Affiliations: [a] Statistics and Reporting Unit, RMIT University, Melbourne, Australia | [b] Department of Econometrics and Business Statistics, Monash University, Melbourne, Australia | [c] School of Economics, Finance and Marketing, RMIT University, Melbourne, Australia
Correspondence:
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Corresponding author: Sivagowry Sriananthakumar, School of Economics, Finance and Marketing, RMIT University, GPO Box 2476 Melbourne 3001, Australia. Tel.: +61 39925 1456; Fax: +61 39925 5986; E-mail:sivagowry.sriananthakumar@rmit.edu.au
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 [33] estimator seems to be the best for the purpose of assessing
asymptotic approximations. Based on the results of this paper,
recommendations on minimum sample sizes required for an accurate normal
approximation of the true distribution of sample means are made.
Keywords: Kullback-Leibler information, central limit theorem, skewness and kurtosis
