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Birnbaum-Saunders and Lognormal Kernel Estimators for Modelling Durations in High Frequency Financial Data

Author

Listed:
  • Xiaodong Jin

    (Department of Mathematics, UNC at Charlotte)

  • Janusz Kawczak

    (Department of Mathematics, UNC at Charlotte)

Abstract

In this article we extend the class of non-negative, asymmetric kernel density estimators and propose Birnbaum-Saunders (BS) and lognormal (LN) kernel density functions. The density functions have bounded support on [0,1). Both BS and LN kernel estimators are free of boundary bias, non-negative, with natural varying shape, and achieve the optimal rate of convergence for the mean integrated squared error. We apply BS and LN kernel density estimators to high frequency intraday time duration data. The comparisons are made on several nonparametric kernel density estimators. BS and LN kernels perform better near the boundary in terms of bias reduction.

Suggested Citation

  • Xiaodong Jin & Janusz Kawczak, 2003. "Birnbaum-Saunders and Lognormal Kernel Estimators for Modelling Durations in High Frequency Financial Data," Annals of Economics and Finance, Society for AEF, vol. 4(1), pages 103-124, May.
    • Handle: RePEc:cuf:journl:y:2003:v:4:i:1:p:103-124
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    References listed on IDEAS

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    1. Olivier SCAILLET, 2001. "Density Estimation Using Inverse and Reciprocal Inverse Guassian Kernels," LIDAM Discussion Papers IRES 2001017, Université catholique de Louvain, Institut de Recherches Economiques et Sociales (IRES).
    2. Fernandes, Marcelo & Grammig, Joachim, 2005. "Nonparametric specification tests for conditional duration models," Journal of Econometrics, Elsevier, vol. 127(1), pages 35-68, July.
    3. Fernandes, Marcelo & Grammig, Joachim, 2006. "A family of autoregressive conditional duration models," Journal of Econometrics, Elsevier, vol. 130(1), pages 1-23, January.
    4. Robert F. Engle & Jeffrey R. Russell, 1998. "Autoregressive Conditional Duration: A New Model for Irregularly Spaced Transaction Data," Econometrica, Econometric Society, vol. 66(5), pages 1127-1162, September.
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    Full references (including those not matched with items on IDEAS)

    Citations

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    Cited by:

    1. Célestin C. Kokonendji & Sobom M. Somé, 2021. "Bayesian Bandwidths in Semiparametric Modelling for Nonnegative Orthant Data with Diagnostics," Stats, MDPI, vol. 4(1), pages 1-22, March.
    2. Xu Li & Juxia Xiao & Weixing Song & Jianhong Shi, 2019. "Local linear regression with reciprocal inverse Gaussian kernel," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 82(6), pages 733-758, August.
    3. Helton Saulo & Jeremias Leão & Víctor Leiva & Robert G. Aykroyd, 2019. "Birnbaum–Saunders autoregressive conditional duration models applied to high-frequency financial data," Statistical Papers, Springer, vol. 60(5), pages 1605-1629, October.
    4. Hirukawa, Masayuki & Sakudo, Mari, 2019. "Another bias correction for asymmetric kernel density estimation with a parametric start," Statistics & Probability Letters, Elsevier, vol. 145(C), pages 158-165.
    5. Igarashi, Gaku & Kakizawa, Yoshihide, 2014. "Re-formulation of inverse Gaussian, reciprocal inverse Gaussian, and Birnbaum–Saunders kernel estimators," Statistics & Probability Letters, Elsevier, vol. 84(C), pages 235-246.
    6. Ouimet, Frédéric & Tolosana-Delgado, Raimon, 2022. "Asymptotic properties of Dirichlet kernel density estimators," Journal of Multivariate Analysis, Elsevier, vol. 187(C).
    7. Gery Geenens, 2021. "Mellin–Meijer kernel density estimation on $${{\mathbb {R}}}^+$$ R +," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 73(5), pages 953-977, October.
    8. Ziane Yasmina & Zougab Nabil & Adjabi Smail, 2021. "Body tail adaptive kernel density estimation for nonnegative heavy-tailed data," Monte Carlo Methods and Applications, De Gruyter, vol. 27(1), pages 57-69, March.
    9. Lemonte, Artur J. & Cribari-Neto, Francisco & Vasconcellos, Klaus L.P., 2007. "Improved statistical inference for the two-parameter Birnbaum-Saunders distribution," Computational Statistics & Data Analysis, Elsevier, vol. 51(9), pages 4656-4681, May.
    10. Yasmina Ziane & Nabil Zougab & Smail Adjabi, 2018. "Birnbaum–Saunders power-exponential kernel density estimation and Bayes local bandwidth selection for nonnegative heavy tailed data," Computational Statistics, Springer, vol. 33(1), pages 299-318, March.
    11. Hirukawa, Masayuki & Sakudo, Mari, 2014. "Nonnegative bias reduction methods for density estimation using asymmetric kernels," Computational Statistics & Data Analysis, Elsevier, vol. 75(C), pages 112-123.
    12. Kakizawa, Yoshihide, 2021. "A class of Birnbaum–Saunders type kernel density estimators for nonnegative data," Computational Statistics & Data Analysis, Elsevier, vol. 161(C).
    13. Gaku Igarashi, 2018. "Multivariate Density Estimation Using a Multivariate Weighted Log-Normal Kernel," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 80(2), pages 247-266, August.
    14. Kakizawa, Yoshihide, 2022. "Multivariate elliptical-based Birnbaum–Saunders kernel density estimation for nonnegative data," Journal of Multivariate Analysis, Elsevier, vol. 187(C).
    15. Pierre Lafaye de Micheaux & Frédéric Ouimet, 2021. "A Study of Seven Asymmetric Kernels for the Estimation of Cumulative Distribution Functions," Mathematics, MDPI, vol. 9(20), pages 1-35, October.
    16. Marchant, Carolina & Bertin, Karine & Leiva, Víctor & Saulo, Helton, 2013. "Generalized Birnbaum–Saunders kernel density estimators and an analysis of financial data," Computational Statistics & Data Analysis, Elsevier, vol. 63(C), pages 1-15.
    17. Naderi, Mehrdad & Hashemi, Farzane & Bekker, Andriette & Jamalizadeh, Ahad, 2020. "Modeling right-skewed financial data streams: A likelihood inference based on the generalized Birnbaum–Saunders mixture model," Applied Mathematics and Computation, Elsevier, vol. 376(C).
    18. Ané, Thierry & Métais, Carole, 2009. "The distribution of realized variances: Marginal behaviors, asymmetric dependence and contagion effects," International Review of Financial Analysis, Elsevier, vol. 18(3), pages 134-150, June.

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    More about this item

    Keywords

    Birnbaum-Saunders kernel; Lognormal kernel; High frequency; ACD model; Durations;
    All these keywords.

    JEL classification:

    • C13 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Estimation: General
    • C14 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Semiparametric and Nonparametric Methods: General
    • C15 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Statistical Simulation Methods: General
    • C41 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods: Special Topics - - - Duration Analysis; Optimal Timing Strategies

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