Rényi entropy of past lifetime from lower $k$-record values

This paper explored the concept of past Rényi entropy within the context of $k$-record values. We began by introducing a representation of the past Rényi entropy for the $n$-th lower $k$-record values, sampled from any continuous distribution function $F,$ concerning the past Rényi entropy of the $n$-th lower $k$-record values sampled from a uniform distribution. Then, we delved into the examination of the monotonicity properties of the past Rényi entropy of $k$-record values. Specifically, we focused on the aging properties of the component lifetimes and investigated how they impacted the monotonicity of the past Rényi entropy. Additionally, we derived an expression for the $n$-th lower $k$-records in terms of the past Rényi entropy, specifically when the first lower $k$-record was less than a specified threshold level, and then investigated several properties of the given formula.