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Approximations of \(\delta \)-Record Probabilities in i.i.d. and Trend Models

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Trends in Mathematical, Information and Data Sciences

Part of the book series: Studies in Systems, Decision and Control ((SSDC,volume 445))

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Abstract

We study the probability of occurrence of \(\delta \)-records in a model with linear trend. While this probability has been studied in the iid case, the existence of a trend makes its analysis much more involved. Asymptotic properties of this probability have been studied in the literature when the number of observations is large. However, no approximations are known when the number of observations is small. We propose a first order approximation as a function of both the values of \(\delta \) and the trend. We assess our results via Montecarlo simulations finding that the approximations are accurate for a small-moderate number of observations.

This work is dedicated to Leandro Pardo, an excellent mathematician, a better person and a great friend. Meeting Leandro is one of the best things that can happen to you in life.

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Acknowledgements

This research was supported by ACE210010, FB210005 basal funds from ANID-Chile and Grant MCIN/ AEI/10.13039/501100011033. ML, RG, FJL and GS are members of the research group Modelos Estocásticos of DGA.

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Authors and Affiliations

  1. Departamento de Métodos Estadísticos, Facultad de Ciencias, Universidad de Zaragoza, Zaragoza, Spain

    Miguel Lafuente, David Ejea, F. Javier López & Gerardo Sanz

  2. Departamento Ingeniería Matemática y Centro de Modelamiento Matemático (IRL 2807, CNRS), Universidad de Chile, Santiago, Chile

    Raúl Gouet

  3. Instituto de Biocomputación y Física de Sistemas Complejos (BIFI), Universidad de Zaragoza, 50018, Zaragoza, Spain

    F. Javier López & Gerardo Sanz

Authors
  1. Miguel Lafuente
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  2. David Ejea
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  3. Raúl Gouet
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  4. F. Javier López
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  5. Gerardo Sanz
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Corresponding author

Correspondence to Miguel Lafuente .

Editor information

Editors and Affiliations

  1. Department of Mathematics and Statistics, McMaster University, Hamilton, ON, Canada

    Narayanaswamy Balakrishnan

  2. Department of Statistics and OR and TM, University of Oviedo, Oviedo, Asturias, Spain

    María Ángeles Gil

  3. Department of Financial and Actuarial Economics & Statistics, Complutense University of Madrid, Madrid, Spain

    Nirian Martín

  4. Department of Statistics, Mathematics and Informatics, Miguel Hernández University of Elche, Elche, Spain

    Domingo Morales

  5. Department of Statistics and Operational Research, Complutense University of Madrid, Madrid, Spain

    María del Carmen Pardo

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Lafuente, M., Ejea, D., Gouet, R., López, F.J., Sanz, G. (2023). Approximations of \(\delta \)-Record Probabilities in i.i.d. and Trend Models. In: Balakrishnan, N., Gil, M.Á., Martín, N., Morales, D., Pardo, M.d.C. (eds) Trends in Mathematical, Information and Data Sciences. Studies in Systems, Decision and Control, vol 445. Springer, Cham. https://doi.org/10.1007/978-3-031-04137-2_8

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  • DOI: https://doi.org/10.1007/978-3-031-04137-2_8

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-04136-5

  • Online ISBN: 978-3-031-04137-2

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