Distribution theory of δ-record values. Case δ≤0

We present the basic distribution theory of δ -record values, R n , δ , δ ≤0, from a sequence of independent and identically distributed random variables from an absolutely continuous parent. We obtain recurrent formulas for the density function of R n , δ and a representation for this random variable that in some sense is similar to Tata’s representation for ordinary records. We also give the probability function of inter- δ -record times and some of its properties. We give examples of our results and also some elements of inference based on δ -records.