The quantile spectral density and comparison based tests for nonlinear time series

In this paper we consider tests for nonlinear time series, which are motivated by the notion of serial dependence. The proposed tests are based on comparisons with the quantile spectral density, which can be considered as a quantile version of the usual spectral density function. The quantile spectral density 'measures' sequential dependence structure of a time series, and is well defined under relatively weak mixing conditions. We propose an estimator for the quantile spectral density and derive its asympototic sampling properties. We use the quantile spectral density to construct a goodness of fit test for time series and explain how this test can also be used for comparing the sequential dependence structure of two time series. The method is illustrated with simulations and some real data examples

Statistical inference for time-varying ARCH processes

In this paper the class of ARCH$(\infty)$ models is generalized to the nonstationary class of ARCH$(\infty)$ models with time-varying coefficients. For fixed time points, a stationary approximation is given leading to the notation ``locally stationary ARCH$(\infty)$ process.'' The asymptotic properties of weighted quasi-likelihood estimators of time-varying ARCH$(p)$ processes ($p<\infty$) are studied, including asymptotic normality. In particular, the extra bias due to nonstationarity of the process is investigated. Moreover, a Taylor expansion of the nonstationary ARCH process in terms of stationary processes is given and it is proved that the time-varying ARCH process can be written as a time-varying Volterra series.Comment: Published at http://dx.doi.org/10.1214/009053606000000227 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

A recursive online algorithm for the estimation of time-varying ARCH parameters

In this paper we propose a recursive online algorithm for estimating the parameters of a time-varying ARCH process. The estimation is done by updating the estimator at time point $t-1$ with observations about the time point $t$ to yield an estimator of the parameter at time point $t$. The sampling properties of this estimator are studied in a non-stationary context -- in particular, asymptotic normality and an expression for the bias due to non-stationarity are established. By running two recursive online algorithms in parallel with different step sizes and taking a linear combination of the estimators, the rate of convergence can be improved for parameter curves from H\"{o}lder classes of order between 1 and 2.Comment: Published at http://dx.doi.org/10.3150/07-BEJ5009 in the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Geology and geochemistry of the Middle Proterozoic Eastern Ghat mobile belt and its comparison with the lower crust of the Southern Peninsular shield

Two prominent rock suites constitute the lithology of the Eastern Ghat mobile belt: (1) the khondalite suite - the metapelites, and (2) the charnockite suite. Later intrusives include ultramafic sequences, anorthosites and granitic gneisses. The chief structural element in the rocks of the Eastern Ghats is a planar fabric (gneissosity), defined by the alignment of platy minerals like flattened quartz, garnet, sillimanite, graphite, etc. The parallelism between the foliation and the lithological layering is related to isoclinal folding. The major structural trend (axial plane foliation trend) observed in the belt is NE-SW. Five major tectonic events have been delineated in the belt. A boundary fault along the western margin of the Eastern Ghats, bordering the low grade terrain has been substantiated by recent gravity and the deep seismic sounding studies. Field evidence shows that the pyroxene granulites (basic granulites) post-date the khondalite suite, but are older than the charnockites as well as the granitic gneisses. Polyphase metamorphism, probably correlatable with different periods of deformation is recorded. The field relations in the Eastern Ghats point to the intense deformation of the terrain, apparently both before, during and after metamorphism