The significance of using quantile-based entropies lies in their ability to capture different characteristics compared to the approach using distribution functions. This article is centered on examining the quantile-based Rényi entropy and its time-dependent versions for record statistics. The additional parameter \(\mu\) in this entropy enables a broader interpretation, with the Shannon entropy being a specific case, enabling the entropy metric to be tailored to the precise properties of the data being studied. The study investigates these entropies across various generalized models that lack a probability density function or a cumulative distribution function and further introduces an alternative representation for these entropy quantities centered on the expected values of other random variables. This, in turn, leads to a finding on the bound of the quantile Rényi residual entropy of the \(n^{th}\) record in terms of the mode of the truncated Gamma distribution. Additionally, the study presents a unique characterization result for Rényi entropy using its residual form and hazard quantile function.
