A General Sharp-Peak Symmetric Distribution: A Smooth Bridge Between L1 and L2 Estimation

This paper introduces a three-parameter symmetric family that provides a smooth transition between least absolute (L1) and least square (L2) estimation. The distribution is defined by the location (µ), scale (σ), and shape (τ) parameters. For small values of τ, the distribution approaches the double exponential (Laplace) distribution; for large values of τ, it approaches the normal distribution. The distribution has a smooth, bounded psi-function, 𝜓(.), that preserves the rank order of the data and can be viewed as a smooth analogue of Huber’s 𝜓(.) function.