  Bairamov et al. (Aust N Z J Stat 47:543-547, 2005) characterize the
exponential distribution in terms of the regression of a function of a record
value with its adjacent record values as covariates. We extend these results to
the case of non-adjacent covariates. We also consider a more general setting
involving monotone transformations. As special cases, we present
characterizations involving weighted arithmetic, geometric, and harmonic means.
