THE EFFICIENCY OF TIME SERIES ESTIMATORS FOR REPEATED SURVEYS

The classical approach to the analysis of data from repeated surveys, based on Patterson (1950), has recently been extended by the work of Blight and Scott (1973) and Scott and Smith (1974), by assuming that a time series relationship exists between the population parameters at different times. The purpose of this paper is to compare the efficiencies of these three approaches by computing the mean square error (MSE) of the estimators of the current mean and of the change in mean on the last two occasions.     

COMPOSITIONAL TIME SERIES ANALYSIS OF MORTALITY PROPORTIONS

Compositional time series are multivariate time series which at each time point are proportions that sum to a constant. Accurate inference for such series which occur in several disciplines such as geology, economics and ecology is important in practice. Usual multivariate statistical procedures ignore the inherent constrained nature of these observations as parts of a whole and may lead to inaccurate estimation and prediction. In this article, a regression model with vector autoregressive moving average (VARMA) errors is fit to the compositional time series after an additive log ratio (ALR) transformation. Inference is carried out in a hierarchical Bayesian framework using Markov chain Monte Carlo techniques. The approach is illustrated on compositional time series of mortality events in Los Angeles in order to investigate dependence of different categories of mortality on air quality.     

COMMON FEATURES IN TIME SERIES WITH BOTH DETERMINISTIC AND STOCHASTIC SEASONALITY

This paper extends the notions of common cycles and common seasonal features to time series having deterministic and stochastic seasonality at different frequencies. The conditions under which quarterly time series with these characteristics have common features are investigated, various representations are presented and statistical inference is discussed. Finally, the analysis is applied to study comovements between different components of consumption and income using UK data.     

PARSIMONIOUS PERIODIC TIME SERIES MODELING

This paper studies techniques for fitting parsimonious periodic time series models to periodic data. Large sample standard errors for the parameter estimates in a periodic autoregressive moving‐average time series model under parametric constraints are derived. Likelihood ratio statistics for hypothesis testing are examined. The techniques are applied in modeling daily temperatures at Griffin, Georgia, USA.     

A Review of Some Modern Approaches to the Problem of Trend Extraction

This article presents a review of some modern approaches to trend extraction for one-dimensional time series, which is one of the major tasks of time series analysis. The trend of a time series is usually defined as a smooth additive component which contains information about the time series global change, and we discuss this and other definitions of the trend. We do not aim to review all the novel approaches, but rather to observe the problem from different viewpoints and from different areas of expertise. The article contributes to understanding the concept of a trend and the problem of its extraction. We present an overview of advantages and disadvantages of the approaches under consideration, which are: the model-based approach (MBA), nonparametric linear filtering, singular spectrum analysis (SSA), and wavelets. The MBA assumes the specification of a stochastic time series model, which is usually either an autoregressive integrated moving average (ARIMA) model or a state space model. The nonparametric filtering methods do not require specification of model and are popular because of their simplicity in application. We discuss the Henderson, LOESS, and Hodrick–Prescott filters and their versions derived by exploiting the Reproducing Kernel Hilbert Space methodology. In addition to these prominent approaches, we consider SSA and wavelet methods. SSA is widespread in the geosciences; its algorithm is similar to that of principal components analysis, but SSA is applied to time series. Wavelet methods are the de facto standard for denoising in signal procession, and recent works revealed their potential in trend analysis.     

Bayesian analysis of time series Poisson data

This paper provides a practical simulation-based Bayesian analysis of parameter-driven models for time series Poisson data with the AR(1) latent process. The posterior distribution is simulated by a Gibbs sampling algorithm. Full conditional posterior distributions of unknown variables in the model are given in convenient forms for the Gibbs sampling algorithm. The case with missing observations is also discussed. The methods are applied to real polio data from 1970 to 1983.     

ELICITING A DIRECTED ACYCLIC GRAPH FOR A MULTIVARIATE TIME SERIES OF VEHICLE COUNTS IN A TRAFFIC NETWORK

The problem of modelling multivariate time series of vehicle counts in traffic networks is considered. It is proposed to use a model called the linear multiregression dynamic model (LMDM). The LMDM is a multivariate Bayesian dynamic model which uses any conditional independence and causal structure across the time series to break down the complex multivariate model into simpler univariate dynamic linear models. The conditional independence and causal structure in the time series can be represented by a directed acyclic graph (DAG). The DAG not only gives a useful pictorial representation of the multivariate structure, but it is also used to build the LMDM. Therefore, eliciting a DAG which gives a realistic representation of the series is a crucial part of the modelling process. A DAG is elicited for the multivariate time series of hourly vehicle counts at the junction of three major roads in the UK. A flow diagram is introduced to give a pictorial representation of the possible vehicle routes through the network. It is shown how this flow diagram, together with a map of the network, can suggest a DAG for the time series suitable for use with an LMDM.     

THE BEHAVIOR OF HEGY TESTS FOR QUARTERLY TIME SERIES WITH SEASONAL MEAN SHIFTS

ABSTRACT

This paper studies the behavior of the HEGY statistics for quarterly data, for seasonal autoregressive unit roots, when the analyzed time series is deterministic seasonal stationary but exhibits a change in the seasonal pattern. We analyze also the HEGY test for the nonseasonal unit root. the data generation process being trend stationary too. Our results show that when the break magnitudes are finite, the HEGY test statistics are not asymptotically biased toward the nonrejection of the seasonal and nonseasonal unit root hypotheses. However, the finite sample power properties may be substantially affected, the behavior of the tests depending on the type of the break.     

WAITING TIME IN THE COUPON-COLLECTOR'S PROBLEM1

The coupon collector's problem is generalized by allowing unequal proportions of the various types of coupons. The upper tail probabilities are used to find the probability distribution of the waiting time. The probability generating function is expressed in terms of the hypergeometric functions and therefrom the mean and the variance are derived.     

THE SOJOURN TIME DISTRIBUTION OF AN ALTERNATING RENEWAL PROCESS

Ordinary, modified, and equilibrium alternating renewal processes are defined in such a way that the distribution of the sojourn time in one state of a two-state system may be considered conditional on the starting state. The double Laplace transform of a non-negative stochastic process is also defined, and used to express a result of Takacs (1957) on sojourn time distributions.     

QUASI-STATIONARY BEHAVIOUR IN THE RANDOM WALK WITH CONTINUOUS TIME

The long-term behaviour of the random walk on the positive integers is discussed. In particular, one distribution describing this behaviour when there is a non-zero drift in one direction, is shown to be of discrete gamma-type. Under the same condition, the effectively ultimate time to absorption is exponential.     

Measuring serial dependence in categorical time series

The analysis of time-indexed categorical data is important in many fields, e.g., in telecommunication network monitoring, manufacturing process control, ecology, etc. Primary interest is in detecting and measuring serial associations and dependencies in such data. For cardinal time series analysis, autocorrelation is a convenient and informative measure of serial association. Yet, for categorical time series analysis an analogous convenient measure and corresponding concepts of weak stationarity have not been provided. For two categorical variables, several ways of measuring association have been suggested. This paper reviews such measures and investigates their properties in a serial context. We discuss concepts of weak stationarity of a categorical time series, in particular of stationarity in association measures. Serial association and weak stationarity are studied in the class of discrete ARMA processes introduced by Jacobs and Lewis (J. Time Ser. Anal. 4(1):19–36, 1983). An intrinsic feature of a time series is that, typically, adjacent observations are dependent. The nature of this dependence among observations of a time series is of considerable practical interest. Time series analysis is concerned with techniques for the analysis of this dependence. (Box et al. 1994p. 1)     

BACKWARD ESTIMATION OF STOCHASTIC PROCESSES WITH FAILURE EVENTS AS TIME ORIGINS

Stochastic processes often exhibit sudden systematic changes in pattern a short time before certain failure events. Examples include increase in medical costs before death and decrease in CD4 counts before AIDS diagnosis. To study such terminal behavior of stochastic processes, a natural and direct way is to align the processes using failure events as time origins. This paper studies backward stochastic processes counting time backward from failure events, and proposes one-sample nonparametric estimation of the mean of backward processes when follow-up is subject to left truncation and right censoring. We will discuss benefits of including prevalent cohort data to enlarge the identifiable region and large sample properties of the proposed estimator with related extensions. A SEER-Medicare linked data set is used to illustrate the proposed methodologies.     

TEACHING STATISTICS: A TALK ABOUT TIME SERIES

When presenting research results to a mixed audience, it is common for seminar speakers either to address the specialists or the uninitiated (thus alienating the other, uncatered-for group of listeners), or to risk falling between these two stools (and so satisfy no one). We suggest a way that the lecturer's dilemma might often be more optimally resolved; and, in so doing, present material which could provide useful ideas for wider application in enlivening formal class-room instruction in time-series analysis. Finally, we pre-empt some of the predicted response that the content of this paper may trigger from certain experts; and discuss the general lack of emphasis on assumptions in statistical teaching.     

Decline and fall: Unemployment among Britain's non-white ethnic communities 1960–1999

The declining employment fortunes of Britain's non-white communities, relative to whites, are explained. To achieve this comparison, some historical official sources of unemployment data are reviewed. The earliest known official time series on unemployment of non-whites dates back to 1960. This historical context is explored, especially the reluctance to make the early data more widely known. An unemployment series for non-white males and females from 1970 to 1999 is derived in two separate ways by splicing together official sources. These series are compared with unemployment of whites to demonstrate a relative increase in unemployment of non-whites.     

Bayesian Analysis of Latent Threshold Dynamic Models

We discuss a general approach to dynamic sparsity modeling in multivariate time series analysis. Time-varying parameters are linked to latent processes that are thresholded to induce zero values adaptively, providing natural mechanisms for dynamic variable inclusion/selection. We discuss Bayesian model specification, analysis and prediction in dynamic regressions, time-varying vector autoregressions, and multivariate volatility models using latent thresholding. Application to a topical macroeconomic time series problem illustrates some of the benefits of the approach in terms of statistical and economic interpretations as well as improved predictions. Supplementary materials for this article are available online.     

Disaggregating Data Using Related Series

Quarterly real GNP and implicit GNP deflator series are derived for the 1948–1970 period using the related series technique of Chow and Lin. These estimated series are compared with the official, revised series; to official, unrevised series; and to univariate proxies using regression, time series, and spectral methods. The derived series possess autocorrelation and turning point characteristics similar to those of the official, revised series. The derived series also deliver structural equation parameter estimates similar to those based on official, revised data.     

Multichannel electroencephalographic analyses via dynamic regression models with time-varying lag–lead structure

Multiple time series of scalp electrical potential activity are generated routinely in electroencephalographic (EEG) studies. Such recordings provide important non-invasive data about brain function in human neuropsychiatric disorders. Analyses of EEG traces aim to isolate characteristics of their spatiotemporal dynamics that may be useful in diagnosis, or may improve the understanding of the underlying neurophysiology or may improve treatment through identifying predictors and indicators of clinical outcomes. We discuss the development and application of non-stationary time series models for multiple EEG series generated from individual subjects in a clinical neuropsychiatric setting. The subjects are depressed patients experiencing generalized tonic–clonic seizures elicited by electroconvulsive therapy (ECT) as antidepressant treatment. Two varieties of models—dynamic latent factor models and dynamic regression models—are introduced and studied. We discuss model motivation and form, and aspects of statistical analysis including parameter identifiability, posterior inference and implementation of these models via Markov chain Monte Carlo techniques. In an application to the analysis of a typical set of 19 EEG series recorded during an ECT seizure at different locations over a patient's scalp, these models reveal time-varying features across the series that are strongly related to the placement of the electrodes. We illustrate various model outputs, the exploration of such time-varying spatial structure and its relevance in the ECT study, and in basic EEG research in general.     

ON THE VARIANCE-COVARIANCE MATRIX IN MARKOVIAN MANPOWER SYSTEMS IN CONTINUOUS TIME

The variability of the grade sizes in Markovian manpower systems in continuous time is considered. The differential equations describing the evolution of the variance-covariance matrix are derived and the associated generator is determined in a closed form. The limiting variability of an expanding system is studied via two theorems which provide conditions for the limit to exist independently of the initial distribution and for the rate of convergence to be exponentially fast. A numerical illustration of the theory is given.