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

We present the basic distribution theory of δ-record values, R <Subscript> n,δ </Subscript>, 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 <Subscript> n,δ </Subscript> 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. Copyright Sociedad de Estadística e Investigación Operativa 2013