The Use of Non-Parametric Methods to Estimate Density Functions of Copulas
Authors:
Munaf Yousif Hmood, Zainab Falih HamzaDOI NO:
https://doi.org/10.26782/jmcms.2019.08.00040Abstract:
Copulas distinguish the dependence among random vectors components as opposed to marginal and joint distributions, which can be directly observed, thus,so the copulas are considered as a hidden dependence among random vectors. Hence , the copulas could be defined as a structure that connects the joint distribution with the marginal distribution based on the non-parametric estimation with the use of the kernel function by the existence of the copula as it is considered as a tool hugely used in the modern statistics and more used in the non-parametric estimations; besides indicating the general characteristics of the estimator and selecting the appropriate bandwidth through the simulation process. A comparison was carried out between transformation estimator and Beta estimator and local likelihood transformation(LLTE) estimator in the estimation of the probability density function , using bimodel normal distribution. The results of simulation showed , according to the measurement of comparison used , that the best method is the method of (LLTE), where V. good estimations and easily to be implemented have been obtained while reducing boundary effect problems.Keywords:
Copula functions,Transformation kernel,Beta kernel,LocalLikelihood transformation Estimator,Refference:
I. A .Charpentier, Fermanian, J.D. and Scaillet,O.,(2007).”The estimation
of copulas: Theory and practice”.
II. A .Sklar., (1959), “Fonctions de répartition à n dimensions et leurs
marges”, Publications de l’Institut de Statistique de l’Université de Paris,
8, 229-231.
III. B.Nelsen, R, (2007).” An introduction to copulas”. Springer Science &
Business Media
IV. C. Genest, and R.J. MacKay, (1986a), “The joy of copulas: Bivariate
distributions with uniform marginals”, The American Statistician, 40,
280-283.
V. C. Genest, and R.J. MacKay,(1986b), Copules Archimédiennes et
familles de lois bidimensionnelles dont les marges sont données, The
Canadian Journal of Statistics, 14, 145-159
VI. C .Loader, 2006. Local regression and likelihood. Springer Science &
Business Media..
VII. G.Geenens, A .Charpentier, and Paindaveine, D. (2014). “Probit
transformation for nonparametric kernel estimation of the copula
density”. arXiv:1404.4414.
VIII. H. Joe., 1997. “Multivariate models and multivariate dependence
concepts”. Chapman and Hall/CRC.
IX. I. Gijbels, and Mielniczuk, J. (1990). “Estimating the density of a copula
function.Communications in Statistics – Theory and Methods”,
19XVIII:445–464.
X. K. Wen, and Wu, X., (2018). “Transformation-Kernel Estimation of
Copula Densities”. Journal of Business & Economic Statistics, (justaccepted),
pp.1-36.
XI. Munaf Yousif, H. (2005). “Comparing nonparametric estimators for
probability density functions”, Ph. D. dissertation, Department of
Statistics, Baghdad University.
XII. R .Lokoman.Yusof,.F,(2018). “Parametric Estimation Methods For
Bivariate Copula In Rainfall Application”, Jurnal Teknologi (Sciences &
Engineering), 81I ,pp1–10
XIII. S.Zhang, and Karunamuni, R.J., (2010). “Boundary performance of the
beta kernel estimators”. Journal of Nonparametric Statistics, 22I, pp.81-
104.
XIV. T .Nagler., (2018). “kdecopula: An R Package for the Kernel Estimation
of Bivariate Copula Densities”. Journal of Statistical Software. Volume
84, Issue 7.XVII
XV. T .Nagler, (2016). “kdecopula: An r package for the kernel estimation of
copula densities”. arXiv preprint arXiv:1603.04229
XVI. W. Scott, D. and Terrell, G. R. (1987). “Biased and unbiased crossvalidation
in density estimation”. Journal of the American Statistical
Association, 82(400):1131–1146.
XVII. W. Silverman, B. (1986).” Density Estimation for Statistics and Data
Analysis”. Chapman and Hall.
XVIII. X. S. Chen, (1999). “Beta kernel estimators for density functions”.
Computational Statistics & Data Analysis, 31XVIII:131–145.
XIX. X. S. Chen, and Huang, T.M., (2007). “Nonparametric estimation of
copula functions for dependence modelling”. Canadian Journal of
Statistics, 35XVIII, pp.265-282.
XX. Yan J (2007). “Enjoy the Joy of Copulas: With a Package copula.”
Journal of Statistical Software, 21VII, 1–21.