Wrapped distributions and measurement errors

Abstract

For a random variableX and α>0 letUα≔nα (X)−X, wherenα(x)=n∈Z iffx∈(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 ofUα and apply them to the case X∼N(θ,σ2). General conditions for the unimodality ofUα are given. The correlation of the measurement errorsX−E (X) andUα (X) is seen to beO (αj) withj depending on the smoothness and asymptotic behavior of the density ofX. This gives a precise sense to the assertion that scale errors upwards and downwards are averagely well-balanced. In the normal case the density ofUα is shown to be constant up to\(o (e^{ - C\alpha ^{ - 2} } )\), as α→0.