In this paper we construct high moment partial sum processes based on residuals of a GARCH model when the mean is known to be 0. We consider partial sums of kth powers of residuals, CUSUM processes and self-normalized partial sum processes. The kth power partial sum process converges to a Brownian process plus a correction term, where the correction term depends on the kth moment µ k of the innovation sequence. If µ k = 0, then the correction term is 0 and, thus, the kth power partial sum process converges weakly to the same Gaussian process as does the kth power partial sum of the i.i.d. innovations sequence. In particular, since µ1 = 0, this holds for the first moment partial sum process, but fails for the second moment partial sum process. We also consider the CUSUM and the self-normalized processes, that is, standardized by the residual sample variance. These behave as if the residuals were asymptotically i.i.d. We also study the joint distribution of the kth and (k + 1)st self-normalized partial sum processes. Applications to change-point problems and goodness-of-fit are considered, in particular, CUSUM statistics for testing GARCH model structure change and the Jarque-Bera omnibus statistic for testing normality of the unobservable innovation distribution of a GARCH model. The use of residuals for constructing a kernel density function estimation of the innovation distribution is discussed.

This paper presents nonparametric tests of independence that can be used to test the independence of p random variables, serial independence for time series, or residuals data. These tests are shown to generalize the classical portmanteau statistics. Applications to both time series and regression residuals are discussed. Academic PressAMS 1990 subject classifications: 62G10; 60F05; 62E20.

We consider a stochastic fire growth model, with the aim of predicting the behaviour of large forest fires. Such a model can describe not only average growth, but also the variability of the growth. Implementing such a model in a computing environment allows one to obtain probability contour plots, burn size distributions, and distributions of time to specified events. Such a model also allows the incorporation of a stochastic spotting mechanism.

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Determining catchment responses to climate signals gives insight into the potential effects of climate change. This study tested the hypothesis that a 28‐year time series of water yields from four headwater catchments in the Turkey Lakes Watershed (TLW), Ontario contains signals of non‐stationary climate change and naturally occurring stationary climate oscillations and that the effects of these signals are greater in catchments with lower rates of change in water loading and lower water storage capacity (small wetlands). Non‐stationary trends explained 0%, 18%, 44%, and 52% of the variance in the water yields of the four catchments. Wavelet analysis using Morlet wavelets identified stationary responses at multiple temporal scales, increasing the amount of variance explained to 56%, 63%, 76%, and 81% when combining stationary and non‐stationary models. The catchment with low water loading and low water storage was most sensitive to non‐stationary and stationary signals, suggesting that these catchments act as sentinels to detect climatic signals. Wavelet coherence analysis revealed correlations between global climate oscillation indices and water yield. The Atlantic Multidecadal Oscillation (AMO) index was strongly correlated with both temperature and precipitation (R2 = 0.46, P < 0.001 and R2 of 0.34, P < 0.001, respectively). Temperature in the TLW increased by 0.067 °C per year from 1981 to 2008, but approximately 0.037 °C of this increase can be explained by the AMO index. While it is likely that anthropogenic climate change impacts water yields, it is important to account for multiple nested climate oscillations to avoid exaggerating its effects. Copyright © 2012 John Wiley & Sons, Ltd.