Resampling-based bias-corrected time series prediction

In this paper, we consider estimation of the mean squared prediction error (MSPE) of the best linear predictor of (possibly) nonlinear functions of finitely many future observations in a stationary time series. We develop a resampling methodology for estimating the MSPE when the unknown parameters in the best linear predictor are estimated. Further, we propose a bias corrected MSPE estimator based on the bootstrap and establish its second order accuracy. Finite sample properties of the method are investigated through a simulation study.     

Bootstrap-based bias corrections for INAR count time series

Integer-valued autoregressive (INAR) processes form a very useful class of processes suitable to model time series of counts. Several practically relevant estimators based on INAR data are known to be systematically biased away from their population values, e.g. sample autocovariances, sample autocorrelations, or the dispersion index. We propose to do bias correction for such estimators by using a recently proposed INAR-type bootstrap scheme that is tailor-made for INAR processes, and which has been proven to be asymptotically consistent under general conditions. This INAR bootstrap allows an implementation with and without parametrically specifying the innovations' distribution. To judge the potential of corresponding bias correction, we compare these bootstraps in simulations to several competitors that include the AR bootstrap and block bootstrap. Finally, we conclude with an illustrative data application.     

On the estimation of an instantaneous transformation for time series

We modify Ramsay's algorithm for estimating monotonic transformations in regression and extend it to autoregression, where strict monotonicity is an essential requirement. Compared with other methods, our method can capture some characteristics that are pertinent to the time series and is much easier to implement. An order selection method is introduced and developed. Some real data sets are analysed.     

Nonparametric multistep-ahead prediction in time series analysis

Summary.  We consider the problem of multistep-ahead prediction in time series analysis by using nonparametric smoothing techniques. Forecasting is always one of the main objectives in time series analysis. Research has shown that non-linear time series models have certain advantages in multistep-ahead forecasting. Traditionally, nonparametric k -step-ahead least squares prediction for non-linear autoregressive AR( d ) models is done by estimating E ( X _{t + k}  | X _t , …,  X _{t − d +1}) via nonparametric smoothing of X _{t + k} on ( X _t , …,  X _{t − d +1}) directly. We propose a multistage nonparametric predictor. We show that the new predictor has smaller asymptotic mean-squared error than the direct smoother, though the convergence rate is the same. Hence, the predictor proposed is more efficient. Some simulation results, advice for practical bandwidth selection and a real data example are provided.     

The echelon Markov and Chebyshev inequalities

Abstract

A multivariate version of the sharp Markov inequality is derived, when associated probabilities are extended to segments of the supports of non-negative random variables, where the probabilities take echelon forms. It is shown that when some positive lower bounds of these probabilities are available, the multivariate Markov inequality without the echelon forms is improved. The corresponding results for Chebyshev’s inequality are also obtained.     

Dynamic changepoint detection in count time series: a particle filter approach

We study Bayesian dynamic models for detecting changepoints in count time series that present structural breaks. As the inferential approach, we develop a parameter learning version of the algorithm proposed by Chopin [Chopin N. Dynamic detection of changepoints in long time series. Annals of the Institute of Statistical Mathematics 2007;59:349–366.], called the Chopin filter with parameter learning, which allows us to estimate the static parameters in the model. In this extension, the static parameters are addressed by using the kernel smoothing approximations proposed by Liu and West [Liu J, West M. Combined parameters and state estimation in simulation-based filtering. In: Doucet A, de Freitas N, Gordon N, editors. Sequential Monte Carlo methods in practice. New York: Springer-Verlag; 2001]. The proposed methodology is then applied to both simulated and real data sets and the time series models include distributions that allow for overdispersion and/or zero inflation. Since our procedure is general, robust and naturally adaptive because the particle filter approach does not require restrictive specifications to ensure its validity and effectiveness, we believe it is a valuable alternative for dealing with the problem of detecting changepoints in count time series. The proposed methodology is also suitable for count time series with no changepoints and for independent count data.     

On nonlinear models for time series

The paper is a review of nonlinear processes used in time series analysis and presents some new original results about stationary distribution of a nonlinear autoregres-sive process of the first order. The following models are considered: nonlinear autoregessive processes, threshold AR processes, threshold MA processes, bilinear models, auto-regressive models with random parameters including double stochastic models, exponential AR models, generalized threshold models and smooth transition autoregressive models, Some tests for linearity of processes are also presented.     

On prediction and mean squared error

Practical questions motivate the search for predictors either of an as yet unobserved random vector, or of a random function of a parameter. An extension of the classical UMVUE theory is presented to cover such situations. In includes a Rao-Blackwell-type theorem, a Cramer-Rao-type inequality, and necessary and sufficient conditions for a predictor to minimize the mean squared error uniformly in the parameter. Applications are considered to the problem of selected means, the species problem, and the examination of some u-v estimates of Robbins (1988).     

Visual analysis of categorical time series

While at least some standard graphical tools do exist for cardinal time series analysis, little research effort has been given directed towards the visualization of categorical time series. The repertoire of such visual methods is nearly exclusively restricted to few isolated proposals from computer science and biology. This article aims at presenting a toolbox of known and newly developed approaches for analysing given categorical time series data visually. Among these tools, especially the rate evolution graph, the circle transformation, pattern histograms and control charts are promising.     

A new class of INAR(1) model for count time series

The present work proposes a new integer valued autoregressive model with Poisson marginal distribution based on the mixing Pegram and dependent Bernoulli thinning operators. Properties of the model are discussed. We consider several methods for estimating the unknown parameters of the model. Also, the classical and Bayesian approaches are used for forecasting. Simulations are performed for the performance of these estimators and forecasting methods. Finally, the analysis of two real data has been presented for illustrative purposes.     

Goodness-of-fit Procedures for Copula Models Based on the Probability Integral Transformation

Abstract.  Wang & Wells [ J. Amer. Statist. Assoc. 95 (2000) 62] describe a non-parametric approach for checking whether the dependence structure of a random sample of censored bivariate data is appropriately modelled by a given family of Archimedean copulas. Their procedure is based on a truncated version of the Kendall process introduced by Genest & Rivest [ J. Amer. Statist. Assoc. 88 (1993) 1034] and later studied by Barbe et al . [ J. Multivariate Anal. 58 (1996) 197]. Although Wang & Wells (2000) determine the asymptotic behaviour of their truncated process, their model selection method is based exclusively on the observed value of its L ²-norm. This paper shows how to compute asymptotic p -values for various goodness-of-fit test statistics based on a non-truncated version of Kendall's process. Conditions for weak convergence are met in the most common copula models, whether Archimedean or not. The empirical behaviour of the proposed goodness-of-fit tests is studied by simulation, and power comparisons are made with a test proposed by Shih [ Biometrika 85 (1998) 189] for the gamma frailty family.     

Volatility model selection for extremes of financial time series

Although both widely used in the financial industry, there is quite often very little justification why GARCH or stochastic volatility is preferred over the other in practice. Most of the relevant literature focuses on the comparison of the fit of various volatility models to a particular data set, which sometimes may be inconclusive due to the statistical similarities of both processes. With an ever growing interest among the financial industry in the risk of extreme price movements, it is natural to consider the selection between both models from an extreme value perspective. By studying the dependence structure of the extreme values of a given series, we are able to clearly distinguish GARCH and stochastic volatility models and to test statistically which one better captures the observed tail behaviour. We illustrate the performance of the method using some stock market returns and find that different volatility models may give a better fit to the upper or lower tails.     

On the nearly nonstationary seasonal time series

A time series is said to be nearly nonstationary if some of its characteristic roots are close to the unit circle. For a seasonal time series, such a notion of near-nonstationarity is studied in a double-array setting. This approach not only furnishes a natural transition between stationarity and nonstationarity, but also unifies the corresponding asymptotic theories in a seasonal-time-series context. The general theory is expressed in terms of functionals of independent diffusion processes. The asymptotic results have applications to estimation and testing in a nearly nonstationary situation and serve as a useful alternative to the common practice of seasonal adjustment.     

On aggregation of strongly dependent time series

We consider cross-sectional aggregation of time series with long-range dependence. This question arises for instance from the statistical analysis of networks where aggregation is defined via routing matrices. Asymptotically, aggregation turns out to increase dependence substantially, transforming a hyperbolic decay of autocorrelations to a slowly varying rate. This effect has direct consequences for statistical inference. For instance, unusually slow rates of convergence for nonparametric trend estimators and nonstandard formulas for optimal bandwidths are obtained. The situation changes, when time-dependent aggregation is applied. Suitably chosen time-dependent aggregation schemes can preserve a hyperbolic rate or even eliminate autocorrelations completely.     

Test of parameter changes in a class of observation-driven models for count time series

Abstract

This paper investigates the parameter-change tests for a class of observation-driven models for count time series. We propose two cumulative sum (CUSUM) test procedures for detection of changes in model parameters. Under regularity conditions, the asymptotic null distributions of the test statistics are established. In addition, the integer-valued generalized autoregressive conditional heteroskedastic (INGARCH) processes with conditional negative binomial distributions are investigated. The developed techniques are examined through simulation studies and also are illustrated using an empirical example.     

Nonparametric prediction of future order statistics

Prediction of censored order statistics from a Type-II censored sample can be done with trivial bounds having perfect confidence. However, given independent samples from the same absolutely continuous distribution, improved bounds can be attained. In this regard, we develop here point prediction based on L-statistics for predicting order statistics (OS) from a future sample as well as for predicting censored OS from a Type-II censored sample. An example is taken to illustrate these ideas, and the limiting case wherein a single independent sample is arbitrarily large is also discussed.     

Forecasting time series with missing data using Holt's model

This paper deals with the prediction of time series with missing data using an alternative formulation for Holt's model with additive errors. This formulation simplifies both the calculus of maximum likelihood estimators of all the unknowns in the model and the calculus of point forecasts. In the presence of missing data, the EM algorithm is used to obtain maximum likelihood estimates and point forecasts. Based on this application we propose a leave-one-out algorithm for the data transformation selection problem which allows us to analyse Holt's model with multiplicative errors. Some numerical results show the performance of these procedures for obtaining robust forecasts.     

On prediction with time dependent arma models

The general nonstationary ARMA model with time dependent coefficients is considered and a necessary and sufficient condition for the existence of the solution of such a model is given. A simple form of the above solution is expressed as a one-sided-moving average, in terms of one-sided Green's functions. Using these results we solve the Prediction Problem and give some expressions for the h(>0) step-ahead linear predictor of such a process. A few examples are added to illustrate the theory.