Henry L. Gray

Global Warming and the Problem of Testing for Trend in Time Series Data

Abstract In recent years a number of statistical tests have been proposed for testing the hypothesis that global warming is occurring. The standard approach is to examine one or two of the more prominent global temperature datasets by letting Yt = a + bt + Et, where Yt represents the temperature at time t and Et represents error from the trend line, and to test the hypothesis that b = 0. Several authors have applied these tests for trend to determine whether or not a significant long-term or deterministic trend exists, and have generally concluded that there is a significant deterministic trend in the data. However, we show that certain autoregressive-moving average (ARMA) models may also be very reasonable models for these data due to the random trends present in their realizations. In this paper, we provide simulation evidence to show that the tests for trend detect a deterministic trend in a relatively high percentage of realizations from a wide range of ARMA models, including those obtained for the te...

A New Approach to ARMA Modeling.

In recent years the Box-Jenkins method has become a popular technique for forecasting future behavior of a time series. Once adecruate computer packages are available for most purposes. un fortunately the problem of determining the appropriate forecast model has, for models of any complexity, been one of the major stumbling blocks to the user of this method. In this paper a satisfactory solution to that problem is obtained and it is demonstrated by numerous examples how this greatly enlarges the class of data sets which can be adequately modeled by autoregressive-moving average models. This new approach is sufficiently unequivocal that most users will find it easy to implement.

Selecting a Model for Detecting the Presence of a Trend

Abstract The authors consider the problem of determining whether the upward trending behavior in the global temperature anomaly series should be forecast to continue. To address this question, the generic problem of determining whether an observed trend in a time series realization is a random (i.e., short-term) trend or a deterministic (i.e., permanent) trend is considered. The importance of making this determination is that forecasts based on these two scenarios are dramatically different. Forecasts based on a series with random trends will not predict the observed trend to continue, while forecasts based on a model with deterministic trend will forecast the trend to continue into the future. In this paper, the authors consider an autoregressive integrated moving average (ARIMA) model and a “deterministic forcing function + autoregressive (AR) noise” model as possible random trend and deterministic trend models, respectively, for realizations displaying trending behavior. A bootstrap-based classificatio...

A New Test for Outlier Detection from a Multivariate Mixture Distribution

Abstract The problem of testing an outlier from a multivariate mixture distribution of several populations has many important applications in practice. One particular example is in monitoring worldwide nuclear testing, where we wish to detect whether an observed seismic event is possibly a nuclear explosion (an outlier) by comparing it with the training samples from mining blasts and earthquakes. The combined population of seismic events from mining blasts and earthquakes can be viewed as a mixture distribution. The classical likelihood ratio test appears to not be applicable in our problem, and in spite of the importance of this problem, little progress has been made in the literature. This article proposes a simple modified likelihood ratio test that overcomes the difficulties in the current problem. Bootstrap techniques are used to approximate the distribution of the test statistic. The advantages of the new test are demonstrated via simulation studies. Some new computational findings are also reported.

On the Jackknife Statistic and Its Relation to UMVU Estimators in the Normal Case

In this paper the Jacklenife method is utilized in conjunction with the bo-blackwell procedure to obtain a specific formula for the UMVU estimator f any analytic function of u and σ2 in the normal case. The formula given a easy to employ and quite suitable for machine calculation

Nonstationary Data Analysis by Time Deformation

ABSTRACT In this article we discuss methodology for analyzing nonstationary time series whose periodic nature changes approximately linearly with time. We make use of the M-stationary process to describe such data sets, and in particular we use the discrete Euler(p) model to obtain forecasts and estimate the spectral characteristics. We discuss the use of the M-spectrum for displaying linear time-varying periodic content in a time series realization in much the same way that the spectrum shows periodic content within a realization of a stationary series. We also introduce the instantaneous frequency and spectrum of an M-stationary process for purposes of describing how frequency changes with time. To illustrate our techniques we use one simulated data set and two bat echolocation signals that show time varying frequency behavior. Our results indicate that for data whose periodic content is changing approximately linearly in time, the Euler model serves as a very good model for spectral analysis, filtering, and forecasting. Additionally, the instantaneous spectrum is shown to provide better representation of the time-varying frequency content in the data than window-based techniques such as the Gabor and wavelet transforms. Finally, it is noted that the results of this article can be extended to processes whose frequencies change like atα, a > 0, −∞ < α < − ∞.

Time-frequency analysis-G(λ)-stationary processes

Methods such as wavelets, short term Fourier transforms and time deformation [Gray and Zhang, 1988. On a class of nonstationary processes. J. Time Ser. Anal. 9(2), 133-154 and Gray, Vijverberg and Woodward, 2005. Nonstationary data analysis by time deformation. Commun. Statist. A, to appear] have been developed to analyze the time-frequency properties of a process where frequency changes with time. When the frequencies of a process are either monotonically increasing or monotonically decreasing with time, a rather general time deformation approach is to apply an appropriate Box-Cox transformation to the time axis for the given signal in order to obtain a new stationary data set. This data set can then be analyzed by standard methods. Processes which are transformed to a stationary process after Box-Cox transformation of the time scale are called G(@l)-stationary processes, where @l is the corresponding parameter of the Box-Cox transformation. In this paper it is shown that the standard concept of stationarity can be viewed as stationarity under a linear transformation of the index set, while transforming time (time deformation) by a non-linear monotonic transformation introduces the concept of stationarity on a non-linear index set. Thus the notion of stationarity is broadened considerably to allow stationarity on different scales. The method is illustrated with analysis of both simulated and real data. Finally, it is shown that such processes can be transformed to stationarity by sampling properly. The software for performing the analysis discussed in this paper can be downloaded from the website http://faculty.smu.edu/hgray/research.htm.

ON G-SPECTRAL ESTIMATION*

Publisher Summary This chapter discusses a new spectral estimator that appears to be of some value in the problem of spectral estimation. This estimator is closely tied to the ARMA processes, it should be understood that this model is not an underlying assumption. Indeed, the underlying assumption is basically that the spectrum of an associated continuous process satisfies the very general requirement that it is continued fraction integrable. Thus, the method should be fairly robust. The chapter presents the definition of G n - and e n -transformations are defined and some simple examples are given to demonstrate their ability to increase the rate of convergence of certain types of sequences.

The G-Spectral Estimator

Abstract In this article a modified definition of the G-spectral estimator is given. It is shown that the resulting estimator is a method of moments autoregressive moving average (ARMA) spectral estimator that does not require an estimate of the moving average parameters. As a result, a new formula for the power spectrum of an ARMA process is given that does not explicitly involve the moving average (MA) parameters. This formula then leads to a closed-form expression for the MA parameters and their corresponding moment estimators.

Minimum variance unblased estimation in the gamma distribution

In this paper a new infinite series UMVU etimator far general functions of the scale parameter in the gamma distribution wita shape parameter known is presented. The formula gives is easy to employ,and also leads to simple approximations to the UMVU. These approximations are shown to perform well is the examples considered.