Rank-based estimation for all-pass time series models

An autoregressive-moving average model in which all roots of the autoregressive polynomial are reciprocals of roots of the moving average polynomial and vice versa is called an all-pass time series model. All-pass models are useful for identifying and modeling noncausal and noninvertible autoregressive-moving average processes. We establish asymptotic normality and consistency for rank-based estimators of all-pass model parameters. The estimators are obtained by minimizing the rank-based residual dispersion function given by Jaeckel [Ann. Math. Statist. 43 (1972) 1449--1458]. These estimators can have the same asymptotic efficiency as maximum likelihood estimators and are robust. The behavior of the estimators for finite samples is studied via simulation and rank estimation is used in the deconvolution of a simulated water gun seismogram.Comment: Published at http://dx.doi.org/10.1214/009053606000001316 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Time Series Modeling with Shape Constraints

This thesis focuses on the development of semiparametric estimation methods for a class of time series models using shape constraints. Many of the existing time series models assume the noise follows some known parametric distributions. Typical examples are the Gaussian and t distributions. Then the model parameters are estimated by maximizing the resultant likelihood function. As an example, the autoregressive moving average (ARMA) models (Brockwell and Davis, 2009) assume Gaussian noise sequence and are estimated under the causal-invertible constraint by maximizing the Gaussian likelihood. Although the same estimates can also be used in the causal-invertible non-Gaussian case, they are not asymptotically optimal (Rosenblatt, 2012). Moreover, for the noncausal/noninvertible cases, the Gaussian likelihood estimation procedure is not applicable, since any second-order based methods cannot distinguish between causal-invertible and noncausal/noninvertible models (Brockwell and Davis,2009). As a result, many estimation methods for noncausal/noninvertible ARMA models assume the noise follows a known non-Gaussian distribution, like a Laplace distribution or a t distribution. To relax this distributional assumption and allow noncausal/noninvertible models, we borrow ideas from nonparametric shape-constraint density estimation and propose a semiparametric estimation procedure for general ARMA models by projecting the underlying noise distribution onto the space of log-concave measures (Cule and Samworth, 2010; Dümbgen et al., 2011). We show the maximum likelihood estimators in this semiparametric setting are consistent. In fact, the MLE is robust to the misspecification of log-concavity in cases where the true distribution of the noise is close to its log-concave projection. We derive a lower bound for the best asymptotic variance of regular estimators at rate sqrt(n) for AR models and construct a semiparametric efficient estimator. We also consider modeling time series of counts with shape constraints. Many of the formulated models for count time series are expressed via a pair of generalized state-space equations. In this set-up, the observation equation specifies the conditional distribution of the observation Yt at time t given a state-variable Xt. For count time series, this conditional distribution is usually specified as coming from a known parametric family such as the Poisson or the Negative Binomial distribution. To relax this formal parametric framework, we introduce a concave shape constraint into the one-parameter exponential family. This essentially amounts to assuming that the reference measure is log-concave. In this fashion, we are able to extend the class of observation-driven models studied in Davis and Liu (2016). Under this formulation, there exists a stationary and ergodic solution to the state-space model. In this new modeling framework, we consider the inference problem of estimating both the parameters of the mean model and the log-concave function, corresponding to the reference measure. We then compute and maximize the likelihood function over both the parameters associated with the mean function and the reference measure subject to a concavity constraint. The estimator of the mean function and the conditional distribution are shown to be consistent and perform well compared to a full parametric model specification. The finite sample behavior of the estimators are studied via simulation and two empirical examples are provided to illustrate the methodology

Estimation of time series models using residuals dependence measures

We propose new estimation methods for time series models, possibly noncausal and/or noninvertible, using serial dependence information from the characteristic function of model residuals. This allows to impose the i.i.d. or martingale difference assumptions on the model errors to identify the unknown location of the roots of the lag polynomials for ARMA models without resorting to higher order moments or distributional assumptions. We consider generalized spectral density and cumulative distribution functions to measure residuals dependence at an increasing number of lags under both assumptions and discuss robust inference to higher order dependence when only mean independence is imposed on model errors. We study the consistency and asymptotic distribution of parameter estimates and discuss efficiency when different restrictions on error dependence are used simultaneously, including serial uncorrelation. Optimal weighting of continuous moment conditions yields maximum likelihood efficiency under independence for unknown error distribution. We investigate numerical implementation and finite sample properties of the new classes of estimates.Financial support from Ministerio de Economía y Competitividad (Spain), grants ECO2017-86009-P and PID2020-114664GB-I00 is gratefully acknowledged

Laboratory for Engineering Man/Machine Systems (LEMS): System identification, model reduction and deconvolution filtering using Fourier based modulating signals and high order statistics

Several important problems in the fields of signal processing and model identification, such as system structure identification, frequency response determination, high order model reduction, high resolution frequency analysis, deconvolution filtering, and etc. Each of these topics involves a wide range of applications and has received considerable attention. Using the Fourier based sinusoidal modulating signals, it is shown that a discrete autoregressive model can be constructed for the least squares identification of continuous systems. Some identification algorithms are presented for both SISO and MIMO systems frequency response determination using only transient data. Also, several new schemes for model reduction were developed. Based upon the complex sinusoidal modulating signals, a parametric least squares algorithm for high resolution frequency estimation is proposed. Numerical examples show that the proposed algorithm gives better performance than the usual. Also, the problem was studied of deconvolution and parameter identification of a general noncausal nonminimum phase ARMA system driven by non-Gaussian stationary random processes. Algorithms are introduced for inverse cumulant estimation, both in the frequency domain via the FFT algorithms and in the domain via the least squares algorithm

Parameter Identification of Image Models by the Recursive Maximum Likelihood Method

This paper considers the problem of identifying the blur parameters of the observed image. It is assumed that the original image is a sample from the homogeneous random field described by a two-dimensional (2-D) semicausal model, and that the point spread function (PSF) characterizing the image blur is symmetric. It is also assumed that the observation noise is negligibly small. By applying the discrete sine transform, we derive a set of nearly uncorrelated ARMA models, which are of non-minimum phase, for the blurred image. Although all-pass components of the MA part of the models can not be estimated, we show that the parameters of the non-minimum phase MA part can be restored by exploiting the fact that the PSF is symmetric. We develop a new algorithm for identifying the blur parameters of the image model from the MA parameters estimated by the recursive maximum likelihood (RML) method. Simulation studies are also included to show the feasibility of the algorithm

Spectral estimation for mixed causal-noncausal autoregressive models

This paper investigates new ways of estimating and identifying causal, noncausal, and mixed causal-noncausal autoregressive models driven by a non-Gaussian error sequence. We do not assume any parametric distribution function for the innovations. Instead, we use the information of higher-order cumulants, combining the spectrum and the bispectrum in a minimum distance estimation. We show how to circumvent the nonlinearity of the parameters and the multimodality in the noncausal and mixed models by selecting the appropriate initial values in the estimation. In addition, we propose a method of identification using a simple comparison criterion based on the global minimum of the estimation function. By means of a Monte Carlo study, we find unbiased estimated parameters and a correct identification as the data depart from normality. We propose an empirical application on eight monthly commodity prices, finding noncausal and mixed causal-noncausal dynamics

Frequency domain minimum distance inference for possibly noninvertible and noncausal arma models.

This article introduces frequency domain minimum distance procedures for performing inference in general, possibly non causal and/or noninvertible, autoregressive moving average (ARMA) models. We use information from higher order moments to achieve identification on the location of the roots of the AR and MA polynomials for non-Gaussian time series. We propose a minimum distance estimator that optimally combines the information contained in second, third, and fourth moments. Contrary to existing estimators, the proposed one is consistent under general assumptions, and may improve on the efficiency of estimators based on only second order moments. Our procedures are also applicable for processes for which either the third or the fourth order spectral density is the zero function.Supported by Ministerio Economía y Competitividad (Spain), Grants ECO2012-31748, ECO2014-57007p and MDM 2014-0431, and Comunidad de Madrid, MadEco-CM (S2015/HUM- 3444). Supported by Asociación Mexicana de Cultura and from the Mexican Consejo Nacional de Ciencia y Tecnología (CONACYT) under project Grant 151624

Optimal Forecasting of Noncausal Autoregressive Time Series

In this paper, we propose a simulation-based method for computing point and density forecasts for univariate noncausal and non-Gaussian autoregressive processes. Numerical methods are needed to forecast such time series because the prediction problem is generally nonlinear and no analytic solution is therefore available. According to a limited simulation experiment, the use of a correct noncausal model can lead to substantial gains in forecast accuracy over the corresponding causal model. An empirical application to U.S. inflation demonstrates the importance of allowing for noncausality in improving point and density forecasts.Noncausal autoregression; density forecast; inflation