Statistical analysis of conditionally binomial nonlinear regression time series with discrete regressors

Kharin, Yu.; Voloshko, V.

doi:10.1090/tpms/1105

Statistical analysis of conditionally binomial nonlinear regression time series with discrete regressors

Authors: Yu. S. Kharin and V. A. Voloshko
Journal: Theor. Probability and Math. Statist. 100 (2020), 181-190
MSC (2010): Primary 62-07, 62J12, 62F12; Secondary 62F03, 62F05, 62M20
DOI: https://doi.org/10.1090/tpms/1105
Published electronically: August 5, 2020
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The model of conditionally binomial nonlinear regression time series with discrete regressors is considered. A new frequencies-based estimator (FBE) of explicit form is constructed for this model. FBE is shown to be consistent, asymptotically normal, asymptotically effective, and to have less restrictive uniqueness assumptions w.r.t. the classical MLE. A fast recursive algorithm is constructed for FBE re-computation under model extension. An asymptotically optimal Wald test and forecasting statistic based on FBE are developed. Computer experiments on simulated data are performed for FBE.

References

Benjamin Kedem and Konstantinos Fokianos, Regression models for time series analysis, Wiley Series in Probability and Statistics, Wiley-Interscience [John Wiley & Sons], Hoboken, NJ, 2002. MR 1933755
Yuriy Kharin, Robustness in statistical forecasting, Robustness and complex data structures, Springer, Heidelberg, 2013, pp. 225–242. MR 3135883, DOI https://doi.org/10.1007/978-3-642-35494-6_14
O. O. Dashkov and O. G. Kukush, Consistency of an orthogonal regression estimator in an implicit linear errors-in-variables model, Teor. Ĭmovīr. Mat. Stat. 97 (2017), 48–57 (Ukrainian, with English, Russian and Ukrainian summaries); English transl., Theory Probab. Math. Statist. 97 (2018), 45–55. MR 3745998, DOI https://doi.org/10.1090/tpms/1047

J. Nelder and R. Wedderburn, Generalized linear models, J. Royal Statistical Society. Series A 35 (1972), no. 3, 370–384.

P. McCullagh and J. A. Nelder, Generalized linear models, Monographs on Statistics and Applied Probability, Chapman & Hall, London, 1989. Second edition [of MR0727836]. MR 3223057

C. H. Weiss, An Introduction to Discrete-Valued Time Series, John Wiley and Sons Ltd, 2018.

Yu. S. Kharin and E. V. Vecherko, Statistical estimation of parameters for binary Markov chain models with embeddings, Discrete Math. Appl. 23 (2013), no. 2, 153–169.

Yu. S. Kharin and E. V. Vecherko, Recognition of embeddings in a binary Markov chain, Diskret. Mat. 27 (2015), no. 3, 123–144 (Russian, with Russian summary); English transl., Discrete Math. Appl. 26 (2016), no. 1, 13–29. MR 3468405, DOI https://doi.org/10.1515/dma-2016-0002

V. A. Voloshko, The steganographic capacity of a one-dimensional Markov covertext, Diskret. Mat. 28 (2016), no. 1, 19–43 (Russian, with Russian summary); English transl., Discrete Math. Appl. 27 (2017), no. 4, 247–268. MR 3527007, DOI https://doi.org/10.1515/dma-2017-0026

Yu. S. Kharin, V. A. Voloshko, and E. A. Medved, Statistical estimation of parameters for binary conditionally nonlinear autoregressive time series, Math. Methods Statist. 27 (2018), no. 2, 103–118. MR 3827356, DOI https://doi.org/10.3103/S1066530718020023

Ludwig Fahrmeir and Heinz Kaufmann, Consistency and asymptotic normality of the maximum likelihood estimator in generalized linear models, Ann. Statist. 13 (1985), no. 1, 342–368. MR 773172, DOI https://doi.org/10.1214/aos/1176346597

B. Noble and J. W. Daniel, Applied Linear Algebra, Prentice-Hall, Englewood Cliffs, 1988.

Albert N. Shiryaev, Probability. 1, 3rd ed., Graduate Texts in Mathematics, vol. 95, Springer, New York, 2016. Translated from the fourth (2007) Russian edition by R. P. Boas and D. M. Chibisov. MR 3467826

Abraham Wald, Tests of statistical hypotheses concerning several parameters when the number of observations is large, Trans. Amer. Math. Soc. 54 (1943), 426–482. MR 12401, DOI https://doi.org/10.1090/S0002-9947-1943-0012401-3

Shelby J. Haberman, Maximum likelihood estimates in exponential response models, Ann. Statist. 5 (1977), no. 5, 815–841. MR 501540

R. W. M. Wedderburn, On the existence and uniqueness of the maximum likelihood estimates for certain generalized linear models, Biometrika 63 (1976), no. 1, 27–32. MR 408092, DOI https://doi.org/10.1093/biomet/63.1.27

M. Bagnoli and T. Bergstrom, Log-Concave Probability and Its Applications, University of Michigan, 1989.

Camille Jordan, Essai sur la géométrie à $n$ dimensions, Bull. Soc. Math. France 3 (1875), 103–174 (French). MR 1503705

Yu. Kharin, Robustness of clustering under outliers, Lecture Notes in Computer Science 1280 (1997), 501–511.

Yu. Kharin and E. Zhuk, Filtering of multivariate samples containing “outliers” for clustering, Pattern Recognition Letters 19 (1998), 1077–1085.

Yuriy Kharin, Robustness of the mean square risk in forecasting of regression time series, Comm. Statist. Theory Methods 40 (2011), no. 16, 2893–2906. MR 2860793, DOI https://doi.org/10.1080/03610926.2011.562774

Alexey Kharin, Performance and robustness evaluation in sequential hypotheses testing, Comm. Statist. Theory Methods 45 (2016), no. 6, 1693–1709. MR 3473943, DOI https://doi.org/10.1080/03610926.2014.944659