## Digitized Theses

1992

Dissertation

#### Degree Name

Doctor of Philosophy

#### Abstract

A time series, {dollar}\{lcub}Z(t)\{rcub}{dollar}, is called integrated stationary if there is a non-negative integer d such that {dollar}\{lcub}W(t) = \bigtriangledown\sp{lcub}d{rcub}Z(t)\{rcub}{dollar} is second-order stationary, where {dollar}\bigtriangledown{dollar} is the backward differencing operator. A special case is the ARIMA (p,d,q) series.;By generalizing Cressie's (1988) results, the generalized and primary increment vectors of order b are defined. These vectors provide a more general way of transforming a homogeneous nonstationary series to stationarity than (repeated) simple differencing through the {dollar}\bigtriangledown\sp{lcub}d{rcub}{dollar} operator. Resulting from the increment-vector methodology are several second-order moment characterizations of an integrated stationary series: the variogram, the generalized covariance function and two types of polyvariogram. The interrelationships and properties of these characterizations are investigated.;The increment-vector methodology also leads to a representation of {dollar}\{lcub}Z(t)\{rcub}{dollar} which reflects the intrinsic stochastic nature of the series and is proved to agree with a special representation defined by Matheron (1973) for spatial processes. From our representation, a theorem for a decomposition of the series is deduced. Using the representation and decomposition theorem we can analyse what happens when overdifferencing takes place and what the divergence rate of the series is (a law of the iterated logarithm is proved).;For two types of polyvariogram of order b, where integer {dollar}b \ge{dollar} max{dollar}\{lcub}0, d- 1\{rcub}{dollar}, the general formulae of the asymptotes are obtained, which show a positive slope when {dollar}b = d - 1{dollar} and zero slope when {dollar}b > d - 1{dollar}. This is the key feature of the polyvariograms which is exploited in the graphical identification of d. The asymptotic distribution and the almost sure convergence rate of the sample polyvariograms are obtained for various b, d and {dollar}\{lcub}W(t)\{rcub}{dollar}. Based on these results, we propose some statistical testing procedures for determining d given an integrated white noise or an integrated ARMA series.

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