High Moments Jarque-Bera Tests for Arbitrary Distribution Functions
Affiliation(s)
1 Laboratoire de Statistique Théorique et Appliquée (LSTA), Université Pierre et Marie Curie, Paris, France. 2 LERSTAD, Université Gaston Berger de Saint-Louis, Saint-Louis, Senegal.
1 Laboratoire de Statistique Théorique et Appliquée (LSTA), Université Pierre et Marie Curie, Paris, France. 2 LERSTAD, Université Gaston Berger de Saint-Louis, Saint-Louis, Senegal.
ABSTRACT
The Jarque-Bera’s fitting test for normality is a celebrated and powerful one. In this paper, we consider general Jarque-Bera tests for any distribution function (df) having at least 4k finite moments for k ≥ 2. The tests use as many moments as possible whereas the JB classical test is supposed to test only skewness and kurtosis for normal variates. But our results unveil the relations between the coeffients in the JB classical test and the moments, showing that it really depends on the first eight moments. This is a new explanation for the powerfulness of such tests. General Chi-square tests for an arbitrary model, not only normal, are also derived. We make use of the modern functional empirical processes approach that makes it easier to handle statistics based on the high moments and allows the generalization of the JB test both in the number of involved moments and in the underlying distribution. Simulation studies are provided and comparison cases with the Kolmogorov-Smirnov’s tests and the classical JB test are given.
The Jarque-Bera’s fitting test for normality is a celebrated and powerful one. In this paper, we consider general Jarque-Bera tests for any distribution function (df) having at least 4k finite moments for k ≥ 2. The tests use as many moments as possible whereas the JB classical test is supposed to test only skewness and kurtosis for normal variates. But our results unveil the relations between the coeffients in the JB classical test and the moments, showing that it really depends on the first eight moments. This is a new explanation for the powerfulness of such tests. General Chi-square tests for an arbitrary model, not only normal, are also derived. We make use of the modern functional empirical processes approach that makes it easier to handle statistics based on the high moments and allows the generalization of the JB test both in the number of involved moments and in the underlying distribution. Simulation studies are provided and comparison cases with the Kolmogorov-Smirnov’s tests and the classical JB test are given.
KEYWORDS
Asymptotic Distribution, Asymptotic Statistical Tests, Normality Tests, Functional Empirical Processes
Asymptotic Distribution, Asymptotic Statistical Tests, Normality Tests, Functional Empirical Processes
Cite this paper
Lo, G. , Thiam, O. and Haidara, M. (2015) High Moments Jarque-Bera Tests for Arbitrary Distribution Functions. Applied Mathematics, 6, 707-716. doi: 10.4236/am.2015.64066.
Lo, G. , Thiam, O. and Haidara, M. (2015) High Moments Jarque-Bera Tests for Arbitrary Distribution Functions. Applied Mathematics, 6, 707-716. doi: 10.4236/am.2015.64066.
References
[1] Jarque, C.M. and Bera, A.K. (1987) A Test for Normality of Observations and Regression Residuals. International Statistical Review, 55, 163-172.
http://dx.doi.org/10.2307/1403192 [2] McNeil, A.J., Frey, R. and Embrechts, P. (2005) Quantitative Risk Management: Concepts, Techniques and Tools. Princeton University Press, Princeton and Oxford. [3] Bowman, K.O. and Shenton, L.R. (1975) Omnibus Contours for Departures from Normality Based on and b2. Biometrika, 62, 243-250. [4] D’Agostino, R.B. (1971) An Omnibus Test for Normality for Moderate and Large Size Samples. Biometrika, 58, 341-348.
http://dx.doi.org/10.1093/biomet/58.2.341 [5] D'Agostino, R.B. and Pearson, E.S. (1973) Tests for Departure from Normality: Empirical Results for the Distributions of b2 and . Biometrika, 60, 613-622. [6] D’Agostino, R.B. and Tietjen, G.L. (1973) Approaches to the Null Distribution of . Biometrika, 60, 169-173.
http://dx.doi.org/10.1093/biomet/60.1.169 [7] van der Vaart, A.W. and Wellner, J.A.(1996) Weak Convergence and Empirical Processes with Applications to Statistics. Springer, New-York. [8] Loeve, M. (1977) Probability Theory I. Springer-Verlag, Berlin.
[1] Jarque, C.M. and Bera, A.K. (1987) A Test for Normality of Observations and Regression Residuals. International Statistical Review, 55, 163-172.
http://dx.doi.org/10.2307/1403192 [2] McNeil, A.J., Frey, R. and Embrechts, P. (2005) Quantitative Risk Management: Concepts, Techniques and Tools. Princeton University Press, Princeton and Oxford. [3] Bowman, K.O. and Shenton, L.R. (1975) Omnibus Contours for Departures from Normality Based on and b2. Biometrika, 62, 243-250. [4] D’Agostino, R.B. (1971) An Omnibus Test for Normality for Moderate and Large Size Samples. Biometrika, 58, 341-348.
http://dx.doi.org/10.1093/biomet/58.2.341 [5] D'Agostino, R.B. and Pearson, E.S. (1973) Tests for Departure from Normality: Empirical Results for the Distributions of b2 and . Biometrika, 60, 613-622. [6] D’Agostino, R.B. and Tietjen, G.L. (1973) Approaches to the Null Distribution of . Biometrika, 60, 169-173.
http://dx.doi.org/10.1093/biomet/60.1.169 [7] van der Vaart, A.W. and Wellner, J.A.(1996) Weak Convergence and Empirical Processes with Applications to Statistics. Springer, New-York. [8] Loeve, M. (1977) Probability Theory I. Springer-Verlag, Berlin.