Inequality for the variance of an asymmetric loss

TL;DR

This paper investigates the variance of asymmetric loss functions in forecast errors, providing a new inequality under specific distributional assumptions to improve understanding of forecast error variability.

Contribution

It introduces an inequality for the variance of asymmetric loss in forecast errors assuming symmetric, non-increasing distributions, and solves a related minimization problem.

Findings

Derived an inequality for the variance of asymmetric loss

Solved a minimization problem under specified distributional assumptions

Provides theoretical insights into forecast error variability

Abstract

We assume that the forecast error follows a probability distribution which is symmetric and monotonically non-increasing on non-negative real numbers, and if there is a mismatch between observed and predicted value, then we suffer a loss. Under the assumptions, we solve a minimization problem with an asymmetric loss function. In addition, we give an inequality for the variance of the loss.