Functional Weak Laws for the Weighted Mean Losses or Gains and Applications
Affiliation(s)
1 LERSTAD, Université Gaston Berger de Saint-Louis, Saint-Louis, Sénégal. 2 LSTA, Université Pierre et Marie Curie, Paris, France. 3 Ecole Normale Supérieure, Dakar, Sénégal.
1 LERSTAD, Université Gaston Berger de Saint-Louis, Saint-Louis, Sénégal. 2 LSTA, Université Pierre et Marie Curie, Paris, France. 3 Ecole Normale Supérieure, Dakar, Sénégal.
ABSTRACT
In this paper, we show that many risk measures arising in Actuarial Sciences, Finance, Medicine, Welfare analysis, etc. are gathered in classes of Weighted Mean Loss or Gain (WMLG) statistics. Some of them are Upper Threshold Based (UTH) or Lower Threshold Based (LTH). These statistics may be time-dependent when the scene is monitored in the time and depend on specific functions w and d. This paper provides time-dependent and uniformly functional weak asymptotic laws that allow temporal and spatial studies of the risk as well as comparison among statistics in terms of dependence and mutual influence. The results are particularized for usual statistics like the Kakwani and Shorrocks ones that are mainly used in welfare analysis. Data-driven applications based on pseudo-panel data are provided.
In this paper, we show that many risk measures arising in Actuarial Sciences, Finance, Medicine, Welfare analysis, etc. are gathered in classes of Weighted Mean Loss or Gain (WMLG) statistics. Some of them are Upper Threshold Based (UTH) or Lower Threshold Based (LTH). These statistics may be time-dependent when the scene is monitored in the time and depend on specific functions w and d. This paper provides time-dependent and uniformly functional weak asymptotic laws that allow temporal and spatial studies of the risk as well as comparison among statistics in terms of dependence and mutual influence. The results are particularized for usual statistics like the Kakwani and Shorrocks ones that are mainly used in welfare analysis. Data-driven applications based on pseudo-panel data are provided.
KEYWORDS
Empirical Process, Time Dependent Process, Weak Theory, Risk Measures, Poverty Index, Loss Function, Economic Welfare
Empirical Process, Time Dependent Process, Weak Theory, Risk Measures, Poverty Index, Loss Function, Economic Welfare
Cite this paper
Lo, G. , Sall, S. and Mergane, P. (2015) Functional Weak Laws for the Weighted Mean Losses or Gains and Applications. Applied Mathematics, 6, 847-863. doi: 10.4236/am.2015.65079.
Lo, G. , Sall, S. and Mergane, P. (2015) Functional Weak Laws for the Weighted Mean Losses or Gains and Applications. Applied Mathematics, 6, 847-863. doi: 10.4236/am.2015.65079.
References
[1] Lo, G.S. (2013) The Generalized Poverty Index. Far East Journal of Theoretical Statistics, 42, 1-22.
http://pphmj.com/journals/fjst.htm [2] Artzner, P., Delbaen, F., Eber, J.M. and Heath, D. (1999) Coherent Measures of Risk. Mathematical Finance, 9, 203-228.
http://dx.doi.org/10.1111/1467-9965.00068 [3] Sen, A.K. (1976) Poverty: An Ordinal Approach to Measurement. Econometrica, 44, 219-231.
http://dx.doi.org/10.2307/1912718 [4] Zheng, B. (1997) Aggregate Poverty Measures. Journal of Economic Surveys, 11, 123-162.
http://dx.doi.org/10.1111/1467-6419.00028 [5] van der Vaart, A.W. and Wellner, J.A. (1996) Weak Convergence and Empirical Processes With Applications to Statistics. Springer, New York. [6] Shorrocks, A.F. (1995) Revisiting the Sen Poverty Index. Econometrica, 63, 1225-1230.
http://dx.doi.org/10.2307/2171728 [7] Thon, D. (1979) On Measuring Poverty. Review of Income and Wealth, 25, 429-440.
http://dx.doi.org/10.1111/j.1475-4991.1979.tb00117.x [8] Kakwani, N. (1980) On a Class of Poverty Measures. Econometrica, 48, 437-446.
http://dx.doi.org/10.2307/1911106 [9] Sall, S.T. and Lo, G.S. (2009) Uniform Weak Convergence of the Time-Dependent Poverty Measure for Continuous Longitudinal Data. Brazilian Journal of Probability and Statistics, 24, 457-467.
http://dx.doi.org/10.1214/08-BJPS101 [10] Lo, G.S. and Sall, S.T. (2009) Uniform Weak Convergence of Non Randomly Weighted Poverty Measures for Longitudinal Data. 57th ISI Session. [11] Lo, G.S. and Sall, S.T. (2010) Asymptotic Representation Theorems for Poverty Indices. Afrika Statistika, 5, 238-244. [12] Lo, G.S. (2010) A Simple Note on Some Empirical Stochastic Process as a Tool in Uniform L-Statistics Weak Laws. Afrika Statistika, 5, 245-251. [13] Shorack, G.R. and Wellner, J.A. (1986) Empirical Processes with Applications to Statistics. Wiley-Interscience, New York.
[1] Lo, G.S. (2013) The Generalized Poverty Index. Far East Journal of Theoretical Statistics, 42, 1-22.
http://pphmj.com/journals/fjst.htm [2] Artzner, P., Delbaen, F., Eber, J.M. and Heath, D. (1999) Coherent Measures of Risk. Mathematical Finance, 9, 203-228.
http://dx.doi.org/10.1111/1467-9965.00068 [3] Sen, A.K. (1976) Poverty: An Ordinal Approach to Measurement. Econometrica, 44, 219-231.
http://dx.doi.org/10.2307/1912718 [4] Zheng, B. (1997) Aggregate Poverty Measures. Journal of Economic Surveys, 11, 123-162.
http://dx.doi.org/10.1111/1467-6419.00028 [5] van der Vaart, A.W. and Wellner, J.A. (1996) Weak Convergence and Empirical Processes With Applications to Statistics. Springer, New York. [6] Shorrocks, A.F. (1995) Revisiting the Sen Poverty Index. Econometrica, 63, 1225-1230.
http://dx.doi.org/10.2307/2171728 [7] Thon, D. (1979) On Measuring Poverty. Review of Income and Wealth, 25, 429-440.
http://dx.doi.org/10.1111/j.1475-4991.1979.tb00117.x [8] Kakwani, N. (1980) On a Class of Poverty Measures. Econometrica, 48, 437-446.
http://dx.doi.org/10.2307/1911106 [9] Sall, S.T. and Lo, G.S. (2009) Uniform Weak Convergence of the Time-Dependent Poverty Measure for Continuous Longitudinal Data. Brazilian Journal of Probability and Statistics, 24, 457-467.
http://dx.doi.org/10.1214/08-BJPS101 [10] Lo, G.S. and Sall, S.T. (2009) Uniform Weak Convergence of Non Randomly Weighted Poverty Measures for Longitudinal Data. 57th ISI Session. [11] Lo, G.S. and Sall, S.T. (2010) Asymptotic Representation Theorems for Poverty Indices. Afrika Statistika, 5, 238-244. [12] Lo, G.S. (2010) A Simple Note on Some Empirical Stochastic Process as a Tool in Uniform L-Statistics Weak Laws. Afrika Statistika, 5, 245-251. [13] Shorack, G.R. and Wellner, J.A. (1986) Empirical Processes with Applications to Statistics. Wiley-Interscience, New York.