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# Applications of Information Measures to Assess Convergence in the Central Limit Theorem

## Author

Listed:
• Ranjani Atukorala

()

• Maxwell L. King

()

• Sivagowry Sriananthakumar

()

## Abstract

The Central Limit Theorem (CLT) is an important result in statistics and econometrics and econometricians often rely on the CLT for inference in practice. Even though, different conditions apply to different kinds of data, the CLT results are believed to be generally available for a range of situations. This paper illustrates the use of the Kullback-Leibler Information (KLI) measure to assess how close an approximating distribution is to a true distribution in the context of investigating how different population distributions affect convergence in the CLT. For this purpose, three different non-parametric methods for estimating the KLI are proposed and investigated. The main findings of this paper are 1) the distribution of the sample means better approximates the normal distribution as the sample size increases, as expected, 2) for any fixed sample size, the distribution of means of samples from skewed distributions converges faster to the normal distribution as the kurtosis increases, 3) at least in the range of values of kurtosis considered, the distribution of means of small samples generated from symmetric distributions is well approximated by the normal distribution, and 4) among the nonparametric methods used, Vasicek's (1976) estimator seems to be the best for the purpose of assessing asymptotic approximations. Based on the results of the paper, recommendations on minimum sample sizes required for an accurate normal approximation of the true distribution of sample means are made.

## Suggested Citation

• Ranjani Atukorala & Maxwell L. King & Sivagowry Sriananthakumar, 2014. "Applications of Information Measures to Assess Convergence in the Central Limit Theorem," Monash Econometrics and Business Statistics Working Papers 29/14, Monash University, Department of Econometrics and Business Statistics.
• Handle: RePEc:msh:ebswps:2014-29
as

## References listed on IDEAS

as
1. Chang, Ching-Hui & Lin, Jyh-Jiuan & Pal, Nabendu & Chiang, Miao-Chen, 2008. "A Note on Improved Approximation of the Binomial Distribution by the Skew-Normal Distribution," The American Statistician, American Statistical Association, vol. 62, pages 167-170, May.
2. White,Halbert, 1996. "Estimation, Inference and Specification Analysis," Cambridge Books, Cambridge University Press, number 9780521574464.
3. Evans, Merran, 1992. "Robustness of size of tests of autocorrelation and heteroscedasticity to nonnormality," Journal of Econometrics, Elsevier, vol. 51(1-2), pages 7-24.
4. Goncalves, Silvia & White, Halbert, 2005. "Bootstrap Standard Error Estimates for Linear Regression," Journal of the American Statistical Association, American Statistical Association, vol. 100, pages 970-979, September.
5. Vuong, Quang H, 1989. "Likelihood Ratio Tests for Model Selection and Non-nested Hypotheses," Econometrica, Econometric Society, vol. 57(2), pages 307-333, March.
6. King, M.L. & Harris, D.C., 1995. "The Applications of the Durbin-Watson Test to the Dynamic Regression Model Under Normal and Non-Normal Errors," Monash Econometrics and Business Statistics Working Papers 6/95, Monash University, Department of Econometrics and Business Statistics.
7. Shilane David & Evans Steven N & Hubbard Alan E., 2010. "Confidence Intervals for Negative Binomial Random Variables of High Dispersion," The International Journal of Biostatistics, De Gruyter, vol. 6(1), pages 1-41, March.
8. White, Halbert, 1982. "Maximum Likelihood Estimation of Misspecified Models," Econometrica, Econometric Society, vol. 50(1), pages 1-25, January.
Full references (including those not matched with items on IDEAS)

### Keywords

Kullback-Leibler Information; Central Limit Theorem; skewness and kurtosis;

### JEL classification:

• C1 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General
• C2 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables
• C4 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods: Special Topics
• C5 - Mathematical and Quantitative Methods - - Econometric Modeling

### NEP fields

This paper has been announced in the following NEP Reports:

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