R. Douglas Martin

Robust Estimation of the First-Order Autoregressive Parameter

Abstract Outliers in time series can adversely affect both the least squares estimates and ordinary M-estimates of autoregressive parameters. Attention is focused here on obtaining robust estimates of the parameter for a first-order autoregressive time series xk The observations are y k = z k + v k, and two models are considered: Model IO, with v k ≡ 0, x k possibly non-Gaussian, and Model AO, with v k nonzero and possibly quite large a small fraction of the time, and x k Gaussian. A class of generalized M-estimates is proposed which has attractive mean-squared-error robustness properties towards both IO and AO type deviations from the Gaussian model.

ROBUST METHODS FOR TIME SERIES

Outliers in time series can wreak havoc with conventional least-squares procedures, just as in the case of ordinary regression. This paper presents two time-series outlier models, points out their ordinary regression analogues and the corresponding outlier patterns, and presents robust alternatives to the least-squares method of fitting autoregressive-moving-average models. The main emphasis is on robust estimation in the presence of additive outliers. This results in the problem having an errors-in-variables aspect. While several methods of robust estimation for this problem are presented, the most attractive approach is an approximate non-Gaussian maximum-likelihood type method which involves the use of a robust non-linear filter/one-sided interpolator with data-dependent scaling. Robust smoothing/two-sided outlier interpolation, forecasting, model selection, and spectral analysis are briefly mentioned, as are the problems of estimating location and dealing with trends, seasonality, and missing data. Some examples of applying the methodology are given.

Modeling flat stretches, bursts, and outliers in time series using mixture transition distribution models

Abstract The class of mixture transition distribution (MTD) time series models is extended to general non-Gaussian time series. In these models the conditional distribution of the current observation given the past is a mixture of conditional distributions given each one of the last p observations. They can capture non-Gaussian and nonlinear features such as flat stretches, bursts of activity, outliers changepoints in a single unified model class. They can also represent time series defined on arbitrary state spaces, univariate or multivariate, continuous, discrete or mixed, which need not even be Euclidean. They perform well in the usual case of Gaussian time series without obvious nonstandard behaviors. The models are simple, analytically tractable, easy to simulate readily estimated. The stationarity and autocorrelation properties of the models are derived. A simple EM algorithm is given and shown to work well for estimation. The models are applied to several real and simulated datasets with satisfacto...

Scalable robust covariance and correlation estimates for data mining

Covariance and correlation estimates have important applications in data mining. In the presence of outliers, classical estimates of covariance and correlation matrices are not reliable. A small fraction of outliers, in some cases even a single outlier, can distort the classical covariance and correlation estimates making them virtually useless. That is, correlations for the vast majority of the data can be very erroneously reported; principal components transformations can be misleading; and multidimensional outlier detection via Mahalanobis distances can fail to detect outliers. There is plenty of statistical literature on robust covariance and correlation matrix estimates with an emphasis on affine-equivariant estimators that possess high breakdown points and small worst case biases. All such estimators have unacceptable exponential complexity in the number of variables and quadratic complexity in the number of observations. In this paper we focus on several variants of robust covariance and correlation matrix estimates with quadratic complexity in the number of variables and linear complexity in the number of observations. These estimators are based on several forms of pairwise robust covariance and correlation estimates. The estimators studied include two fast estimators based on coordinate-wise robust transformations embedded in an overall procedure recently proposed by [14]. We show that the estimators have attractive robustness properties, and give an example that uses one of the estimators in the new Insightful Miner data mining product.

Introduction to modern portfolio optimization with NUOPT and S-PLUS

Linear and Quadratic Programming.- General Optimization With Simple.- Advanced Issues in Mean-Variance Optimization.- Resampling and Portfolio Choice.- Scenario Optimization: Addressing Non-normality.- Robust Statistical Methods for Portfolio Construction.- Bayes Methods.

Robust Estimation for Time Series Autoregressions

Publisher Summary This chapter presents some theory and methodology of robust estimation for time series having two distinctive types of outliers. Research on robust estimation in the time series context has lagged behind, and perhaps understandably so in view of the increased difficulties imposed by dependency and the considerable diversity in qualitative features of time series data sets. For time series parameter, estimation problems, efficiency robustness, and min–max robustness are concepts directly applicable. Influence curves for parameter estimates may also be defined without special difficulties. A greater care is needed in defining breakdown points as the detailed nature of the failure mechanism may be quite important. A major problem that remains is that of providing an appropriate and workable definition of qualitative robustness in the time series context. For time series, the desire for a complete probabilistic description of either a nearly-Gaussian process with outliers, or the corresponding asymptotic distribution of parameter estimates, will often dictate that one specify more than a single finite-dimensional distribution of the process. It is only in special circumstances that the asymptotic distribution of the estimate will depend only upon a single univariate distribution or a single multivariate distribution.

Asymptotically Min—Max Bias Robust M-Estimates of Scale for Positive Random Variables

Abstract Asymptotically min—max bias robust M-estimates of scale are obtained for positive random variables with e-contaminated distributions, using any one of a broad class of loss functions of positive and negative bias. Any such estimate is a scaled order statistic, with the order statistic and scaling constant determined by e, the nominal distribution, and the loss function. Some calculations for specific cases show that for a large range of e the min—max order statistic solution has a breakdown point near .5 and is well approximated by a scaled median. Some calculations and Monte Carlo results indicate that the asymptotic result has considerable finite sample-size relevance by virtue of the squared bias being at least as large as the variance for rather modest sample sizes (depending on e) and by virtue of attractive mean squared error performance in the presence of even rather small fractions of contamination.

Robust Bayesian Model Selection for Autoregressive Processes with Additive Outliers

Abstract Autoregressive (AR) models of order k are often used for forecasting and control of time series, as well as for the estimation of functionals such as the spectrum. Here we propose a method that consists of calculating the posterior probabilities of the competing AR(k) models in a way that is robust to outliers, and then obtaining the predictive distributions of quantities of interest, such as future observations and the spectrum, as a weighted average of the predictive distributions conditional on each model. This method is based on the idea of robust Bayes factors, calculated by replacing the likelihood for the nominal model by a robust likelihood. It draws on and synthesizes several recent research advances, namely robust filtering and the Laplace method for integrals, modified to take account of the finite range of the parameters. The method performs well in simulation experiments and on real and artificial data. Software is available from StatLib.

Robust Portfolio Construction

Outliers in asset returns factors are a frequently occurring phenomenon across all asset classes and can have an adverse influence on the performance of mean–variance optimized (MVO) portfolios. This occurs by virtue of the unbounded influence that outliers can have on the mean returns and covariance matrix estimates (alternatively, correlations and variances estimates) that are inputs are optimizer inputs. A possible solution to the problem of such outlier sensitivity of MVO is to use robust estimates of mean returns and covariance matrices in place of the classical estimates of these quantities thereby providing robust MVO portfolios. We show that the differences occurring between classical and robust estimates for these portfolios are such as to be of considerable concern to a portfolio manager. It turns out that robust distances based on a robust covariance matrix can provide reliable identification of multidimensional outliers in both portfolio returns and the exposures matrix of a fundamental factor model, something that is not possible with one-dimensional Winsorization. Multidimensional visualization combined with clustering methods is also useful for returns outlier identification. The question of using robust and classical MVO vs. optimization-based fat-tailed skewed distribution fits and downside risk measure is briefly discussed. Some other applications of robust methods in portfolio management are described, and we point out some future research that is needed on the topic.

Small sample behavior of robust stochastic approximation and iterated weighted least squares estimates for location

A class of robuet estimates of location uses the Robbins-Monro stochastic approximation algorithm as a basis. These estimates, called SA-estimates, have been proposed by Martin (1972), who established an asymptotic min-max result like that of P. Huber (1964) for M-estimates. Here we study in detail the small sample behavior of robust SA-estimates for location at sample sizes N = 10, 20, 40 using Monte Carlo swindle techniques. Results are presented for point estimate efficiencies and for error rates and expected confidence interval lengths obtained by studentizing through use of the natural estimate of the asymptotic variance formula. Unlike M-estimates, SA-estimates are not invariant under permutations of the data order. Thus our study included one-step M-estimates and iterated- weighted least squares estimates. This was done not only for comparison purposes, but also because the latter are of interest in their own right. The results show that SA-estimate losses due to the lack of invariance under permut...