among these criteria there are the least squares criterion, the least absolute deviations criterion, the maximum likelihood criterion in the problem of shift parameter estimating with a fixed scale parameter for independent identically-distributed random variables with the generalized Gaussian distribution and the generalized Cauchy distribution, the generalized maximum likelihood criterion with the Meshalkin's and Demidenko's cost functions, as well as the criterion of the maximum of the histogram. In addition, the several variants of the limiting passage from the criterion of minimum of the quasi-extent to the criterion of minimum of the strict extent are presented, where the latter gives the statement of the NP-hard problem of minimizing the quasi-norm of l0-space for the discrete case. It is emphasized that most of these criteria lead to the formulation of the optimization problem with a non-convex and non-unimodal objective function. It is also indicated that the presented results do not depend on the type and parameters of the used mathematical data model. The obtained limiting cases allow to conclude that the minimum-extent criterion is an universal tool to formulate the problems of data modeling and data processing for different conditions of data observation.

### References

Vovk S. M. Kryterii minimumu protiazhnosti / S. M. Vovk // Systemni tekhnolohii. Rehionalnyi mizhvuzivskyi zbirnyk naukovykh prats. Vypusk 1 (120). – Dnipro, 2019. – C. 19 – 25.

Vovk S. M. Postanovka zadach obrabotki dannyih na osnove kriteriya minimuma protyazhennosti / S. M. Vovk // Radioelektronika, informatyka, upravlinnia. – 2019. – N.1 – S. 157–166.

Vovk S. M. Metod obrobky danykh v umovakh skladnoho shumovoho otochennia / S. M. Vovk, V. V. Hnatushenko // Naukovyi visnyk Tavriiskoho derzhavnoho ahrotekhnolohichnoho universytetu. –2018. –Vyp. 8. – T.2. – S. 1–9.

Vovk S.M. General approach to building the methods of filtering based on the minimum duration principle / S. M. Vovk // Radioelectronics and Communications Systems. – 2016. – V. 59. – N. 7 – P. 281-292.

Wolberg J. Data Analysis Using the Method of Least Squares: Extracting the Most Information from Experiments / J. Wolberg. – Berlin: Springer-Verlag, 2005. – 250 p.

Aysal T. C. Meridian filtering for robust signal processing / T. C. Aysal, K. E. Barner // IEEE Tr. Signal Processing. – 2007. – V. 55. – N. 8. – P. 3949–3962.

Millar R. B. Maximum Likelihood Estimation and Inference: With Examples in R, SAS and ADMB / R. B. Millar. – New York: Wiley, 2011.– 376 p.

Carrillo R. E. Generalized Cauchy distribution based robust estimation / R. E. Carrillo, T. C. Aysal, K. E. Barner // Proc. of Int. Conf. Acoustic, Speech and Signal Processing, ICASSP 2008, Las Vegas, 2008. – P. 3389–3392.

Huber P. Robust statistics. 2nd ed. / P. Huber, E. M. Ronchetti. – Hoboken: Wiley, 2009. – 370 p.

Shevlyakov G. L. Robustness in data analysis: criteria and methods / G. L. Shevlyakov, N. O. Vilchevski. – Utrecht: VSP. – 2002. – 310 p.

Demidenko E. Z. Optimizatsiya i regressiya / E. Z. Demidenko. – M.: Nauka. – 1989. – 296 s.