Computational Statistics & Data Analysis 49 (2005) 741 – 756 www.elsevier.com/locate/csda A large deviation approach to normality testing J. Sigut ∗ , J. Piñeiro, L. Moreno, J. Estevez, R. Aguilar, R. Marichal Department of Physics, Electronics and Systems, Facultad de Fisica, University of La Laguna,Tenerife, La Laguna, 38200, Spain Received 3 September 2003; received in revised form 25 May 2004; accepted 26 May 2004 Available online 23 June 2004 Abstract A new approach to the classical problem of determining whether or not a set of data has been sampled from a univariate normal distribution is considered. The problem is posed in a pattern recognition framework and the concept of an asymptotically optimal expert is introduced. Some results of Large Deviation theory are used in the selection of experts so that their combined abilities lead to a suitable discrimination procedure. The fact that good asymptotical properties can be extended to finite samples with small sizes is the basis for the rest of the work. The performance of the proposed procedure is also compared to that of some known normality tests. An additional advantage of the procedure is the possibility of computing a reliable measure of support such as the Bayesian posterior probabilities as opposed to the P -values. Furthermore, some extra information can be obtained concerning the type of deviation from normality which is present in the data. © 2004 Elsevier B.V.All rights reserved. Keywords: Normality testing; Large Deviation theory; Probabilistic inference; Bayesian posterior probabilities; Kullback–Leibler distance 1. Introduction It is widely known that the assumption of normality underlies many important techniques in statistical analysis. The analytical simplicity of the normal distribution is one of the main reasons why this distribution is chosen. ∗ Corresponding author. Tel.: +34-922318283; fax: +34-922318288. E-mail address: sigut@cyc.ull.es (J. Sigut) 0167-9473/$ - see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.csda.2004.05.037