Comparing Quasi Newton BFGS and Nelder Mead Algorithm for Box-Cox Transformation

Abstract

In parametric statistics test, there are some assumptions which must be fulfilled in order to make a valid conclusion. Some common assumptions in parametric statistics test are homoscedasticity or homogeneity of variance and assumption of normality. Many of us in social sciences deal with data that do not conform to assumptions of normality. To analyze non normal data with statistical test that require assumptions of normality, data should be transformed to a normal distribution. Box-Cox transformation is well known method and popular among statistician to get the normality (univariate and multivariate normal) because it doesn’t require knowledge about data characteristics or trial and error to transform any data. Box-Cox method use estimation of parameter λ to do transformation, λ ̂ is obtained by maximize the Box-Cox function. Numerical method like quasi newton BFGS is commonly used to maximize the Box-Cox function because this method has fast convergence property but it may fail to convergence in some circumstance. In other hand, direct optimization algorithm like Nelder Mead may still be convergence in case which quasi newton BFGS fail to convergence but this algorithm generally has slower convergence property. In this paper, we will compare efficiency in term of function evaluation and processing time of the two algorithms to maximize Box-Cox function on some scenario.