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# Nonparametric Multivariate Density Estimation: Case Study of Cauchy Mixture Model

Department of Applied Mathematics, Faculty of Mathematics and Natural Sciences, Kaunas University of Technology, 44249 Kaunas, Lithuania
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Mathematics 2021, 9(21), 2717; https://doi.org/10.3390/math9212717
Received: 1 September 2021 / Revised: 18 October 2021 / Accepted: 21 October 2021 / Published: 26 October 2021
Estimation of probability density functions (pdf) is considered an essential part of statistical modelling. Heteroskedasticity and outliers are the problems that make data analysis harder. The Cauchy mixture model helps us to cover both of them. This paper studies five different significant types of non-parametric multivariate density estimation techniques algorithmically and empirically. At the same time, we do not make assumptions about the origin of data from any known parametric families of distribution. The method of the inversion formula is made when the cluster of noise is involved in the general mixture model. The effectiveness of the method is demonstrated through a simulation study. The relationship between the accuracy of evaluation and complicated multidimensional Cauchy mixture models (CMM) is analyzed using the Monte Carlo method. For larger dimensions (d ~ 5) and small samples (n ~ 50), the adaptive kernel method is recommended. If the sample is n ~ 100, it is recommended to use a modified inversion formula (MIDE). It is better for larger samples with overlapping distributions to use a semi-parametric kernel estimation and more isolated distribution-modified inversion methods. For the mean absolute percentage error, it is recommended to use a semi-parametric kernel estimation when the sample has overlapping distributions. In the smaller dimensions (d = 2) and a sample is with overlapping distributions, it is recommended to use the semi-parametric kernel method (SKDE) and for isolated distributions, it is recommended to use modified inversion formula (MIDE). The inversion formula algorithm shows that with noise cluster, the results of the inversion formula improved significantly. View Full-Text
MDPI and ACS Style

Ruzgas, T.; Lukauskas, M.; Čepkauskas, G. Nonparametric Multivariate Density Estimation: Case Study of Cauchy Mixture Model. Mathematics 2021, 9, 2717. https://doi.org/10.3390/math9212717

AMA Style

Ruzgas T, Lukauskas M, Čepkauskas G. Nonparametric Multivariate Density Estimation: Case Study of Cauchy Mixture Model. Mathematics. 2021; 9(21):2717. https://doi.org/10.3390/math9212717

Chicago/Turabian Style

Ruzgas, Tomas, Mantas Lukauskas, and Gedmantas Čepkauskas. 2021. "Nonparametric Multivariate Density Estimation: Case Study of Cauchy Mixture Model" Mathematics 9, no. 21: 2717. https://doi.org/10.3390/math9212717

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