Robustness to outliers in location–scale parameter model using log-regularly varying distributions

Desgagné, Alain

doi:10.1214/15-aos1316

Cited by 20 publications

(43 citation statements)

References 15 publications

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“…Proof. See Desgagné (2015). Proposition 2.2 essentially implies that the conditional density of an outlier (1/σ) f ((y − x T β)/σ) asymptotically behaves like f (y) as y → ∞.…”

Section: Log-regularly Varying Distributionsmentioning

confidence: 99%

“…Definition 2.2 implies that any density f with tails behaving like |z| −1 (log |z|) −θ with θ > 1 is a LRVD. Some examples like the LPTN distribution are given in Desgagné (2015). The most important property of this class of distributions follows from Definition 2.1: the asymptotic location-scale invariance of their density, as stated in Proposition 2.2.…”

Section: Log-regularly Varying Distributionsmentioning

confidence: 99%

“…The second step towards the objective is therefore to generalise the results of Desgagné (2015) to linear regression. In fact, it is a generalisation of the results of Desgagné and Gagnon (2017), which are essentially an application of those of Desgagné (2015) in simple linear regression through the origin for robust estimation of ratios. This second step represents our key theoretical contribution.…”

Section: Introductionmentioning

confidence: 99%

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A New Bayesian Approach to Robustness Against Outliers in Linear Regression

Gagnon¹,

Desgagné²,

Bédard³

2020

Bayesian Anal.

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Linear regression is ubiquitous in statistical analysis. It is well understood that conflicting sources of information may contaminate the inference when the classical normality of errors is assumed. The contamination caused by the light normal tails follows from an undesirable effect: the posterior concentrates in an area in between the different sources with a large enough scaling to incorporate them all. The theory of conflict resolution in Bayesian statistics (O'Hagan and Pericchi (2012)) recommends to address this problem by limiting the impact of outliers to obtain conclusions consistent with the bulk of the data. In this paper, we propose a model with super heavy-tailed errors to achieve this. We prove that it is wholly robust, meaning that the impact of outliers gradually vanishes as they move further and further away form the general trend. The super heavy-tailed density is similar to the normal outside of the tails, which gives rise to an efficient estimation procedure. In addition, estimates are easily computed. This is highlighted via a detailed user guide, where all steps are explained through a case study. The performance is shown using simulation. All required code is given.MSC 2010 subject classifications: Primary 62F35; secondary 62J05.

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“…Proof. See Desgagné (2015). Proposition 2.2 essentially implies that the conditional density of an outlier (1/σ) f ((y − x T β)/σ) asymptotically behaves like f (y) as y → ∞.…”

Section: Log-regularly Varying Distributionsmentioning

confidence: 99%

Section: Log-regularly Varying Distributionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

A New Bayesian Approach to Robustness Against Outliers in Linear Regression

Gagnon¹,

Desgagné²,

Bédard³

2020

Bayesian Anal.

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“…A new technique emerged to gain robustness against outliers in parametric modelling: replace the traditional distribution assumption (which is a normal assumption in the problems previously studied) by a super-heavy-tailed distribution assumption (Desgagné, 2015;Desgagné and Gagnon, 2019;Gagnon, Desgagné and Bédard, 2020a;Gagnon, Bédard and Desgagné, 2021). The ra-tionale is that this latter assumption is more adapted to the eventual presence of outliers by giving higher probabilities to extreme values.…”

Section: Wholly-robust Linear Regressionmentioning

confidence: 99%

Informed reversible jump algorithms

Gagnon

2021

Electron. J. Statist.

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Incorporating information about the target distribution in proposal mechanisms generally improves Markov chain Monte Carlo algorithms. For instance, it has proved successful to incorporate gradient information in fixed-dimensional algorithms such as Hamiltonian Monte Carlo. In trans-dimensional algorithms, Green (2003) recommended to sample the parameter proposals during model switches from normal distributions with informative means and covariance matrices. These proposal distributions can be viewed as asymptotic approximations to the parameter distributions, where the limit is with regard to the sample size. Models are typically proposed using uninformed uniform distributions. In this paper, we build on the approach of Zanella (2020) for discrete spaces to incorporate information about neighbouring models. We rely on approximations to posterior model probabilities that are asymptotically exact. We prove that, in some scenarios, the samplers combining this approach with that of Green (2003) behave like ideal ones that use the exact model probabilities and sample from the correct parameter distributions, in the large-sample regime. We show that the implementation of the proposed samplers is straightforward in some cases. The methodology is applied to a real-data example. The code is available online. ‡

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“…The first step in performing an approximate robust PCA is to obtain C by stan- Relying on a robust locationscale model as in [7], with an LPTN error distribution and ρ := 0.95, ensures that…”

Section: Robust Principal Component Analysismentioning

confidence: 99%

An automatic robust Bayesian approach to principal component regression

Gagnon

Bédard

Desgagné

2020

Journal of Applied Statistics

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Principal component regression uses principal components as regressors. It is particularly useful in prediction settings with high-dimensional covariates. The existing literature treating of Bayesian approaches is relatively sparse. We introduce a Bayesian approach that is robust to outliers in both the dependent variable and the covariates. Outliers can be thought of as observations that are not in line with the general trend. The proposed approach automatically penalises these observations so that their impact on the posterior gradually vanishes as they move further and further away from the general trend, leading to whole robustness. The predictions produced are thus consistent with the bulk of the data. The approach also exploits the geometry of principal components to efficiently identify those that are significant. Individual predictions obtained from the resulting models are consolidated according to model-averaging mechanisms to account for model uncertainty. The approach is evaluated on real data and compared to its nonrobust Bayesian counterpart, the traditional frequentist approach, and a commonly employed robust frequentist method. Detailed guidelines to automate the entire statistical procedure are provided. All required code is made available, see ArXiv:1711.06341.ther summarised using two different linear regression models that respectively yield a robust regression line (in green) and an ordinary least squares regression line (in red). Given that different summaries lead to different data points and therefore different regressions, one might however wonder whether the available or natural summaries are necessarily suitable for the tasks at hand.Principal component regression (PCR) is the name given to a linear regression model using principal components (PCs) as regressors. It is based on a principal component analysis (PCA), which is commonly used to summarise the information contained in covariates. The principle is to find new axes in the covariate space by exploiting the correlation structure between the covariates, and then encode the covariate observations in that new coordinate system. The resulting variables, called principal components (PCs), are linearly independent and have the remarkable property that the first q PCs retain the maximum amount of information carried by the original observations (compared to any other q-dimensional summary). Regrouping correlated variables to produce linearly independent ones is appealing in a linear regression context, as strongly correlated variables are known to carry redundant information, leading to unstable estimations. Companies within the same economic sector in stock market indices like S&P 500 and S&P/TSX are an example of such correlated variables. Linear independence also allows visualising the relationship between the dependent variable and the PCs by plotting the dependent variable against each of the PCs.Due to the loss in the interpretability of the inference results engendered by transforming covariates, PCR is mainly used in a predictio...

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Robustness to outliers in location–scale parameter model using log-regularly varying distributions

Cited by 20 publications

References 15 publications

A New Bayesian Approach to Robustness Against Outliers in Linear Regression

A New Bayesian Approach to Robustness Against Outliers in Linear Regression

Informed reversible jump algorithms

An automatic robust Bayesian approach to principal component regression

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