Wrapped distributions and measurement errors

Abstract

For a random variable X and α>0 let Uα{colon equals}nα (X)-X, where nα(x)=n∈Z iff x∈(nα-α/2, nα+α/2]. Random variables of this type are important in the theory of measurement errors. We derive formulas for the distribution of Uα and apply them to the case X∼N(θ,σ2). General conditions for the unimodality of Uα are given. The correlation of the measurement errors X-E (X) and Uα (X) is seen to be O (αj) with j depending on the smoothness and asymptotic behavior of the density of X. This gives a precise sense to the assertion that scale errors upwards and downwards are averagely well-balanced. In the normal case the density of Uα is shown to be constant up to {Mathematical expression}, as α→0. © 1984 Physica-Verlag.