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We prove absolute regularity ($\beta$-mixing) for nonstationary and multivariate versions of two popular classes of integer-valued processes. We show how this result can be used to prove asymptotic normality of a least squares estimator of an involved model parameter.

We propose a new model for nonstationary integer-valued time series which is particularly suitable for data with a strong trend. In contrast to popular Poisson-INGARCH models, but in line with classical GARCH models, we propose to pick the conditional distributions from nearly scale invariant families where the mean absolute value and the standard deviation are of the same order of magnitude. As an important prerequisite for applications in statistics, we prove absolute regularity of the count process with exponentially decaying coefficients.

We consider a nonparametric version of the integer-valued GARCH(1,1) model for time series of counts. The link function in the recursion for the variances is not specified by finite-dimensional parameters, but we impose nonparametric smoothness conditions. We propose a least squares estimator for this function and show that it is consistent with a rate that we conjecture to be nearly optimal., Comment: 24 pages

The first motivation of this paper is to study stationarity and ergodic properties for a general class of time series models defined conditional on an exogenous covariates process. The dynamic of these models is given by an autoregressive latent process which forms a Markov chain in random environments. Contrarily to existing contributions in the field of Markov chains in random environments, the state space is not discrete and we do not use small set type assumptions or uniform contraction conditions for the random Markov kernels. Our assumptions are quite general and allows to deal with models that are not fully contractive, such as threshold autoregressive processes. Using a coupling approach, we study the existence of a limit, in Wasserstein metric, for the backward iterations of the chain. We also derive ergodic properties for the corresponding skew-product Markov chain. Our results are illustrated with many examples of autoregressive processes widely used in statistics or in econometrics, including GARCH type processes, count autoregressions and categorical time series.

We consider integer-valued GARCH processes, where the count variable conditioned on past values of the count and state variables follows a so-called Skellam distribution. Using arguments for contractive Markov chains we prove that the process has a unique stationary regime. Furthermore, we show asymptotic regularity ($\beta$-mixing) with geometrically decaying coefficients for the count process. These probabilistic results are complemented by a statistical analysis, a few simulations as well as an application to recent COVID-19 data.

We consider a general monotone regression estimation where we allow for independent and dependent regressors. We propose a modification of the classical isotonic least squares estimator and establish its rate of convergence for the integrated $L_1$-loss function. The methodology captures the shape of the data without assuming additivity or a parametric form for the regression function. Furthermore, the degree of smoothing is chosen automatically and no auxiliary tuning is required for the theoretical analysis. Some simulations and two real data illustrations complement the study of the proposed estimator.

We consider a class of observation-driven Poisson count processes where the current value of the accompanying intensity process depends on previous values of both processes. We show under a contractive condition that the bivariate process has a unique stationary distribution and that a stationary version of the count process is absolutely regular. Moreover, since the intensities can be written as measurable functionals of the count variables, we conclude that the bivariate process is ergodic. As an important application of these results, we show how a test method previously used in the case of independent Poisson data can be used in the case of Poisson count processes., Comment: Published in at http://dx.doi.org/10.3150/10-BEJ313 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)

New goodness-of-fit tests for Markovian models in time series analysis are developed which are based on the difference between a fully nonparametric estimate of the one-step transition distribution function of the observed process and that of the model class postulated under the null hypothesis. The model specification under the null allows for Markovian models, the transition mechanisms of which depend on an unknown vector of parameters and an unspecified distribution of i.i.d. innovations. Asymptotic properties of the test statistic are derived and the critical values of the test are found using appropriate bootstrap schemes. General properties of the bootstrap for Markovian processes are derived. A new central limit theorem for triangular arrays of weakly dependent random variables is obtained. For the proof of stochastic equicontinuity of multidimensional empirical processes, we use a simple approach based on an anisotropic tiling of the space. The finite-sample behavior of the proposed test is illustrated by some numerical examples and a real-data application is given., Comment: Published in at http://dx.doi.org/10.3150/07-BEJ6055 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)

We suppose that a L\'evy process is observed at discrete time points. A rather general construction of minimum-distance estimators is shown to give consistent estimators of the L\'evy-Khinchine characteristics as the number of observations tends to infinity, keeping the observation distance fixed. For a specific $C^2$-criterion this estimator is rate-optimal. The connection with deconvolution and inverse problems is explained. A key step in the proof is a uniform control on the deviations of the empirical characteristic function on the whole real line., Comment: 24 pages, 2 figures

It is proved that nonparametric autoregression is asymptotically equivalent in the sense of Le Cam's deficiency distance to nonparametric regression with random design as well as with regular nonrandom design., Comment: Published at http://dx.doi.org/10.1214/009053606000000560 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)