Abstract
Given a sequence of random variables {X n ,n≥1} and δ∈ℝ, an observation X n is a δ-record if X n >max {X 1,…,X n−1}+δ. We obtain, for δ≤0, weak and strong laws of large numbers for the counting process of δ-records among the first n observations from a sequence of independent identically distributed random variables, with common distribution F, possibly discontinuous. We provide examples of our results in the context of common probability distributions. Finally, we show how δ-records can be used for maximum likelihood estimation.
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In Memoriam. Miguel San Miguel.
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Gouet, R., López, F.J. & Sanz, G. On δ-record observations: asymptotic rates for the counting process and elements of maximum likelihood estimation. TEST 21, 188–214 (2012). https://doi.org/10.1007/s11749-011-0242-6
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DOI: https://doi.org/10.1007/s11749-011-0242-6