Fisher information matrix of binary time series. (English) Zbl 1427.62096

Summary: 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.


62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
62B10 Statistical aspects of information-theoretic topics
62J12 Generalized linear models (logistic models)


Fahrmeir; HIBITS
Full Text: DOI arXiv


[1] Agresti, A., Min, Y.: On small-sample confidence intervals for parameters in discrete distributions. Biometrics 57(3), 963-971 (2001) · Zbl 1209.62041
[2] Barrett, K., Barman, S.M., Boitano, S., Brooks, H.: Ganong’s review of medical physiology. Lange Medical Publications (2009)
[3] Billingsley, P.: Statistical Inference for Markov Process. University of Chicago Press, Chicago (1961) · Zbl 0106.34201
[4] Bonney, E.: Logistic regression for dependent binary observations. Biometrics 43, 951-973 (2004) · Zbl 0707.62153
[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)
[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)
[7] Davis, R., Dunsmuir, W., Wang, Y.: On autocorrelation in a Poisson regression model. Biometrika 87(3), 491-505 (2000) · Zbl 0956.62075
[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] Diggle, J., Liang, K.: Zeger, L: Analysis of Longitudinal Data. Oxford University Press, Oxford (1994)
[10] Dodge, Y.: The Oxford Dictionary of Statistical Terms. OUP, Oxford (2003) · Zbl 1027.62001
[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)
[12] Fahrmeir, L., Kaufmann, H.: Regression models for nonstationary categorical time series. Time Ser. Anal. 8, 147-160 (1987) · Zbl 0616.62116
[13] Fahrmeir, L., Tutz, G.: Multivariate Statistical Modelling Based on Generalized Linear. Models. Springer, New York (1994) · Zbl 0809.62064
[14] Fokianos, K., Kedem, B.: Prediction and classification of non-stationary categorical time series. J. Multivariate AnaL 67, 277-296 (1998) · Zbl 0919.62105
[15] Fokianos, K., Kedem, B.: Regression theory for categorical time series. Stat. Sci. 18(3), 357-376 (2003) · Zbl 1055.62095
[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) · Zbl 1422.62219
[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)
[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] Hauck, Jr, Donner, A.: Walds test as applied to hypotheses in logit analysis. J. Am. Stat. Assoc. 72, 851-853 (1977) · Zbl 0375.62022
[20] Holmes, T., Rahe, R.: The social readjustment rating scale. J. Psychosom. Res. 11(2), 213-218 (1967)
[21] Katz, R.: On some criteria for estimating the order of a Markov chain. Technometrics 23(3), 243-249 (1981) · Zbl 0485.62086
[22] Kaufmann, H.: Regression models for nonstationary time series: asymptotic estimation theory. Ann. Stat. 15, 79-98 (1987) · Zbl 0614.62111
[23] Kedem, B.: Time Series Analysis by Higher Order Crossings. IEEE Press, New York (1994) · Zbl 0818.62077
[24] Kedem, B.: Binary Time Series. Marcel Dekker, New York (1980) · Zbl 0424.62062
[25] Kedem, B., Fokianos, K.: Regression Models for Time Series Analysis. Wiley, New York (2002) · Zbl 1011.62089
[26] Keenan, D.: A time series analysis of binary data. J. Am. Stat. Assoc. 77(380), 816-821 (1982) · Zbl 0507.62079
[27] Meyn, S., Tweedie, R.: Markov Chains and Stochastic Stability. Springer, London (2012) · Zbl 0925.60001
[28] Muenz, L., Rubinstein, L.: Markov models for covariate dependence of binary sequences. Biometrics 41, 91-101 (1985)
[29] Newcombe, R.G.: Interval estimation for the difference between independent proportions: comparison of eleven methods. Stat. Med. 17, 873-890 (1998)
[30] Startz, R.: Binomial autoregressive moving average models with an application to U.S. recessions. J. Bus. Econ. Stat. 26(1), 1-8 (2012)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.