Distribution theory of \(\delta\)-record values. Case \(\delta\leq 0\) (Q384770)

Assuming \(X_1, X_2, \ldots\) to be a sequence of independent and identically distributed random variables, \(M_j = \max\{X_1, \dots, X_j\}\), \(j \geq 1\), \(T_{1, \delta} = 1\), \(T_{k, \delta} = \min\{j > T_{k-1, \delta}: X_j > M_{j-1} + \delta\}\), \(k > 1,\) the \(k\)th \(\delta\)-record of the sequence is the random variable \(R_{k, \delta} = X_{T_{k, \delta}}\), \(k \geq 1\). The case \(\delta = 0\) corresponds to the ordinary upper records. Assuming \(\delta < 0\), the authors obtain recurrent formula for the density of \(R_{k, \delta} \), study the distribution of inter-\(\delta\)-record times, give a representation for \(\delta\)-records, and show the abundance of records among \(\delta\)-records. The distribution theory studied is applied to some common distributions like exponential, Gumbel and uniform and some basic inferential aspects for \(\delta\)-records are considered.