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A note on the Bickel-Rosenblatt test in autoregressive time series

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  • Bachmann, Dirk
  • Dette, Holger

Abstract

In a recent paper Lee and Na [2002. Statist. Probab. Lett. 56(1), 23-25] introduced a test for the parametric form of the distribution of the innovations in autoregressive models, which is based on the integrated squared error of the nonparametric density estimate from the residuals and a smoothed version of the parametric fit of the density. They derived the asymptotic distribution under the null-hypothesis, which is the same as for the classical Bickel-Rosenblatt [1973. Ann. Statist. 1, 1071-1095] test for the distribution of i.i.d. observations. In this note we first extend the results of Bickel and Rosenblatt to the case of fixed alternatives, for which asymptotic normality is still true but with a different rate of convergence. As a by-product we also provide an alternative proof of the Bickel and Rosenblatt result under substantially weaker assumptions on the kernel density estimate. As a further application we derive the asymptotic behaviour of Lee and Na's statistic in autoregressive models under fixed alternatives. The results can be used for the calculation of the probability of the type II error if the Bickel-Rosenblatt test is used to check the parametric form of the error distribution or to test interval hypotheses in this context.

Suggested Citation

  • Bachmann, Dirk & Dette, Holger, 2005. "A note on the Bickel-Rosenblatt test in autoregressive time series," Statistics & Probability Letters, Elsevier, vol. 74(3), pages 221-234, October.
    • Handle: RePEc:eee:stapro:v:74:y:2005:i:3:p:221-234
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    References listed on IDEAS

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    1. Lee, Sangyeol & Na, Seongryong, 2002. "On the Bickel-Rosenblatt test for first-order autoregressive models," Statistics & Probability Letters, Elsevier, vol. 56(1), pages 23-35, January.
    2. Sellke T. & Bayarri M. J. & Berger J. O., 2001. "Calibration of rho Values for Testing Precise Null Hypotheses," The American Statistician, American Statistical Association, vol. 55, pages 62-71, February.
    3. Hall, Peter, 1984. "Central limit theorem for integrated square error of multivariate nonparametric density estimators," Journal of Multivariate Analysis, Elsevier, vol. 14(1), pages 1-16, February.
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    Cited by:

    1. Hira Koul & Nao Mimoto & Donatas Surgailis, 2013. "Goodness-of-fit tests for long memory moving average marginal density," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 76(2), pages 205-224, February.
    2. Gao, Min & Yang, Wenzhi & Wu, Shipeng & Yu, Wei, 2022. "Asymptotic normality of residual density estimator in stationary and explosive autoregressive models," Computational Statistics & Data Analysis, Elsevier, vol. 175(C).
    3. Wenceslao González-Manteiga & Rosa Crujeiras, 2013. "An updated review of Goodness-of-Fit tests for regression models," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 22(3), pages 361-411, September.
    4. Mimoto, Nao, 2008. "Convergence in distribution for the sup-norm of a kernel density estimator for GARCH innovations," Statistics & Probability Letters, Elsevier, vol. 78(7), pages 915-923, May.
    5. Holzmann, Hajo, 2008. "Testing parametric models in the presence of instrumental variables," Statistics & Probability Letters, Elsevier, vol. 78(6), pages 629-636, April.
    6. Cheng, Fuxia, 2018. "Glivenko–Cantelli Theorem for the kernel error distribution estimator in the first-order autoregressive model," Statistics & Probability Letters, Elsevier, vol. 139(C), pages 95-102.
    7. Fuxia Cheng & Hira L. Koul, 2023. "An analog of Bickel–Rosenblatt test for fitting an error density in the two phase linear regression model," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 86(1), pages 27-56, January.
    8. Hira Koul & Nao Mimoto, 2012. "A goodness-of-fit test for GARCH innovation density," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 75(1), pages 127-149, January.
    9. Nadine Hilgert & Bruno Portier, 2012. "Strong uniform consistency and asymptotic normality of a kernel based error density estimator in functional autoregressive models," Statistical Inference for Stochastic Processes, Springer, vol. 15(2), pages 105-125, July.

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