Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

Convergence of Probability Measures

In this book, Andrew Harvey sets out to provide a unified and comprehensive theory of structural time series models. Unlike the traditional ARIMA models, structural time series models consist explicitly of unobserved components, such as trends and seasonals, which have a direct interpretation. As a result the model selection methodology associated with structural models is much closer to econometric methodology. The link with econometrics is made even closer by the natural way in which the models can be extended to include explanatory variables and to cope with multivariate time series. From the technical point of view, state space models and the Kalman filter play a key role in the statistical treatment of structural time series models. The book includes a detailed treatment of the Kalman filter. This technique was originally developed in control engineering, but is becoming increasingly important in fields such as economics and operations research. This book is concerned primarily with modelling economic and social time series, and with addressing the special problems which the treatment of such series poses. The properties of the models and the methodological techniques used to select them are illustrated with various applications. These range from the modellling of trends and cycles in US macroeconomic time series to to an evaluation of the effects of seat belt legislation in the UK.

Forecasting, Structural Time Series Models and the Kalman Filter

https://www.robjhyndman.com/papers/ijf25.pdf

25 years of time series forecasting

The Analysis of Variance.

(1992). Time Series—A Biostatistical Introduction. Technometrics: Vol. 34, No. 2, pp. 229-230.

Time Series—A Biostatistical Introduction

The statistical analysis of functional data is a growing need in many research areas. In particular, a robust methodology is important to study curves, which are the output of many experiments in applied statistics. As a starting point for this robust analysis, we propose, analyze, and apply a new definition of depth for functional observations based on the graphic representation of the curves. Given a collection of functions, it establishes the “centrality” of an observation and provides a natural center-outward ordering of the sample curves. Robust statistics, such as the median function or a trimmed mean function, can be defined from this depth definition. Its finite-dimensional version provides a new depth for multivariate data that is computationally feasible and useful for studying high-dimensional observations. Thus, this new depth is also suitable for complex observations such as microarray data, images, and those arising in some recent marketing and financial studies. Natural properties of these ...

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On the Concept of Depth for Functional Data

Bootstrap prediction for returns and volatilities in GARCH models

A half-region depth for functional data

SUMMARY We propose a new method to visualize and detect shape outliers in samples of curves. In functional data analysis, we observe curves defined over a given real interval and shape outliers may be defined as those curves that exhibit a different shape from the rest of the sample. Whereas magnitude outliers, that is, curves that lie outside the range of the majority of the data, are in general easy to identify, shape outliers are often masked among the rest of the curves and thus difficult to detect. In this article, we exploit the relationship between two measures of depth for functional data to help to visualize curves in terms of shape and to develop an algorithm for shape outlier detection. We illustrate the use of the visualization tool, the outliergram, through several examples and analyze the performance of the algorithm on a simulation study. Finally, we apply our method to assess cluster quality in a real set of time course microarray data.

Shape outlier detection and visualization for functional data: the outliergram.

In this study, we propose a new bootstrap strategy to obtain prediction intervals for autoregressive integrated moving-average processes. Its main advantage over other bootstrap methods previously proposed for autoregressive integrated processes is that variability due to parameter estimation can be incorporated into prediction intervals without requiring the backward representation of the process. Consequently, the procedure is very flexible and can be extended to processes even if their backward representation is not available. Furthermore, its implementation is very simple. The asymptotic properties of the bootstrap prediction densities are obtained. Extensive finite-sample Monte Carlo experiments are carried out to compare the performance of the proposed strategy vs. alternative procedures. The behaviour of our proposal equals or outperforms the alternatives in most of the cases. Furthermore, our bootstrap strategy is also applied for the first time to obtain the prediction density of processes with moving-average components.

Juan Romo

Papers

On the Concept of Depth for Functional Data

Bootstrap prediction for returns and volatilities in GARCH models

A half-region depth for functional data

Shape outlier detection and visualization for functional data: the outliergram.

Bootstrap Predictive Inference for ARIMA Processes