Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-28T04:15:36.641Z Has data issue: false hasContentIssue false
  • English
  • Français

A NEW CHARACTERIZATION OF THE NORMAL DISTRIBUTION AND TEST FOR NORMALITY

Published online by Cambridge University Press:  19 May 2015

Anil K. Bera ,
Antonio F. Galvao ,
Liang Wang  and
Zhijie Xiao
Anil K. Bera
Affiliation:
University of Illinois
Antonio F. Galvao
Affiliation:
University of Iowa
Liang Wang
Affiliation:
University of Wisconsin-Milwaukee
Zhijie Xiao*
Affiliation:
Boston College and Shandong University
*
*Address correspondence to Zhijie Xiao, Department of Economics, Boston College, Chestnut Hill, MA 02467; e-mail: xiaoz@bc.edu.
Get access Rights & Permissions [Opens in a new window]

Abstract

We study the asymptotic covariance function of the sample mean and quantile, and derive a new and surprising characterization of the normal distribution: the asymptotic covariance between the sample mean and quantile is constant across all quantiles, if and only if the underlying distribution is normal. This is a powerful result and facilitates statistical inference. Utilizing this result, we develop a new omnibus test for normality based on the quantile-mean covariance process. Compared to existing normality tests, the proposed testing procedure has several important attractive features. Monte Carlo evidence shows that the proposed test possesses good finite sample properties. In addition to the formal test, we suggest a graphical procedure that is easy to implement and visualize in practice. Finally, we illustrate the use of the suggested techniques with an application to stock return datasets.

Type
ARTICLES
Information
Econometric Theory , Volume 32 , Issue 5 , October 2016 , pp. 1216 - 1252
Copyright
Copyright © Cambridge University Press 2015 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Andrews, D.W.K. (1993) Tests for parameter instability and structural change with unknown change point. Econometrica 61, 821–856.Google Scholar
Andrews, D.W.K. & Ploberger, W. (1994) Optimal tests when a nuisance parameter is present only under the alternative. Econometrica 62, 1383–1414.Google Scholar
Aparicio, F. & Estrada, J. (1997) Empirical distributions of stock returns: Scandinavian securities markets, 1990–95, Discussion Paper, Department of Statistics and Econometrics, Carlos III University, Madrid, Spain.Google Scholar
Azzalini, A. (1985) A class of distributions which includes the normal ones. Scandinavian Journal of Statistics 12, 171–178.Google Scholar
Bai, J. & Ng, S. (2005) Tests for skewness, kurtosis, and normality for time series data. Journal of Business and Economic Statistics 23, 49–60.Google Scholar
Bera, A.K., Galvao, A.F., Wang, L., & Xiao, Z. (2014) A new characterization of the normal distribution and test for normality, http://dx.doi.org/10.2139/ssrn.CrossRefGoogle Scholar
Billingsley, P. (1999) Convergence of Probability Measures, 2nd ed. Wiley.Google Scholar
Boldin, M.V. (1982) Estimation of the distribution of noise in an autoregressive scheme, Theory of Probability and its Applications 27, 866–871.Google Scholar
Bontemps, C., & Meddahi, N. (2005) Testing normality: A GMM approach. Journal of Econometrics 124, 149–186.Google Scholar
Bowman, K.O., & Shenton, L.R. (1975) Omnibus test contours for departures from normality based on $\sqrt {b_1 } $ and b 2 . Biometrika 62, 243–250.Google Scholar
Chikuse, Y. (1990) Functional forms of characteristic functions and characterizations of multivariate distributions. Econometric Theory 6, 445–458.Google Scholar
Clark, P. (1973) A subordinated stochastic process model with finite variance for speculative prices. Econometrica 41, 135–155.Google Scholar
Corrado, C., & Su, T. (1996) Skewness and kurtosis in S&P 500 index returns implied by option prices. Journal of Financial Research 19, 175–192.Google Scholar
Epps, T.W. & Pulley, L.B. (1983) A test for normality based on the empirical characteristic function. Biometrika 70, 723–726.Google Scholar
Epps, T.W., Singleton, K., & Pulley, L.B. (1982) A test of separate families of distributions based on the empirical moment generating function. Biometrika 69, 391–399.Google Scholar
Fama, E. (1965) The behavior of stock market prices. Journal of Business 38, 34–105.Google Scholar
Ferguson, T.S. (1999) Asymptotic joint distribution of sample mean and a sample quantile, http://www.math.ucla.edu/tom/papers/unpublished/meanmed.pdf.Google Scholar
Fisher, R.A. (1930) The moments of the distribution for normal samples of measures of departure from normality. Proceedings of the Royal Society, Series A 130, 16–28.Google Scholar
Geary, R.C. (1947) Testing for normality. Biometrika 34, 209–242.Google Scholar
Gray, J. & French, D. (1990) Empirical comparisons of distributional models for stock index returns. Journal of Business Finance & Accounting 17, 451–459.Google Scholar
Hall, A. (1990) Lagrange multiplier tests for normality against semiparametric alternatives. Journal of Business and Economic Statistics 8, 417–426.Google Scholar
Jarque, C. & Bera, A. (1980) Efficient tests for normality, homoscedasticity and serial independence of regression residuals. Econometric Letters 6, 255–259.CrossRefGoogle Scholar
Kagan, A.M., Linnik, I., & Rao, C.R. (1973) Characterization Problems in Mathematical Statistics. Wiley.Google Scholar
Kiefer, N.M., & Salmon, M. (1983) Testing normality in econometric models. Economics Letters 11, 123–127.Google Scholar
Kilian, L. & Demiroglu, U. (2000) Residual-based tests for normality in autoregressive: Asymptotic theory and simulation evidence, Journal of Business and Economic Statistics 18, 40–50.Google Scholar
Koenker, R. & Bassett, G.W. (1978) Regression quantiles. Econometrica 46, 33–49.Google Scholar
Kolmogorov, A. (1933) Sulla determinazione empiricadi una legge di distribuzione. Giornale dell’Istituto Italiano degli Attuari 4, 83–91.Google Scholar
Kotz, S., Kozubowski, T., & Podgorski, K. (2001) The Laplace Distribution and Generalizations: A Revisit with Applications to Communications, Economics, Engineering, and Finance. Birkhauser.Google Scholar
Koutrouvelis, I.A. (1980) A goodness-of-fit test of simple hypotheses based on the empirical characteristic function. Biometrika 67, 238–240.Google Scholar
Koutrouvelis, I.A. & Kellermeier, J. (1981) A goodness-of-fit test based on the empirical characteristic function when parameters must be estimated, Journal of the Royal Statistical Society, Series B 43, 173–176.Google Scholar
Laha, R.G. (1957) On a characterization of the normal distribution from properties of suitable linear statistics. Annals of Mathematical Statistics 28, 126–139.Google Scholar
Laplace, P. (1810) Memoire sur les approximations des formules qui sont fonctions de tres grands nombers et sur leur application aux probabilites, Mémoires de l’Acadmie royale des Sciences de Paris, pp. 353–415, 559–565, Reprinted in Laplace (1878–1972), vol. 12, pp. 301–353.Google Scholar
Laplace, P. (1818) Deuxieme Supplement a la Theorie Analytique des Probabilites. Courcier, Reprinted (1847) in Oeuvres de Laplace 7, pp. 569–623. Imprimerie Royale; (1886) in Oeuvres Completes de Laplace 7, pp. 531–80. Gauthier-Villars.Google Scholar
Lilliefors, H. (1967) On the Kolmogorov-Smirnov test for normality with mean and variance unknown. Journal of the American Statistical Association 62, 399–402.Google Scholar
Lobato, I.N. & Velasco, C. (2004) A simple test of normality for time series. Econometric Theory 20, 671–689.CrossRefGoogle Scholar
Lukacs, E. (1942) A characterization of the normal distribution. Annals of Mathematical Statistics 13, 91–93.Google Scholar
Mandelbrot, B. (1963) The variation of certain speculative prices. Journal of Business 36, 394–419.Google Scholar
Mathai, A.M. & Pederzoli, G. (1978) Characterizations of the Normal Probability Law. Halsted Press.Google Scholar
Peters, E. (1991) Chaos and Order in the Capital Markets. A New View of Cycles, Prices, and Market Volatility. Wiley.Google Scholar
Petrov, V.V. (1995) Limit Theorems of Probability Theory. Clarendon Press.Google Scholar
Premaratne, G. & Bera, A. (2005) A test for symmetry with leptokurtic financial data. Journal of Financial Econometrics 3, 169–187.Google Scholar
Rao, J.N.K. (1958) A characterization of the normal distribution. Annals of Mathematical Statistics, 29, 914–919.Google Scholar
Richardson, M. & Smith, T. (1993) A test for multivariate normality in stock returns. Journal of Business 66, 295–321.Google Scholar
Riedel, M. (1985) On a characterization of the normal distribution by means of identically distributed linear forms. Journal of Multivariate Analysis 16, 241–252.Google Scholar
Shapiro, S.S. & Francia, R.S. (1972) An approximate analysis of variance test for normality. Journal of the American Statistical Association 67, 215–216.Google Scholar
Shapiro, S.S. & Wilk, M.B. (1965) An analysis of variance test for normality (complete samples). Biometrika 52, 591–611.CrossRefGoogle Scholar
Smirnov, N.V. (1939) Sur les Ecarts de la Courbe de Distribution Empirique, French/Russian Su Matematiceskii Sbornik N.S. 6, 3–26.Google Scholar
Stigler, S.M. (1973) Laplace, Fisher, and the discovery of the concept of sufficiency. Biometrika 60, 439–445.Google Scholar
Thode, H.C. (2002) Testing for Normality. Marcel Dekker.Google Scholar
Tukey, J. (1965) What part of the sample contains the information? Proceedings of the National Academy of Sciences 53, 127–134.Google Scholar
van der Klaauw, B. & Koning, R. H. (2003) Testing the normality assumption in the sample selection model with an application to travel demand. Journal of Business and Economic Statistics 21, 31–42.Google Scholar
Wellner, J.A. (1992) Empirical process in action: A review. International Statistical Review 60, 247–256.Google Scholar
White, H. & MacDonald, G.M. (1980) Some large sample tests for nonnormality in the linear regression model, Journal of the American Statistical Association 75, 16–28.Google Scholar
Xu, J.L. (1998) On a characterization of the normal distribution by a property of order statistics, Sankhyā 60, 145–149.Google Scholar
Zheng, J.X. (2000) A Consistent test of conditional parametric distributions. Econometric Theory 16, 667–691.Google Scholar