Fisher information matrix of binary time series


A common approach to analyzing categorical correlated time series data is to fit a generalized linear model (GLM) with past data as covariate inputs. There remain challenges to conducting inference for time series with short length. By treating the historical data as covariate inputs, standard errors of estimates of GLM parameters computed from the empirical Fisher information do not fully account the auto-correlation in the data. To overcome this serious limitation, we derive the exact conditional Fisher information matrix of a general logistic autoregressive model with endogenous covariates for any series length T. Moreover, we also develop an iterative computational formula that allows for relatively easy implementation of the proposed estimator. Our simulation studies show that confidence intervals derived using the exact Fisher information matrix tend to be narrower than those utilizing the empirical Fisher information matrix while maintaining type I error rates at or below nominal levels. Further, we establish that, as T tends to infinity, the exact Fisher information matrix approaches the asymptotic Fisher information matrix previously derived for binary time series data. The developed exact conditional Fisher information matrix is applied to time-series data on respiratory rate among a cohort of expectant mothers where it is found to provide narrower confidence intervals for functionals of scientific interest and lead to greater statistical power when compared to the empirical Fisher information matrix.

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  1. 1.

    Agresti, A., Min, Y.: On small-sample confidence intervals for parameters in discrete distributions. Biometrics 57(3), 963–971 (2001)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Barrett, K., Barman, S.M., Boitano, S., Brooks, H.: Ganong’s review of medical physiology. Lange Medical Publications (2009)

  3. 3.

    Billingsley, P.: Statistical Inference for Markov Process. University of Chicago Press, Chicago (1961)

    MATH  Google Scholar 

  4. 4.

    Bonney, E.: Logistic regression for dependent binary observations. Biometrics 43, 951–973 (2004)

    Article  Google Scholar 

  5. 5.

    Chen, P., Jiao, J., Xu, M., Gao, X., Bischak, C.: Promoting active student travel: a longitudinal study. J. Transport Geogr. 70, 265–274 (2018)

    Article  Google Scholar 

  6. 6.

    Chen, P., Sun, F., Wang, Z., Gao, X., Jiao, J., Tao, Z.: Built environment effects on bike crash frequency and risk in Beijing. J. Saf. Res. 64, 135–143 (2018)

    Article  Google Scholar 

  7. 7.

    Davis, R., Dunsmuir, W., Wang, Y.: On autocorrelation in a Poisson regression model. Biometrika 87(3), 491–505 (2000)

    MathSciNet  Article  Google Scholar 

  8. 8.

    de Vries, O.S., Fidler, V., Kuipers, W., Hunink, M.: Fitting multistate transition models with autoregressive logistic regression: supervised exercise in intermittent claudication. Med. Decis. Mak. 18(1), 52–60 (1998)

  9. 9.

    Diggle, J., Liang, K.: Zeger, L: Analysis of Longitudinal Data. Oxford University Press, Oxford (1994)

  10. 10.

    Dodge, Y.: The Oxford Dictionary of Statistical Terms. OUP, Oxford (2003)

    MATH  Google Scholar 

  11. 11.

    Entringer, S., Epel, E., Lin, J., Blackburn, E., Bussa, C., Shahbaba, S., Gillen, D., Venkataramanan, R., Simhan, H., Wadhwa, P.: Maternal folate concentration in early pregnancy and newborn telomere length. Ann. Nutr. Metab. 66, 202–208 (2015)

    Article  Google Scholar 

  12. 12.

    Fahrmeir, L., Kaufmann, H.: Regression models for nonstationary categorical time series. Time Ser. Anal. 8, 147–160 (1987)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Fahrmeir, L., Tutz, G.: Multivariate Statistical Modelling Based on Generalized Linear. Models. Springer, New York (1994)

  14. 14.

    Fokianos, K., Kedem, B.: Prediction and classification of non-stationary categorical time series. J. Multivariate AnaL 67, 277–296 (1998)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Fokianos, K., Kedem, B.: Regression theory for categorical time series. Stat. Sci. 18(3), 357–376 (2003)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Gao, X., Shahbaba, B., Ombao, H.: Modeling Binary Time Series Using Gaussian Processes with Application to Predicting Sleep States. arXiv:1711.05466 (2017) (arXiv preprint)

  17. 17.

    Gouveia, S., Scotto, M.G., Weiß, C.H., Ferreira, P.J.S.: Binary auto-regressive geometric modelling in a DNA context. J. R. Stat. Soc. Ser. C (Appl. Stat.) 66(2), 253–271 (2017)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Guo, Y., Wang, Y., Marin, T., Kirk, E., Patel, R., Josephson, C.: Statistical methods for characterizing transfusion-related changes in regional oxygenation using near-infrared spectroscopy (NIRS) in preterm infants. arXiv:1801.08153 (2018) (arXiv preprint)

  19. 19.

    Hauck, Jr, Donner, A.: Walds test as applied to hypotheses in logit analysis. J. Am. Stat. Assoc. 72, 851–853 (1977)

    MathSciNet  MATH  Google Scholar 

  20. 20.

    Holmes, T., Rahe, R.: The social readjustment rating scale. J. Psychosom. Res. 11(2), 213–218 (1967)

    Article  Google Scholar 

  21. 21.

    Katz, R.: On some criteria for estimating the order of a Markov chain. Technometrics 23(3), 243–249 (1981)

    MathSciNet  Article  Google Scholar 

  22. 22.

    Kaufmann, H.: Regression models for nonstationary time series: asymptotic estimation theory. Ann. Stat. 15, 79–98 (1987)

    MathSciNet  Article  Google Scholar 

  23. 23.

    Kedem, B.: Time Series Analysis by Higher Order Crossings. IEEE Press, New York (1994)

  24. 24.

    Kedem, B.: Binary Time Series. Marcel Dekker, New York (1980)

  25. 25.

    Kedem, B., Fokianos, K.: Regression Models for Time Series Analysis. Wiley, New York (2002)

    Book  Google Scholar 

  26. 26.

    Keenan, D.: A time series analysis of binary data. J. Am. Stat. Assoc. 77(380), 816–821 (1982)

    MathSciNet  Article  Google Scholar 

  27. 27.

    Meyn, S., Tweedie, R.: Markov Chains and Stochastic Stability. Springer, London (2012)

    MATH  Google Scholar 

  28. 28.

    Muenz, L., Rubinstein, L.: Markov models for covariate dependence of binary sequences. Biometrics 41, 91–101 (1985)

    MathSciNet  Article  Google Scholar 

  29. 29.

    Newcombe, R.G.: Interval estimation for the difference between independent proportions: comparison of eleven methods. Stat. Med. 17, 873–890 (1998)

    Article  Google Scholar 

  30. 30.

    Startz, R.: Binomial autoregressive moving average models with an application to U.S. recessions. J. Bus. Econ. Stat. 26(1), 1–8 (2012)

    MathSciNet  Article  Google Scholar 

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Data collection for the maternal cohort was supported by US PHS (NIH) grant RO1 HD-060628 to Pathik D. Wadhwa.

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Correspondence to Xu Gao.

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Gao, X., Gillen, D. & Ombao, H. Fisher information matrix of binary time series. METRON 76, 287–304 (2018).

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  • Binary time series
  • Correlated binary data
  • Empirical Fisher information
  • Exact Fisher information matrix
  • Logistic autoregressive model