## A perturbation analysis of Markov chains models with time-varying parameters.(English)Zbl 1455.60092

Summary: We study some regularity properties in locally stationary Markov models which are fundamental for controlling the bias of nonparametric kernel estimators. In particular, we provide an alternative to the standard notion of derivative process developed in the literature and that can be used for studying a wide class of Markov processes. To this end, for some families of $$V$$-geometrically ergodic Markov kernels indexed by a real parameter $$u$$, we give conditions under which the invariant probability distribution is differentiable with respect to $$u$$, in the sense of signed measures. Our results also complete the existing literature for the perturbation analysis of Markov chains, in particular when exponential moments are not finite. Our conditions are checked on several original examples of locally stationary processes such as integer-valued autoregressive processes, categorical time series or threshold autoregressive processes.

### MSC:

 60J05 Discrete-time Markov processes on general state spaces 62G05 Nonparametric estimation

### Keywords:

local stationarity; time-inhomogeneous Markov chains
Full Text:

### References:

 [1] Al-Osh, M.A. and Alzaid, A.A. (1987). First-order integer-valued autoregressive (INAR $$(1))$$ process. J. Time Ser. Anal. 8 261-275. · Zbl 0617.62096 [2] Dahlhaus, R. (1997). Fitting time series models to nonstationary processes. Ann. Statist. 25 1-37. · Zbl 0871.62080 [3] Dahlhaus, R. (2012). Locally stationary processes. In Handbook of Statistics 30 351-413. Elsevier. [4] Dahlhaus, R., Richter, S. and Wu, W.B. (2019). Towards a general theory for nonlinear locally stationary processes. Bernoulli 25 1013-1044. · Zbl 1427.60057 [5] Dahlhaus, R. and Subba Rao, S. (2006). Statistical inference for time-varying ARCH processes. Ann. Statist. 34 1075-1114. · Zbl 1113.62099 [6] Davis, R.A., Dunsmuir, W.T.M. and Streett, S.B. (2003). Observation-driven models for Poisson counts. Biometrika 90 777-790. · Zbl 1436.62418 [7] Davis, R.A. and Liu, H. (2016). Theory and inference for a class of nonlinear models with application to time series of counts. Statist. Sinica 26 1673-1707. · Zbl 1356.62137 [8] Douc, R., Doukhan, P. and Moulines, E. (2013). Ergodicity of observation-driven time series models and consistency of the maximum likelihood estimator. Stochastic Process. Appl. 123 2620-2647. · Zbl 1285.62104 [9] Douc, R., Moulines, E. and Stoffer, D.S. (2014). Nonlinear Time Series. Chapman & Hall/CRC Texts in Statistical Science Series. Boca Raton, FL: CRC Press/CRC. Theory, methods, and applications with R examples. · Zbl 1306.62026 [10] Du, J.G. and Li, Y. (1991). The integer-valued autoregressive $$(\text{INAR}(p))$$ model. J. Time Ser. Anal. 12 129-142. · Zbl 0727.62084 [11] Durbin, J. and Koopman, S.J. (2000). Time series analysis of non-Gaussian observations based on state space models from both classical and Bayesian perspectives. J. R. Stat. Soc. Ser. B Stat. Methodol. 62 3-56. · Zbl 0945.62084 [12] Fan, J. and Gijbels, I. (1996). Local Polynomial Modelling and Its Applications. Monographs on Statistics and Applied Probability 66. London: CRC Press. · Zbl 0873.62037 [13] Ferland, R., Latour, A. and Oraichi, D. (2006). Integer-valued GARCH process. J. Time Ser. Anal. 27 923-942. · Zbl 1150.62046 [14] Ferré, D., Hervé, L. and Ledoux, J. (2013). Regular perturbation of $$V$$-geometrically ergodic Markov chains. J. Appl. Probab. 50 184-194. · Zbl 1276.60075 [15] Fokianos, K. and Truquet, L. (2019). On categorical time series models with covariates. Stochastic Process. Appl. 129 3446-3462. · Zbl 1431.62370 [16] Fryzlewicz, P., Sapatinas, T. and Subba Rao, S. (2008). Normalized least-squares estimation in time-varying ARCH models. Ann. Statist. 36 742-786. · Zbl 1133.62071 [17] Glynn, P.W. and L’Ecuyer, P. (1995). Likelihood ratio gradient estimation for stochastic recursions. Adv. in Appl. Probab. 27 1019-1053. · Zbl 0835.62071 [18] Hairer, M. and Mattingly, J.C. (2011). Yet another look at Harris’ ergodic theorem for Markov chains. In Seminar on Stochastic Analysis, Random Fields and Applications VI. Progress in Probability 63 109-117. Basel: Birkhäuser/Springer Basel AG. · Zbl 1248.60082 [19] Heidergott, B. and Hordijk, A. (2003). Taylor series expansions for stationary Markov chains. Adv. in Appl. Probab. 35 1046-1070. · Zbl 1043.60056 [20] Heidergott, B., Hordijk, A. and Weisshaupt, H. (2006). Measure-valued differentiation for stationary Markov chains. Math. Oper. Res. 31 154-172. · Zbl 1278.90428 [21] Hervé, L. and Pène, F. (2010). The Nagaev-Guivarc’h method via the Keller-Liverani theorem. Bull. Soc. Math. France 138 415-489. · Zbl 1205.60133 [22] Kartashov, N.V. (1985). Inequalities in stability and ergodicity theorems for Markov chains with a general phase space. I. Theory Probab. Appl. 30 247-259. · Zbl 0657.60088 [23] Kushner, H.J. and Yin, G.G. (2003). Stochastic Approximation and Recursive Algorithms and Applications, 2nd ed. Applications of Mathematics (New York) 35. New York: Springer. Stochastic Modelling and Applied Probability. · Zbl 1026.62084 [24] Latour, A. (1997). The multivariate $$\text{GINAR}(p)$$ process. Adv. in Appl. Probab. 29 228-248. · Zbl 0871.62073 [25] McKenzie, E. (1986). Autoregressive moving-average processes with negative-binomial and geometric marginal distributions. Adv. in Appl. Probab. 18 679-705. · Zbl 0603.62100 [26] Meyn, S. and Tweedie, R.L. (2009). Markov Chains and Stochastic Stability, 2nd ed. Cambridge: Cambridge Univ. Press. With a prologue by Peter W. Glynn. · Zbl 1165.60001 [27] Moulines, E., Priouret, P. and Roueff, F. (2005). On recursive estimation for time varying autoregressive processes. Ann. Statist. 33 2610-2654. · Zbl 1084.62089 [28] Pflug, G.Ch. (1992). Gradient estimates for the performance of Markov chains and discrete event processes. Ann. Oper. Res. 39 173-194. · Zbl 0766.60088 [29] Pflug, G.Ch. (1996). Optimization of Stochastic Models: The interface between simulation and optimization. The Kluwer International Series in Engineering and Computer Science 373. Boston, MA: Kluwer Academic. [30] Richter, S. and Dahlhaus, R. (2019). Cross validation for locally stationary processes. Ann. Statist. 47 2145-2173. · Zbl 1433.62267 [31] Rudolf, D. and Schweizer, N. (2018). Perturbation theory for Markov chains via Wasserstein distance. Bernoulli 24 2610-2639. · Zbl 1465.60065 [32] Schweitzer, P.J. (1968). Perturbation theory and finite Markov chains. J. Appl. Probability 5 401-413. · Zbl 0196.19803 [33] Subba Rao, S. (2006). On some nonstationary, nonlinear random processes and their stationary approximations. Adv. in Appl. Probab. 38 1155-1172. · Zbl 1103.62085 [34] Truquet, L. (2017). Parameter stability and semiparametric inference in time varying auto-regressive conditional heteroscedasticity models. J. R. Stat. Soc. Ser. B. Stat. Methodol. 79 1391-1414. · Zbl 06840438 [35] Truquet, L. (2019). Local stationarity and time-inhomogeneous Markov chains. Ann. Statist. 47 2023-2050. · Zbl 1429.62372 [36] Truquet, L. (2020). Supplement to “A perturbation analysis of Markov chains models with time-varying parameters.” https://doi.org/10.3150/20-BEJ1210SUPP [37] Tsybakov, A.B. (2009). Introduction to Nonparametric Estimation. Springer Series in Statistics. New York: Springer. Revised and extended from the 2004 French original, Translated by Vladimir Zaiats. · Zbl 1176.62032 [38] Vázquez-Abad, F.J. and Kushner, H.J. (1992). Estimation of the derivative of a stationary measure with respect to a control parameter. J. Appl. Probab. 29 343-352. · Zbl 0792.62076 [39] Weiß, C.H. (2018). An Introduction to Discrete-valued Time Series. New York: Wiley. · Zbl 1407.62009 [40] Zhang, T. · Zbl 1257.62049
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