Asymptotic analysis and approximations of parameterized univariate Gaussian splitting

Abstract Splitting a Gaussian into a Gaussian mixture is a common tactic for managing the impact of a non-linear transformation on the Gaussian. To obtain a mixture model that approximates the Gaussian accurately we minimise the squared Euclidean norm between the two. For homoscedastic mixtures with fixed spacing between mixands, we obtain asymptotic approximations for this statistical distance under various limits of the mixture parameters. Based on these expressions, we derive an approximation for the mixand spacing that yields the minimal statistical distance. Extensive numerical investigations show our asymptotic results have good predictive power on the behaviour of the splitting algorithm. We further show the asymptotic results can be used to design and parameterize a splitting algorithm with close to optimal accuracy.