Squared-residual autocorrelations have been found useful in detecting nonlinear types of statistical dependence in the residuals of fitted autoregressive-moving average (ARMA) models (Granger and Andersen, 1978; Miller, 1979). In this note it is shown that the normalized squared-residual autocorrelations are asymptotically unit multivariate normal. The results of a simulation experiment confirming the small-sample validity of the proposed tests is reported.

Summary The large‐sample distribution of the multivariate residual autocorrelations in the vector arma model is derived. This result is somewhat less complicated for the vector autoregressive model. A new multivariate portmanteau test for checking the adequacy of fitted vector arma models is developed. A simulation study shows that a simple modification of the portmanteau test improves its accuracy in small samples.

Abstract. An overview of model building with periodic autoregression (PAR) models is given emphasizing the three stages of model development: identification, estimation and diagnostic checking. New results on the distribution of residual autocorrelations and suitable diagnostic checks are derived. The validity of these checks is demonstrated by simulation. The methodology discussed is illustrated with an application. It is pointed out that the PAR approach to model development offers some important advantages over the more general approach using periodic autoregressive moving-average (PARMA) models.I have written S functions for the periodic autoregressive modelling methods discussed in my paper. Complete S style documentation for each function is provided. To obtain, e-mail the following message: send pear from S to statlib@temper.stat.cmu.edu or use anonymous ftp to connect to fisher.stats.uwo.ca and download the shar archive file, pear.sh, located in the directory pub/pear.

Previous research related to the controversial Hurst phenomenon is reviewed and evaluated. Because of the inherent statistical properties of the rescaled adjusted range (RAR) statistic it is suggested that research primarily be devoted to this statistic rather than to the various definitions of the Hurst coefficient. Simulation studies reveal that for independently distributed random variables the RAR does not significantly depend on the underlying distribution of the random variables but is a function of the sample size. For modeling correlated data the statistical attributes of a discrete fractional Gaussian noise (FGN) process are studied and also improved. An efficient maximum likelihood estimation technique is developed for the FGN model, and it is shown how the residuals of the fitted model can be calculated and then subjected to diagnostic checks. An exact simulation procedure is developed for simulating FGN in such a way that synthetic traces from the model lie outside the Brownian domain. The Akaike information criterion (AIC) is suggested as a method for choosing between a FGN and a Box-Jenkins model. For the six annual river flow series that are considered the AIC selects the best Box-Jenkins model in preference to the FGN process for each data set. Because Box-Jenkins models can be shown to preserve the historical RAR, in many practical applications it may be advantageous to use a Box-Jenkins model instead of a FGN process.

Summary The large sample distribution of the residual autocorrelations in the arma model is derived. The main advantage of this derivation over that of Box and Pierce (1970) is that it extends directly to more general situations. Generalizations of the derived distribution are presented for the residual autocorrelations in the multiplicative seasonal arma model and for the autocorrelations of a subseries of the residuals.

Box‐Jenkins modeling of time series data can be improved and simplified by adhering to contemporary modeling procedures. This paper gives the theory and techniques of the application of many recent advances that have been made at the identification, estimation, and diagnostic check stages of model development. The inverse autocorrelation function and the inverse partial autocorrelation function are demonstrated to be useful identification tools for both nonseasonal and seasonal models. Parameters can be estimated more efficiently by employing the modified sum of squares technique. At the estimation stage it is also possible to obtain a maximum likelihood estimate for a Box‐Cox power transformation. The Akaike information criterion is introduced to formalize mathematically the concept of model parsimony. When checking for model adequacy, knowing the distribution of the residual autocorrelation allows for a sensitive test for residual whiteness. Diagnostic checks are given for verifying the assumption of homoscedasticity of the model residuals. In practice, heteroscedasticity and nonnormality of the residuals can often be removed by a Box‐Cox transformation.

Ischemic strokes account for ≈87% of all types of strokes; the distribution of ischemic stroke subtypes varies in different parts of the world.1 In Asia and South America, small vessel disease is the most prominent ischemic stroke subtype, whereas in Europe and the United States there is regional and ethnic variation in the distribution of stroke subtypes and their risk factors.2 Correctly identifying the cause of stroke is important for selection of the appropriate therapy to best reduce the risk of recurrence. 3 This may be particularly important in patients with minor or moderate stroke/transient ischemic attack (TIA), because they are less disabled, so will have more to lose from a recurrent stroke. With changing patterns of practice, in particular, increasing use of statins in the past decade, the distribution of risk factors and stroke subtypes in this population is expected to change over time. Changes in stroke subtypes resulting from these changes in practice can be expected to lead to changes in how physicians view the likelihood of different causes of stroke among their patients and plan strategies for investigation of their patients.The motivation for this study was the clinical suspicion, on the part of 2 senior stroke neurologists at our center, that cardioembolic strokes seemed to be increasing as a proportion of new patients referred to our local urgent TIA clinic. Our primary objective was to determine secular trends in ischemic stroke subtypes. We hypothesized that with more intensive management of atherosclerotic risk factors, there will have been a decrease in atherosclerotic risk factors and a decrease in large artery atherosclerosis and small vessel disease, and, in consequence, a proportional increase in cardioembolic stroke/TIA. Methods Study Setting and TimelineThis was a retrospective cohort study of patients diagnosed with minor or moderate stroke/TIA at the urgent TIA Clinic at University Hospital, a designated regional stroke hospital in London, Ontario. Based on Census reports from Statistics Canada, 599 538 residents were living in the referral area in 2006, and 619 881 residents were recorded in 2011. According to the 2011 census, 82% of the population of London are white, 2.7% Latin American, 2.6% Arab, 2.4% black, 2.2% South Asian, 2.0% Chinese, 1.9% Aboriginal, 1% Southeast Asian, 0.8% West Asian, 0.8% Korean, 0.6% Filipino, and 0.7% belong to other groups. In the surrounding farming area a higher Background and Purpose-Early diagnosis and treatment of a stroke improves patient outcomes, and knowledge of the cause of the initial event is crucial to identification of the appropriate therapy to maximally reduce risk of recurrence. Assumptions based on historical frequency of ischemic subtypes may need revision if stroke subtypes are changing as a result of recent changes in therapy, such as increased use of statins. Methods-We analyzed secular trends in stroke risk factors and ischemic stroke subtypes among patients with transient ischemic attack or minor or moderate stroke refer...

A general trend analysis methodology is developed for detecting and modelling trends in water quality time series measured in rivers and streams. The procedure is specifically designed for use with typically ill-behaved river quality series characterized by problematic features such as non-normal positively skewed populations, irregularly spaced instantaneous observations, seasonal periodicities, and dependence among water quality variables and riverflows. In order to analyze these "messy" environmental data in a systematic and rigorous fashion, the overall trend analysis approach is divided into the two main categories of graphical studies and trend tests. Within these two main steps, specific graphical, parametric and nonparametric statistical techniques are utilized. Graphical methods used in the procedure include time series plots, robust regression smooths, as well as box and whisker graphs. Nonparametric techniques include the Mann-Kendall and Kruskal-Wallis tests. Additionally, a test based on Spearman's partial rank correlation is introduced as a means for eliminating seasonal effects when testing for the presence of a trend. The efficacy of the trend analysis methodology is explained and demonstrated by applying it to water quality time series observed in the Saugeen and Grand Rivers of Southwestern Ontario, Can ad a.