Linearity of regression for weak records, revisited

TL;DR

This paper revisits the problem of characterizing distributions via linear regression of weak records, identifying limitations in previous methods and clarifying when the reduction approach is valid or fails.

Contribution

It critically analyzes Lopez-Blazquez's reduction method, clarifies its validity range, and shows where the characterization problem remains unsolved.

Findings

The reduction method is valid when 0<β₁<1.

The operator is injective for β₁≥1 when s=2,3,4.

The method fails for β₁≥1 when s≥5.

Abstract

Since many years characterization of distribution by linearity of regression of non-adjacent weak records E(W_{i+s}|W_i) = \beta_1 W_i+\beta_0 for discrete observations has been known to be a difficult question. Lopez- Blazquez (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 major aw in the proof given in that paper. This aw 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 0<\beta_1<1. We show that when \beta_1>=1 the operator is injective when s = 2, 3, 4. Therefore in these cases the method proposed by Lopez-Blazquez is valid. We also show that the operator is not injective when \beta_1 >=1 and s >= 5. Consequently, in this case the reduction methodology does not work and thus the characterization problem remains open.