IDEAS home Printed from https://ideas.repec.org/a/eee/phsmap/v392y2013i5p1080-1089.html
   My bibliography  Save this article

On the sampling distribution of Allan factor estimator for a homogeneous Poisson process and its use to test inhomogeneities at multiple scales

Author

Listed:
  • Serinaldi, Francesco
  • Kilsby, Chris G.

Abstract

The Allan factor (AF) is a statistic widely used to assess if the rate of occurrences of an event tends to cluster and show persistence in a range of space and/or time scales. For a homogeneous Poisson process, the relationship between AF and the space/time is expected to be constant, thus denoting the lack of clustering and persistence in the occurrence process. However, in time series analysis, conclusions about the persistence of the underlying process have been usually drawn by visual inspection of the diagrams of AF estimates versus scale, without applying any formal statistical test. This study investigates the sampling distribution function of the AF estimator when the underlying process is homogeneous Poissonian. Monte Carlo simulations show that the distribution of the AF estimator is described by a gamma distribution whose mean and variance can be deduced by the delta method. Therefore, the derived analytical distribution of the AF estimator can be used to build a formal statistical test to evaluate the significance of the AF fluctuation against the Poissonian hypothesis across a range of space/time scales. As an example, we apply the AF-based test to analyse the time series of the number of rainfall observations exceeding fixed high thresholds in order to study the properties of the rate of occurrence of the extreme values over a wide range of time scales.

Suggested Citation

  • Serinaldi, Francesco & Kilsby, Chris G., 2013. "On the sampling distribution of Allan factor estimator for a homogeneous Poisson process and its use to test inhomogeneities at multiple scales," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 392(5), pages 1080-1089.
    • Handle: RePEc:eee:phsmap:v:392:y:2013:i:5:p:1080-1089
    DOI: 10.1016/j.physa.2012.11.015
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0378437112009806
    Download Restriction: Full text for ScienceDirect subscribers only. Journal offers the option of making the article available online on Science direct for a fee of $3,000
    File URL: https://libkey.io/10.1016/j.physa.2012.11.015?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Telesca, Luciano & Mohamed, Abuo El-Ela Amin & ElGabry, Mohamed & El-hady, Sherif & Abou Elenean, Kamal M., 2012. "Time dynamics in the point process modeling of seismicity of Aswan area (Egypt)," Chaos, Solitons & Fractals, Elsevier, vol. 45(1), pages 47-55.
    2. C. Anteneodo & R. D. Malmgren & D. R. Chialvo, 2010. "Poissonian bursts in e-mail correspondence," The European Physical Journal B: Condensed Matter and Complex Systems, Springer;EDP Sciences, vol. 75(3), pages 389-394, June.
    3. Stasinopoulos, D. Mikis & Rigby, Robert A., 2007. "Generalized Additive Models for Location Scale and Shape (GAMLSS) in R," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 23(i07).
    4. Yuan, Ying & Zhuang, Xin-tian & Liu, Zhi-ying & Huang, Wei-qiang, 2012. "Time-clustering behavior of sharp fluctuation sequences in Chinese stock markets," Chaos, Solitons & Fractals, Elsevier, vol. 45(6), pages 838-845.
    5. Serinaldi, Francesco, 2011. "Distributional modeling and short-term forecasting of electricity prices by Generalized Additive Models for Location, Scale and Shape," Energy Economics, Elsevier, vol. 33(6), pages 1216-1226.
    6. Telesca, Luciano & Lovallo, Michele, 2008. "Analysis of the temporal properties in car accident time series," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 387(13), pages 3299-3304.
    7. Telesca, Luciano & Bernardi, Marina & Rovelli, Cinzia, 2005. "Intra-cluster and inter-cluster time correlations in lightning sequences," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 356(2), pages 655-661.
    8. R. A. Rigby & D. M. Stasinopoulos, 2005. "Generalized additive models for location, scale and shape," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 54(3), pages 507-554, June.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Micha{l} Narajewski & Florian Ziel, 2020. "Ensemble Forecasting for Intraday Electricity Prices: Simulating Trajectories," Papers 2005.01365, arXiv.org, revised Aug 2020.
    2. Angelica Gianfreda & Derek Bunn, 2018. "A Stochastic Latent Moment Model for Electricity Price Formation," BEMPS - Bozen Economics & Management Paper Series BEMPS46, Faculty of Economics and Management at the Free University of Bozen.
    3. Narajewski, Michał & Ziel, Florian, 2020. "Ensemble forecasting for intraday electricity prices: Simulating trajectories," Applied Energy, Elsevier, vol. 279(C).
    4. Antonio Bello & Derek Bunn & Javier Reneses & Antonio Muñoz, 2016. "Parametric Density Recalibration of a Fundamental Market Model to Forecast Electricity Prices," Energies, MDPI, vol. 9(11), pages 1-15, November.
    5. Yixuan Wang & Jianzhu Li & Ping Feng & Rong Hu, 2015. "A Time-Dependent Drought Index for Non-Stationary Precipitation Series," Water Resources Management: An International Journal, Published for the European Water Resources Association (EWRA), Springer;European Water Resources Association (EWRA), vol. 29(15), pages 5631-5647, December.
    6. Panayi, Efstathios & Peters, Gareth W. & Danielsson, Jon & Zigrand, Jean-Pierre, 2018. "Designating market maker behaviour in limit order book markets," Econometrics and Statistics, Elsevier, vol. 5(C), pages 20-44.
    7. Gauss Cordeiro & Josemar Rodrigues & Mário Castro, 2012. "The exponential COM-Poisson distribution," Statistical Papers, Springer, vol. 53(3), pages 653-664, August.
    8. Christian Kleiber & Achim Zeileis, 2016. "Visualizing Count Data Regressions Using Rootograms," The American Statistician, Taylor & Francis Journals, vol. 70(3), pages 296-303, July.
    9. Matteo Malavasi & Gareth W. Peters & Pavel V. Shevchenko & Stefan Truck & Jiwook Jang & Georgy Sofronov, 2021. "Cyber Risk Frequency, Severity and Insurance Viability," Papers 2111.03366, arXiv.org, revised Mar 2022.
    10. Lucio Masserini & Matilde Bini & Monica Pratesi, 2017. "Effectiveness of non-selective evaluation test scores for predicting first-year performance in university career: a zero-inflated beta regression approach," Quality & Quantity: International Journal of Methodology, Springer, vol. 51(2), pages 693-708, March.
    11. Tong, Edward N.C. & Mues, Christophe & Thomas, Lyn, 2013. "A zero-adjusted gamma model for mortgage loan loss given default," International Journal of Forecasting, Elsevier, vol. 29(4), pages 548-562.
    12. Alexander Silbersdorff & Kai Sebastian Schneider, 2019. "Distributional Regression Techniques in Socioeconomic Research on the Inequality of Health with an Application on the Relationship between Mental Health and Income," IJERPH, MDPI, vol. 16(20), pages 1-28, October.
    13. Tong, Edward N.C. & Mues, Christophe & Brown, Iain & Thomas, Lyn C., 2016. "Exposure at default models with and without the credit conversion factor," European Journal of Operational Research, Elsevier, vol. 252(3), pages 910-920.
    14. D. Chiru Naik & Sagar Rohidas Chavan & P. Sonali, 2023. "Incorporating the climate oscillations in the computation of meteorological drought over India," Natural Hazards: Journal of the International Society for the Prevention and Mitigation of Natural Hazards, Springer;International Society for the Prevention and Mitigation of Natural Hazards, vol. 117(3), pages 2617-2646, July.
    15. Shuhui Guo & Lihua Xiong & Jie Chen & Shenglian Guo & Jun Xia & Ling Zeng & Chong-Yu Xu, 2023. "Nonstationary Regional Flood Frequency Analysis Based on the Bayesian Method," Water Resources Management: An International Journal, Published for the European Water Resources Association (EWRA), Springer;European Water Resources Association (EWRA), vol. 37(2), pages 659-681, January.
    16. Maike Hohberg & Katja Landau & Thomas Kneib & Stephan Klasen & Walter Zucchini, 2018. "Vulnerability to poverty revisited: Flexible modeling and better predictive performance," The Journal of Economic Inequality, Springer;Society for the Study of Economic Inequality, vol. 16(3), pages 439-454, September.
    17. Marcelo Cajias, 2017. "Is there room for another hedonic model? –The advantages of the GAMLSS approach in real estate research," ERES eres2017_226, European Real Estate Society (ERES).
    18. Kuntz, Laura-Chloé, 2020. "Beta dispersion and market timing," Journal of Empirical Finance, Elsevier, vol. 59(C), pages 235-256.
    19. Epstein, Leonardo D. & Inostroza-Quezada, Ignacio E. & Goodstein, Ronald C. & Choi, S. Chan, 2021. "Dynamic effects of store promotions on purchase conversion: Expanding technology applications with innovative analytics," Journal of Business Research, Elsevier, vol. 128(C), pages 279-289.
    20. Serinaldi, Francesco, 2011. "Distributional modeling and short-term forecasting of electricity prices by Generalized Additive Models for Location, Scale and Shape," Energy Economics, Elsevier, vol. 33(6), pages 1216-1226.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:phsmap:v:392:y:2013:i:5:p:1080-1089. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.journals.elsevier.com/physica-a-statistical-mechpplications/ .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.