Letter to the Editor—A Simple Proof of a Theorem in Exponential Smoothing

Abstract

A simple proof is given of a theorem in exponential smoothing that is sometimes referred to as “the fundamental theorem”. This theorem states that a polynomial, of degree N at most, minimizes the exponentially-discounted sum of squares of the errors if and only if it satisfies the set of N + 1 equations obtained by equating the i th iteration of the exponentially-discounted average of the sequence to the corresponding average of the polynomial, i = 1, 2, …, N + 1.