Bivariate Radialsymmetric Kernel Density Estimation of Well-being Distribution and Poverty Index

Youssou Ciss; Mamadou Djitan Sinclair; Mohamed Dinah Bangoura; Aboubakary Diakhaby

doi:doi:10.11648/j.ajtas.20261505.11

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Bivariate Radialsymmetric Kernel Density Estimation of Well-being Distribution and Poverty Index

Youssou Ciss

, Mamadou Djitan Sinclair, Mohamed Dinah Bangoura, Aboubakary Diakhaby

Published in American Journal of Theoretical and Applied Statistics (Volume 15, Issue 5)

Received: 9 June 2026 Accepted: 24 July 2026 Published: 5 September 2026

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Abstract

In two previous papers, we proposed two bivariate product kernel estimators for the bi-dimensional extension of the Foster, Greer and Thorbecke (FGT) index based respectively on classical and adaptive kernel. The Foster, Greer and Thorbecke (FGT) index was introduced in the literature for the purpose of a dominance approach to multidimensional poverty. The poverty measure utilized in this dominance method is fundamentally a generalization, from one to two dimensions, of this Foster, Greer and Thorbecke index with separate poverty aversion parameters for each dimension. This statistical method (the product kernel estimator) has the following main disadvantes: the curse of dimensionality, the complex selection of the bandwidth, and high computational costs. In this study, we focus on a bivariate radialsymmetric kernel estimator for the bi-dimensional extension of the Foster, Greer and Thorbecke index. Our new bivariate kernel estimator is developed with a classical bivariate kernel of Parzen-Rosenblatt of a probability density function (pdf) utilizing Riemann sums. We next provide complete asymptotic behaviour by establishing both almost-sure uniform and uniform mean square consistencies for the bivariate radialsymmetric kernel estimator. A simulation study indicates that the proposed bivariate kernel estimator performs favorably for small samples comparatively to the bivariate multiplicative kernel estimator.

Published in	American Journal of Theoretical and Applied Statistics (Volume 15, Issue 5)
DOI	10.11648/j.ajtas.20261505.11
Page(s)	177-201
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2026. Published by Science Publishing Group

Keywords

Bidimensional Poverty Index, Foster-Greer-Thorbecke (FGT) Measure, Bivariate Radialsymmetric Kernel Density Estimation, Nonparametric Estimation, Uniform Almost-sure Consistency, Riemann Sums, Well-being Distribution, Rate of Convergence

References

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[6]	Davidson, R., & Flachaire, E. (2007). Asymptotic and bootstrap inference for inequality and poverty measures. Journal of Econometrics, 141(1), 141-166, 2007. https://doi.org/10.1016/j.jeconom.2007.01.009
[7]	Duclos, J. Y., Sahn, D. E. and Younger, S. D. (2006). Robust multidimensional poverty comparison. Economic Journal, 113:943-968, 2006. https://doi.org/10.1111/j.1468-0297.2006.01118.x
[8]	Duclos, J. Y., Sahn, D. E. and Younger, S. D. (2006). Robust multidimensional spatial poverty comparisons in Ghana, Madagascar, and Uganda. World Bank Econ. Rev., 20(1), 91-113, 2006.
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[12]	Foster, J. E. and Shorrocks, A.F. (1988) Inequality and Poverty Orderings. European Economic Review, 32, 654-662, 1988.
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[14]	H"ardle, W., M"uller, M. (1997). Multivariate and Semiparametric Kernel Regression. Sonderforschungsbereich 373: Quantification and Simulation of Economic Processes., 1997,26.
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APA Style

Ciss, Y., Sinclair, M. D., Bangoura, M. D., Diakhaby, A. (2026). Bivariate Radialsymmetric Kernel Density Estimation of Well-being Distribution and Poverty Index. American Journal of Theoretical and Applied Statistics, 15(5), 177-201. https://doi.org/10.11648/j.ajtas.20261505.11

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ACS Style

Ciss, Y.; Sinclair, M. D.; Bangoura, M. D.; Diakhaby, A. Bivariate Radialsymmetric Kernel Density Estimation of Well-being Distribution and Poverty Index. Am. J. Theor. Appl. Stat. 2026, 15(5), 177-201. doi: 10.11648/j.ajtas.20261505.11

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AMA Style

Ciss Y, Sinclair MD, Bangoura MD, Diakhaby A. Bivariate Radialsymmetric Kernel Density Estimation of Well-being Distribution and Poverty Index. Am J Theor Appl Stat. 2026;15(5):177-201. doi: 10.11648/j.ajtas.20261505.11

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@article{10.11648/j.ajtas.20261505.11,
  author = {Youssou Ciss and Mamadou Djitan Sinclair and Mohamed Dinah Bangoura and Aboubakary Diakhaby},
  title = {Bivariate Radialsymmetric Kernel Density Estimation of Well-being Distribution and Poverty Index},
  journal = {American Journal of Theoretical and Applied Statistics},
  volume = {15},
  number = {5},
  pages = {177-201},
  doi = {10.11648/j.ajtas.20261505.11},
  url = {https://doi.org/10.11648/j.ajtas.20261505.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajtas.20261505.11},
  abstract = {In two previous papers, we proposed two bivariate product kernel estimators for the bi-dimensional extension of the Foster, Greer and Thorbecke (FGT) index  based respectively on classical and adaptive kernel. The Foster, Greer and Thorbecke (FGT) index was introduced in the literature for the purpose of a dominance approach to multidimensional poverty. The poverty measure utilized in this dominance method is fundamentally a generalization, from one to two dimensions, of this Foster, Greer and Thorbecke index with separate poverty aversion parameters for each dimension. This statistical method (the product kernel estimator) has the following main disadvantes: the curse of dimensionality, the complex selection of the bandwidth, and high computational costs.  In this  study, we focus on a bivariate radialsymmetric kernel estimator for the bi-dimensional extension of the Foster, Greer and Thorbecke index. Our new bivariate kernel estimator is developed with a classical bivariate kernel of Parzen-Rosenblatt of a probability density function (pdf) utilizing Riemann sums. We next provide complete asymptotic  behaviour by establishing both almost-sure uniform and  uniform mean square consistencies for the bivariate radialsymmetric kernel estimator. A simulation study indicates that the proposed bivariate kernel estimator performs favorably for small samples comparatively to the bivariate multiplicative kernel estimator.},
 year = {2026}
}

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TY  - JOUR
T1  - Bivariate Radialsymmetric Kernel Density Estimation of Well-being Distribution and Poverty Index
AU  - Youssou Ciss
AU  - Mamadou Djitan Sinclair
AU  - Mohamed Dinah Bangoura
AU  - Aboubakary Diakhaby
Y1  - 2026/09/05
PY  - 2026
N1  - https://doi.org/10.11648/j.ajtas.20261505.11
DO  - 10.11648/j.ajtas.20261505.11
T2  - American Journal of Theoretical and Applied Statistics
JF  - American Journal of Theoretical and Applied Statistics
JO  - American Journal of Theoretical and Applied Statistics
SP  - 177
EP  - 201
PB  - Science Publishing Group
SN  - 2326-9006
UR  - https://doi.org/10.11648/j.ajtas.20261505.11
AB  - In two previous papers, we proposed two bivariate product kernel estimators for the bi-dimensional extension of the Foster, Greer and Thorbecke (FGT) index  based respectively on classical and adaptive kernel. The Foster, Greer and Thorbecke (FGT) index was introduced in the literature for the purpose of a dominance approach to multidimensional poverty. The poverty measure utilized in this dominance method is fundamentally a generalization, from one to two dimensions, of this Foster, Greer and Thorbecke index with separate poverty aversion parameters for each dimension. This statistical method (the product kernel estimator) has the following main disadvantes: the curse of dimensionality, the complex selection of the bandwidth, and high computational costs.  In this  study, we focus on a bivariate radialsymmetric kernel estimator for the bi-dimensional extension of the Foster, Greer and Thorbecke index. Our new bivariate kernel estimator is developed with a classical bivariate kernel of Parzen-Rosenblatt of a probability density function (pdf) utilizing Riemann sums. We next provide complete asymptotic  behaviour by establishing both almost-sure uniform and  uniform mean square consistencies for the bivariate radialsymmetric kernel estimator. A simulation study indicates that the proposed bivariate kernel estimator performs favorably for small samples comparatively to the bivariate multiplicative kernel estimator.
VL  - 15
IS  - 5
ER  -

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Author Information

Youssou Ciss

West African Institute of Mathematics, Gamal Abdel Nasser University, Conakry, Guinea

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http://orcid.org/0009-0007-1533-3931
Mamadou Djitan Sinclair

West African Institute of Mathematics, Gamal Abdel Nasser University, Conakry, Guinea

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Mohamed Dinah Bangoura

Department of Mathematics, Gamal Abdel Nasser University, Conakry, Guinea

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Aboubakary Diakhaby

West African Institute of Mathematics, Gamal Abdel Nasser University, Conakry, Guinea

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APA Style

Ciss, Y., Sinclair, M. D., Bangoura, M. D., Diakhaby, A. (2026). Bivariate Radialsymmetric Kernel Density Estimation of Well-being Distribution and Poverty Index. American Journal of Theoretical and Applied Statistics, 15(5), 177-201. https://doi.org/10.11648/j.ajtas.20261505.11

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ACS Style

Ciss, Y.; Sinclair, M. D.; Bangoura, M. D.; Diakhaby, A. Bivariate Radialsymmetric Kernel Density Estimation of Well-being Distribution and Poverty Index. Am. J. Theor. Appl. Stat. 2026, 15(5), 177-201. doi: 10.11648/j.ajtas.20261505.11

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AMA Style

Ciss Y, Sinclair MD, Bangoura MD, Diakhaby A. Bivariate Radialsymmetric Kernel Density Estimation of Well-being Distribution and Poverty Index. Am J Theor Appl Stat. 2026;15(5):177-201. doi: 10.11648/j.ajtas.20261505.11

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@article{10.11648/j.ajtas.20261505.11,
  author = {Youssou Ciss and Mamadou Djitan Sinclair and Mohamed Dinah Bangoura and Aboubakary Diakhaby},
  title = {Bivariate Radialsymmetric Kernel Density Estimation of Well-being Distribution and Poverty Index},
  journal = {American Journal of Theoretical and Applied Statistics},
  volume = {15},
  number = {5},
  pages = {177-201},
  doi = {10.11648/j.ajtas.20261505.11},
  url = {https://doi.org/10.11648/j.ajtas.20261505.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajtas.20261505.11},
  abstract = {In two previous papers, we proposed two bivariate product kernel estimators for the bi-dimensional extension of the Foster, Greer and Thorbecke (FGT) index  based respectively on classical and adaptive kernel. The Foster, Greer and Thorbecke (FGT) index was introduced in the literature for the purpose of a dominance approach to multidimensional poverty. The poverty measure utilized in this dominance method is fundamentally a generalization, from one to two dimensions, of this Foster, Greer and Thorbecke index with separate poverty aversion parameters for each dimension. This statistical method (the product kernel estimator) has the following main disadvantes: the curse of dimensionality, the complex selection of the bandwidth, and high computational costs.  In this  study, we focus on a bivariate radialsymmetric kernel estimator for the bi-dimensional extension of the Foster, Greer and Thorbecke index. Our new bivariate kernel estimator is developed with a classical bivariate kernel of Parzen-Rosenblatt of a probability density function (pdf) utilizing Riemann sums. We next provide complete asymptotic  behaviour by establishing both almost-sure uniform and  uniform mean square consistencies for the bivariate radialsymmetric kernel estimator. A simulation study indicates that the proposed bivariate kernel estimator performs favorably for small samples comparatively to the bivariate multiplicative kernel estimator.},
 year = {2026}
}

Copy | Download

TY  - JOUR
T1  - Bivariate Radialsymmetric Kernel Density Estimation of Well-being Distribution and Poverty Index
AU  - Youssou Ciss
AU  - Mamadou Djitan Sinclair
AU  - Mohamed Dinah Bangoura
AU  - Aboubakary Diakhaby
Y1  - 2026/09/05
PY  - 2026
N1  - https://doi.org/10.11648/j.ajtas.20261505.11
DO  - 10.11648/j.ajtas.20261505.11
T2  - American Journal of Theoretical and Applied Statistics
JF  - American Journal of Theoretical and Applied Statistics
JO  - American Journal of Theoretical and Applied Statistics
SP  - 177
EP  - 201
PB  - Science Publishing Group
SN  - 2326-9006
UR  - https://doi.org/10.11648/j.ajtas.20261505.11
AB  - In two previous papers, we proposed two bivariate product kernel estimators for the bi-dimensional extension of the Foster, Greer and Thorbecke (FGT) index  based respectively on classical and adaptive kernel. The Foster, Greer and Thorbecke (FGT) index was introduced in the literature for the purpose of a dominance approach to multidimensional poverty. The poverty measure utilized in this dominance method is fundamentally a generalization, from one to two dimensions, of this Foster, Greer and Thorbecke index with separate poverty aversion parameters for each dimension. This statistical method (the product kernel estimator) has the following main disadvantes: the curse of dimensionality, the complex selection of the bandwidth, and high computational costs.  In this  study, we focus on a bivariate radialsymmetric kernel estimator for the bi-dimensional extension of the Foster, Greer and Thorbecke index. Our new bivariate kernel estimator is developed with a classical bivariate kernel of Parzen-Rosenblatt of a probability density function (pdf) utilizing Riemann sums. We next provide complete asymptotic  behaviour by establishing both almost-sure uniform and  uniform mean square consistencies for the bivariate radialsymmetric kernel estimator. A simulation study indicates that the proposed bivariate kernel estimator performs favorably for small samples comparatively to the bivariate multiplicative kernel estimator.
VL  - 15
IS  - 5
ER  -

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