Non-Gaussian Time Series:

My interest in non-Gaussian linear time series models goes back to my dissertation research, on the structure of innovations for non-Gaussian linear processes. Even the well-known autoregressive moving average models become a much richer class when driven by non-Gaussian noise. In contrast to the Gaussian case, it is possible to distinguish among causal/noncausal and invertible/noninvertible systems in the non-Gaussian case. Time-irreversibility is the norm for non-Gaussian linear processes, and the processes can exhibit many other interesting behaviors. For example, in the analysis of returns on financial assets such as stocks, it is common to observe lack of serial correlation, heavy-tailed marginal distributions, and volatility clustering. Typically, nonlinear models with time-dependent conditional variances, such as ARCH and stochastic volatility models, are suggested for such time series. It is perhaps less well known that linear, non-Gaussian models can display exactly this behavior.

The linear models which display this behavior are all-pass models: autoregressive-moving average models in which all of the roots of the autoregressive polynomial are reciprocals of roots of the moving average polynomial and vice versa. All-pass models generate uncorrelated (white noise) time series, but these series are not independent in the non-Gaussian case. If the process is driven with heavy-tailed noise, then its marginal distribution will also have heavy tails, and the process will exhibit volatility clustering.

In Breidt, Davis, and Trindade (2001), we considered the problem of estimation of all-pass models using a least absolute deviations (LAD) criterion. With graduate student Beth Andrews, Richard Davis and I are currently extending this work in a number of ways, including LAD estimation for heavy-tailed processes, minimum dispersion estimation, and maximum likelihood estimation. We also plan to study properties of the sample autocorrelation function of transformed all-pass series in the heavy-tailed case. These results may lead to identification and preliminary estimation methods for all-pass. Ultimately, we plan to use all-pass modelling in a sequential method for identifying and estimating noncausal and/or noninvertible ARMA models.

Environmental Monitoring:

Environmental monitoring has been a long-standing interest of mine. My masters project involved establishing ambient water quality criteria for a part of the Delaware Water Gap, in association with the National Park Service's Water Resources Division. As a faculty member at Iowa State, I was part of a team responsible for the design and analysis of the USDA's National Resources Inventory (NRI), a nationwide survey of land resources, based on 300,000 primary sampling units. I am currently involved in funded research to improve the quality of estimates from the US Forest Service's Forest Inventory and Analysis (FIA) program, through effective use of remotely-sensed data, and through combination with other inventory data (such as NRI). Further, I am part of a research team at CSU working on the Space-Time Aquatic Resources Modeling and Analysis Program (STARMAP), and I am part of a parallel program at Oregon State. Both programs are supported by STAR grants from the Environmental Protection Agency (EPA). The two programs are concerned with design and analysis of surveys of aquatic resources, particularly the EPA's EMAP surface waters studies. Further, I've been working with scientists in CSU's Natural Resources Ecology Lab on problems of carbon sequestration in agricultural soils. All of these projects fit in with a new NSF-funded IGERT called PRIMES (PRogram for Interdisciplinary Mathematics, Statistics, and Ecology), which offers generous support for graduate students interested in quantitative ecology. I was one of the principal writers of this proposal and I serve on the PRIMES Council which directs the program.

Nonparametric Regression in Surveys:

Most of the studies described above involve probability surveys. The development of new theory and methodology for complex surveys has been a major research theme for me. The field of survey sampling often has a nonparametric flavor because the operational constraints of large-scale, complex surveys make case-by-case parametric modeling impractical in many instances. With my colleague Jean Opsomer at Iowa State, I have been working on nonparametric methods for the efficient use of auxiliary information in complex surveys. Our current research extends the work of Breidt and Opsomer (2000), in which we proposed a new class of model-assisted estimators based on local polynomial regression. The resulting estimators are nonparametric versions of the standard, generalized regression estimators. The nonparametric estimator shares most of the desirable properties of the parametric regression estimators, but under much weaker assumptions on the superpopulation model. Theoretical and simulation experiments indicate that the estimators are more efficient than parametrically specified regression estimators when the model regression function is incorrectly specified, while being approximately as efficient when the parametric specification is correct.

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