and is the same as in the Britton, Fisher and Whitley’s formulation. The parameter ξ equals the difference between the distribution’s mean and mode and can be viewed as unnormed measure of skewness. The three parameterizations are mathematically equivalent, meaning that there is a strict relationship between the parameters and that it is possible to go from one parameterization to another. The following relationships hold: Multivariate Extensions The multivariate generalization of the split normal distribution was proposed by Villani and Larsson. They assume that each of the principal components has univariate split normal distribution with a different set of parameters μ, σ2 and σ1. Estimation of parameters John proposes to estimate the parameters using maximum likelihood method. He shows that the likelihood function can be expressed in an intensive form, in which the scale parameters σ1 and σ2 are a function of the location parameter μ. The likelihood in its intensive form is: and has to be maximized numerically with respect to a single parameter μ only. Given the maximum likelihood estimator the other parameters take values: where N is the number of observations. Villani and Larsson propose to use either maximum likelihood method or bayesian estimation and provide some analytical results for either univariate and multivariate case. Applications The split normal distribution has been used mainly in econometrics and time series. A remarkable area of application is the construction of the fan chart, a representation of the inflation forecast distribution reported by inflation targeting central banks around the globe. References Continuous distributions Normal distribution