Divergence Measures Estimation and Its Asymptotic Normality Theory Using Wavelets Empirical Processes I

We deal with the normality asymptotic theory of empirical divergences measures based on wavelets in a series of three papers. In this first paper, we provide the asymptotic theory of the general of ϕ -divergences measures, which includes the most common divergence measures : Renyi and Tsallis famili...

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Veröffentlicht in:Journal of statistical theory and applications Jg. 17; H. 1; S. 158 - 171
Hauptverfasser: Ba, Amadou Diadié, LO, Gane Samb, Ba, Diam
Format: Journal Article
Sprache:Englisch
Veröffentlicht: Dordrecht Springer Netherlands 01.03.2018
Springer Nature B.V
Springer
Schlagworte:
Asymptotic methods
Asymptotic normality
Asymptotic properties
Besov spaces
Divergence
Divergence measures estimation
Estimators
Function space
Normality
Probability density functions
Research Article
Statistical analysis
Statistical tests
Wavelet theory
wavelets empirical processes
62G05
62G07
Divergence measures estimation
Asymptotic normality
62G20
Wavelet theory
Besov spaces
wavelets empirical processes
ISSN:1538-7887, 2214-1766, 1538-7887
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Abstract We deal with the normality asymptotic theory of empirical divergences measures based on wavelets in a series of three papers. In this first paper, we provide the asymptotic theory of the general of ϕ -divergences measures, which includes the most common divergence measures : Renyi and Tsallis families and the Kullback-Leibler measures. Instead of using the Parzen nonparametric estimators of the probability density functions whose discrepancy is estimated, we use the wavelets approach and the geometry of Besov spaces. One-sided and two-sided statistical tests are derived. This paper is devoted to the foundations the general asymptotic theory and the exposition of the mains theoretical tools concerning the ϕ -forms, while proofs and next detailed and applied results will be given in the two subsequent papers which deal important key divergence measures and symmetrized estimators.
AbstractList We deal with the normality asymptotic theory of empirical divergences measures based on wavelets in a series of three papers. In this first paper, we provide the asymptotic theory of the general of ϕ-divergences measures, which includes the most common divergence measures : Renyi and Tsallis families and the Kullback-Leibler measures. Instead of using the Parzen nonparametric estimators of the probability density functions whose discrepancy is estimated, we use the wavelets approach and the geometry of Besov spaces. One-sided and two-sided statistical tests are derived. This paper is devoted to the foundations the general asymptotic theory and the exposition of the mains theoretical tools concerning the ϕ-forms, while proofs and next detailed and applied results will be given in the two subsequent papers which deal important key divergence measures and symmetrized estimators.
We deal with the normality asymptotic theory of empirical divergences measures based on wavelets in a series of three papers. In this first paper, we provide the asymptotic theory of the general of ϕ -divergences measures, which includes the most common divergence measures : Renyi and Tsallis families and the Kullback-Leibler measures. Instead of using the Parzen nonparametric estimators of the probability density functions whose discrepancy is estimated, we use the wavelets approach and the geometry of Besov spaces. One-sided and two-sided statistical tests are derived. This paper is devoted to the foundations the general asymptotic theory and the exposition of the mains theoretical tools concerning the ϕ -forms, while proofs and next detailed and applied results will be given in the two subsequent papers which deal important key divergence measures and symmetrized estimators.
Author LO, Gane Samb
Ba, Diam
Ba, Amadou Diadié
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Keywords 62G05
62G07
Divergence measures estimation
Asymptotic normality
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Wavelet theory
Besov spaces
wavelets empirical processes
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Blatter, C. (1998) Wavelets, a Primer. A. K. Peters, Natick. MA.
Kullback, S. and Leibler, R.(1951). On information and sufficiency. The Annals of Mathematical Statistics Vol.22,(1), pp 79–86.
Daubechies, I.(1992). Ten Lectures on Wavelets. Society for Industrial and Applied Mathematics, Philadelphia.
Hall, P. (1987). On Kullback-Leibler loss and density estimation. The Annals of Statistics, Vol.15(4), pp.1491–1519.
Liu, H., Lafferty, J., and Wasserman, L.(2012). Exponential concentration inequality for mutual information estimation . In Neural Information Processing Systems (NIPS).
Topsoe, F. (2000), Some inequalities for information divergence and related measures of discrimination, IEEE Transactions on Informations Theory, vol.46, pp.1602–1609.
Moon, K.R. and Hero, III. A.O. , (2014). Ensemble estimation of multivariate f-divergence. in IEEE Internatonal Symposium on Information Theory, pp. 356–360.
Poczós, B. and Jeff, S.(2011). On the estimation of α—Divergences. In International Conference on Artificial Intelligence and Statistics, pp 609–617.
Giné, E. and Nickl, R.(2009). Uniform limit theorems for wavelet density estimators. The Annals of Probability, Vol.37(4), pp.1605–1646.
Cichocki, A. and Amari, S.(2010). Families of Alpha-Beta-and Gamma-Divergences: Flexible and Robust Measures of Similarities. Entropy, Vol.12(6), pp 1532–1568.
Sricharan, K., Wei, D., and Hero, A. O. Ensemble estimators for multivariate entropy estimation. arXiv:1203.5829, 2012.
Ba, A. D., LO, Lo, G.S and Ba, Diam B. (2017) Divergence Measures Estimation and Its Asymptotic Normality Theory Using Wavelets Empirical Processes. ArXiv:1704.04536
Love, M.(1972). Probabily Theory I 4th Edition. Springer.
Kallberg D. and Seleznjev O. 2012. Estimation of entropy-type integral functionals. arXiv:1209.2544.
Hardle, W., Kerkyacharian, G., Picard, D., and Tsybakov, A.(1998). Wavelets, Approximation, and Statistical Applications. Lecture Notes in Statistics.
Dhaker H., Ngom P ., Deme E. and Mendy Pierre (2016). Kernel-Type Estimators of Divergence Measures and Its Strong Uniform Consistency. American Journal of Theoretical and Applied Statistics. Vol. 5 (1), pp. 13–22. doi: 10.11648/j.ajtas.20160501.13
Evren, A. (2012). Some Applications of Kullback-Leibler and Jeffreys’ Divergences in Multinomial Populations. Journal of Selcuk University natural and Applied Science, Vol.1(4), pp 48–58.
Akshay K., Kirthevasan K., Poczos B., and Wasserman, L.(2014). Nonparametric Estimation of Rényi Divergence and Friends. Journal of Machine Learning Research Workshop and conference Proceedings, 32. Vol.3, pp. 2.
Valiron, G. (1966). Théorie des fonctions. Masson, Paris Milan Melbourne.
Singh S. and Poczos, B. (2014). Generalized Exponential Concentration Inequality for Rényi Divergence Estimation. Journal of Machine Learning Research. Vol.6. Carnegie Mellon University.
References_xml – reference: Singh S. and Poczos, B. (2014). Generalized Exponential Concentration Inequality for Rényi Divergence Estimation. Journal of Machine Learning Research. Vol.6. Carnegie Mellon University.
– reference: Liu, H., Lafferty, J., and Wasserman, L.(2012). Exponential concentration inequality for mutual information estimation . In Neural Information Processing Systems (NIPS).
– reference: Love, M.(1972). Probabily Theory I 4th Edition. Springer.
– reference: Giné, E. and Nickl, R.(2009). Uniform limit theorems for wavelet density estimators. The Annals of Probability, Vol.37(4), pp.1605–1646.
– reference: Ba, A. D., LO, Lo, G.S and Ba, Diam B. (2017) Divergence Measures Estimation and Its Asymptotic Normality Theory Using Wavelets Empirical Processes. ArXiv:1704.04536
– reference: Krishnamurthy A., Kandasamy K., Poczós B. and and Wasserman L.(2015) To appear in Proceedings of the 18th International Con- ference on Artificial Intelligence and Statistics (AISTATS) 2015, San Diego, CA, USA. JMLR: W&CP volume 38. Copyright 2015 by the authors.
– reference: Kallberg D. and Seleznjev O. 2012. Estimation of entropy-type integral functionals. arXiv:1209.2544.
– reference: Hardle, W., Kerkyacharian, G., Picard, D., and Tsybakov, A.(1998). Wavelets, Approximation, and Statistical Applications. Lecture Notes in Statistics.
– reference: Poczós, B. and Jeff, S.(2011). On the estimation of α—Divergences. In International Conference on Artificial Intelligence and Statistics, pp 609–617.
– reference: Valiron, G. (1966). Théorie des fonctions. Masson, Paris Milan Melbourne.
– reference: Blatter, C. (1998) Wavelets, a Primer. A. K. Peters, Natick. MA.
– reference: Topsoe, F. (2000), Some inequalities for information divergence and related measures of discrimination, IEEE Transactions on Informations Theory, vol.46, pp.1602–1609.
– reference: Evren, A. (2012). Some Applications of Kullback-Leibler and Jeffreys’ Divergences in Multinomial Populations. Journal of Selcuk University natural and Applied Science, Vol.1(4), pp 48–58.
– reference: Kullback, S. and Leibler, R.(1951). On information and sufficiency. The Annals of Mathematical Statistics Vol.22,(1), pp 79–86.
– reference: Akshay K., Kirthevasan K., Poczos B., and Wasserman, L.(2014). Nonparametric Estimation of Rényi Divergence and Friends. Journal of Machine Learning Research Workshop and conference Proceedings, 32. Vol.3, pp. 2.
– reference: Dhaker H., Ngom P ., Deme E. and Mendy Pierre (2016). Kernel-Type Estimators of Divergence Measures and Its Strong Uniform Consistency. American Journal of Theoretical and Applied Statistics. Vol. 5 (1), pp. 13–22. doi: 10.11648/j.ajtas.20160501.13
– reference: Sricharan, K., Wei, D., and Hero, A. O. Ensemble estimators for multivariate entropy estimation. arXiv:1203.5829, 2012.
– reference: Moon, K.R. and Hero, III. A.O. , (2014). Ensemble estimation of multivariate f-divergence. in IEEE Internatonal Symposium on Information Theory, pp. 356–360.
– reference: Hall, P. (1987). On Kullback-Leibler loss and density estimation. The Annals of Statistics, Vol.15(4), pp.1491–1519.
– reference: Daubechies, I.(1992). Ten Lectures on Wavelets. Society for Industrial and Applied Mathematics, Philadelphia.
– reference: Cichocki, A. and Amari, S.(2010). Families of Alpha-Beta-and Gamma-Divergences: Flexible and Robust Measures of Similarities. Entropy, Vol.12(6), pp 1532–1568.
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SubjectTerms Asymptotic methods
Asymptotic normality
Asymptotic properties
Besov spaces
Divergence
Divergence measures estimation
Estimators
Function space
Normality
Probability density functions
Research Article
Statistical analysis
Statistical tests
Wavelet theory
wavelets empirical processes
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