Optimality and Robustness in Statistical Forecasting. Yu. Kharin1. 1 Belarussian State University, Fr. Skoriny av., 4, BY 220050 Minsk, Belarus. Keywords: ...
Optimality and Robustness in Statistical Forecasting Yu. Kharin1 1
Belarussian State University, Fr. Skoriny av., 4, BY 220050 Minsk, Belarus
Keywords: Robustness, Forecasting, Time Series, Distortions.
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Introduction
Statistical forecasts are based only in part upon the observations; an equally important base is formed by prior assumptions, i.e. by hypothetical models of data. In practice, however, the classical hypothetical models are usually distorted. Unfortunately, the traditionally used “optimal” statistical procedures, which minimize the mean square risk of forecasting under hypothetical models, lose their optimality and have uncontrollably increasing risk under a little distorted hypothetical models (see Kharin (1996), Kharin (2001), Kharin (2003), Kharin and Zenevich (1999), Stockinger and Dutter (1987)). This paper is devoted to the topical problems of robustness analysis and construction of robust forecasting procedures for time series under different types of distortions.
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Main Results
Let xt ∈ R, t ∈ Z be an observed time series, X1T = (x1 , . . . , xT )0 ∈ RT be the vector of observations, T be the length of the observation period, xT +τ ∈ R be an unknown random variable to be forecasted, τ ≥ 1 be a “forecasting horizon”. The probability model of the observed time series under ε-distortions is described by a family of probability measures Pε,θ0 = n o
ε n Pn,θ : n ∈ N , where θ0 ∈ Θ ⊆ Rm is a true unknown value of the vector param0 (A), A ∈ B eter, ε ∈ [0, ε+ ] is a distortion level, and ε+ ≥ 0 is its maximal admissible value. For a statistical forecast x ˆT +τ , defined by a statistic x ˆT +τ = fT,τ (X1T ), we introduce the 2 point (mean square) risk ρε (fT,τ ; θ) = E{(ˆ xT +τ − xT +τ ) } ≥ 0, the integral risk rε (fT,τ ) = R ρ (f ; θ)π(θ)dθ ≥ 0 (where π(·) is a weighted function), and the guaranteed (upper) risk ε T,τ Θ ∗ r+ (fT,τ ) = sup0≤ε≤ε+ rε (fT,τ ). The robust forecasting statistic x ˆ∗T +τ = fT,τ (X1T ) is determined by ∗ the minimax criterion: r+ (fT,τ ) = inf fT ,τ (·) r+ (fT,τ ). In the paper we evaluate these robustness characteristics and propose new robust forecasting statistics for some types of distortions which are usual in practice and can be represented in terms of Pε,θ0 . For the trend model and for the regression model under outliers, where 0 < ε < 21 is the probability of outlier appearance, we propose the Local Median – forecasting procedure. We give its “breakdown point” and asymptotic properties. For the trend model under functional ε-distortions in l2 -metrics we construct a new plug-in forecasting procedure based on the M-estimator θˆ with the special choice of the loss function. For the AR(p)-model under specification errors and nonhomogeneities in the innovation process we find the critical value ε∗+ for the distortion level in the LS-forecasting and construct a robustified LS-procedure. For the VAR-model with missing values we propose statistical forecasting procedures under different levels of the prior information and compare their performance. Analytical results are illustrated by results of numerical modeling and also by real dendrological data forecasting of the increase in spruce-groves in our country. The author would like to thank the Organizers of the Workshop RMED’03 for the invitation and for the financial support of participation.
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Optimality and Robustness in Statistical Forecasting
References Yu. Kharin. Robustness in Statistical Pattern Recognition. Kluwer Academic Publishers, Dordrecht / Boston / London, 1996. Yu. Kharin. Robustness in forecasting of time series. In S. Aivazian et al., editors, Computer Data Analysis and Modeling, Vol. 1, pages 203–211. BSU, Minsk, 2001. Yu. Kharin. Robustness analysis in forecasting of time series. In R. Dutter et al., editors, Developments in Robust Statistics, pages 180–193. Springer, Heidelberg, 2003. Yu. Kharin, and D. Zenevich. Robustness of statistical forecasting by autoregression model under distortions. Theory of Stochastic Processes, 5(3-4), 84–91, 1999. N. Stockinger, and R. Dutter. Robust time series analysis: a survey. Kybernetika, 23, 1–91, 1987.
