Linearity of regression for weak records, revisited

Rafal Karczewski, Jacek Wesołowski

Abstract:

Since many years characterization of distribution by linearity of regression of non-adjacent weak
records E(Wi+s|Wi) = β1Wi + β0 for discrete observations has been known to be a difficult question. L ́opez-
Bl ́azquez (2004) proposed an interesting idea of reducing it to the adjacent case and claimed to have the
characterization problem completely solved. We will explain that, unfortunately, there is a flaw in the
proof given in that paper. This flaw is related to fact that in some situations the operator responsible for
reduction of the non-adjacent case to the adjacent one is not injective. The operator is trivially injective
when β1 ∈ (0, 1). We show that when β1 ≥ 1 the operator is injective when s = 2, 3, 4. Therefore in these
cases the method proposed by L ́opez-Bl ́azquez is valid. We also show that the operator is not injective
when β1 ≥ 1 and s ≥ 5. Consequently, in this case the reduction methodology does not work and thus the
characterization problem remains open.