arXiv subjects

Alain Desgagné

Publications and source records attributed to Alain Desgagné.

12 recordsLinked to original sources

arXiv2026

Reconciling Bayesian and frequentist approaches to robustness against outliers

Heavy-tailed models are used as a way to gain robustness against outliers in Bayesian analyses. In frequentist analyses, M-estimators are often employed. In this paper, the two approaches are tentatively reconciled by considering M-estimators as maximum likelihood estimators of heavy-tailed models. From this perspective, it is realized that a fundamental difference exists as frequentists, contrarily to Bayesians, do not require these heavy-tailed models to be proper. For instance, a popular robust estimator in linear regression, Tukey's biweight M-estimator, does not correspond to a proper heavy-tailed model. Thus, a Bayesian practitioner does not have access to the same range of tools as a frequentist practitioner. It is shown through two real-data linear regression analyses that the former may in consequence obtain significantly different estimation results than the latter, where the difference is due to a more pronounced influence by the outliers in the former case. It is highlighted that a way to give these practitioners access to the same range of tools is for the Bayesian to adopt the generalized Bayesian framework of Bissiri et al. (2016) which allows the use of improper models, in combination with proper prior distributions yielding proper generalized posterior distributions. A complete reconciliation of the Bayesian and frequentist approaches to robustness is then achieved. An extensive theoretical study of the generalized Bayesian counterpart of Tukey's biweight M-estimator is provided, which includes a robustness characterization result, a strong consistency result and a Bernstein--von Mises result, the latter allowing to calibrate the generalized posterior distribution for meaningful uncertainty quantification. After adopting the generalized Bayesian framework, the Bayesian practitioner obtains similar results as the frequentist practitioner in the aforementioned examples.

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arXiv2026

Omnibus goodness-of-fit tests for univariate continuous distributions based on trigonometric moments

We propose a new omnibus goodness-of-fit test based on trigonometric moments of probability-integral-transformed data. The test builds on the framework of the LK test introduced by Langholz and Kronmal [J. Amer. Statist. Assoc. 86 (1991), 1077-1084], but fully exploits the covariance structure of the associated trigonometric statistics. As a result, our test statistic converges under the null hypothesis to a $χ_2^2$ distribution, even in the presence of nuisance parameters, yielding a well-calibrated rejection region. We derive the exact asymptotic covariance matrix required for normalization and propose a unified approach to computing the LK normalizing scalar. The applicability of both the proposed test and the LK test is substantially expanded by providing implementation details for 11 families of continuous distributions, covering most commonly used parametric models. Simulation studies demonstrate accurate empirical size, close to the nominal level, and strong power properties, yielding fully plug-and-play procedures. Further insight is provided by an analysis under local alternatives. The methodology is illustrated using surface temperature forecast errors from a numerical weather prediction model.

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arXiv2025

Normal approximations for the multivariate inverse Gaussian distribution and asymmetric kernel smoothing on $d$-dimensional half-spaces

This paper introduces a novel density estimator supported on $d$-dimensional half-spaces. It stands out as the first asymmetric kernel density estimator for half-spaces in the literature. Using the multivariate inverse Gaussian (MIG) density from Minami (2003) as the kernel and incorporating locally adaptive parameters, the estimator achieves desirable boundary properties. To analyze its mean integrated squared error (MISE) and asymptotic normality, a local limit theorem and probability metric bounds are established between the MIG and the corresponding multivariate Gaussian distribution with the same mean vector and covariance matrix, which may also be of independent interest. Additionally, a new algorithm for generating MIG random vectors is developed, proving to be faster and more accurate than Minami's algorithm based on a Brownian first-hitting location representation. This algorithm is then used to discuss and compare optimal MISE and likelihood cross-validation bandwidths for the estimator in a simulation study under various target distributions. As an application, the MIG asymmetric kernel is used to smooth the posterior distribution of a generalized Pareto model fitted to large electromagnetic storms.

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arXiv2024

Asymptotics for non-degenerate multivariate $U$-statistics with estimated nuisance parameters under the null and local alternative hypotheses

The large-sample behavior of non-degenerate multivariate $U$-statistics of arbitrary degree is investigated under the assumption that their kernel depends on parameters that can be estimated consistently. Mild regularity conditions are provided which guarantee that once properly normalized, such statistics are asymptotically multivariate Gaussian both under the null hypothesis and sequences of local alternatives. The work of Randles (1982, Ann. Statist.) is extended in three ways: the data and the kernel values can be multivariate rather than univariate, the limiting behavior under local alternatives is studied for the first time, and the effect of knowing some of the nuisance parameters is quantified. These results can be applied to a broad range of goodness-of-fit testing contexts, as shown in two specific examples.

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arXiv2022

Goodness-of-fit tests for Laplace, Gaussian and exponential power distributions based on $λ$-th power skewness and kurtosis

Temperature data, like many other measurements in quantitative fields, are usually modeled using a normal distribution. However, some distributions can offer a better fit while avoiding underestimation of tail event probabilities. To this point, we extend Pearson's notions of skewness and kurtosis to build a powerful family of goodness-of-fit tests based on Rao's score for the exponential power distribution $\mathrm{EPD}_λ(μ,σ)$, including tests for normality and Laplacity when $λ$ is set to 1 or 2. We find the asymptotic distribution of our test statistic, which is the sum of the squares of two $Z$-scores, under the null and under local alternatives. We also develop an innovative regression strategy to obtain $Z$-scores that are nearly independent and distributed as standard Gaussians, resulting in a $χ_2^2$ distribution valid for any sample size (up to very high precision for $n\geq 20$). The case $λ=1$ leads to a powerful test of fit for the Laplace($μ,σ$) distribution, whose empirical power is superior to all $39$ competitors in the literature, over a wide range of $400$ alternatives. Theoretical proofs in this case are particularly challenging and substantial. We applied our tests to three temperature datasets. The new tests are implemented in the R package PoweR.

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arXiv2022

A comprehensive empirical power comparison of univariate goodness-of-fit tests for the Laplace distribution

In this paper we present the results from an empirical power comparison of 40 goodness-of-fit tests for the univariate Laplace distribution, carried out using Monte Carlo simulations with sample sizes $n = 20, 50, 100, 200$, significance levels $α= 0.01, 0.05, 0.10$, and 400 alternatives consisting of asymmetric and symmetric light/heavy-tailed distributions taken as special cases from 11 models. In addition to the unmatched scope of our study, an interesting contribution is the proposal of an innovative design for the selection of alternatives. The 400 alternatives consist of 20 specific cases of 20 submodels drawn from the main 11 models. For each submodel, the 20 specific cases corresponded to parameter values chosen to cover the full power range. An analysis of the results leads to a recommendation of the best tests for five different groupings of the alternative distributions. A real-data example is also presented, where an appropriate test for the goodness-of-fit of the univariate Laplace distribution is applied to weekly log-returns of Amazon stock over a recent four-year period.

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arXiv2020

An automatic robust Bayesian approach to principal component regression

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, corresponding to a concept in Bayesian statistics called 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.

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arXiv2019

Efficient and Robust Estimation of Linear Regression with Normal Errors

Linear regression with normally distributed errors - including particular cases such as ANOVA, Student's t-test or location-scale inference - is a widely used statistical procedure. In this case the ordinary least squares estimator possesses remarkable properties but is very sensitive to outliers. Several robust alternatives have been proposed, but there is still significant room for improvement. This paper thus proposes an original method of estimation that offers the best efficiency simultaneously in the absence and the presence of outliers, both for the estimation of the regression coefficients and the scale parameter. The approach first consists in broadening the normal assumption of the errors to a mixture of the normal and the filtered-log-Pareto (FLP), an original distribution designed to represent the outliers. The expectation-maximization (EM) algorithm is then adapted and we obtain the N-FLP estimators of the regression coefficients, the scale parameter and the proportion of outliers, along with probabilities of each observation being an outlier. The performance of the N-FLP estimators is compared with the best alternatives in an extensive Monte Carlo simulation. The paper demonstrates that this method of estimation can also be used for a complete robust inference, including confidence intervals, hypothesis testing and model selection.

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arXiv2019

A New Bayesian Approach to Robustness Against Outliers in Linear Regression

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 simulated case study. The performance is shown using simulation. All required code is given.

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arXiv2019

Weak Convergence and Optimal Tuning of the Reversible Jump Algorithm

The reversible jump algorithm is a useful Markov chain Monte Carlo method introduced by Green (1995) that allows switches between subspaces of differing dimensionality, and therefore, model selection. Although this method is now increasingly used in key areas (e.g. biology and finance), it remains a challenge to implement it. In this paper, we focus on a simple sampling context in order to obtain theoretical results that lead to an optimal tuning procedure for the considered reversible jump algorithm, and consequently, to easy implementation. The key result is the weak convergence of the sequence of stochastic processes engendered by the algorithm. It represents the main contribution of this paper as it is, to our knowledge, the first weak convergence result for the reversible jump algorithm. The sampler updating the parameters according to a random walk, this result allows to retrieve the well-known 0.234 rule for finding the optimal scaling. It also leads to an answer to the question: "with what probability should a parameter update be proposed comparatively to a model switch at each iteration?"

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arXiv2018

Bayesian Robustness to Outliers in Linear Regression and Ratio Estimation

Whole robustness is a nice property to have for statistical models. It implies that the impact of outliers gradually vanishes as they approach plus or minus infinity. So far, the Bayesian literature provides results that ensure whole robustness for the location-scale model. In this paper, we make two contributions. First, we generalise the results to attain whole robustness in simple linear regression through the origin, which is a necessary step towards results for general linear regression models. We allow the variance of the error term to depend on the explanatory variable. This flexibility leads to the second contribution: we provide a simple Bayesian approach to robustly estimate finite population means and ratios. The strategy to attain whole robustness is simple since it lies in replacing the traditional normal assumption on the error term by a super heavy-tailed distribution assumption. As a result, users can estimate the parameters as usual, using the posterior distribution.

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arXiv2015

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

Estimating the location and scale parameters is common in statistics, using, for instance, the well-known sample mean and standard deviation. However, inference can be contaminated by the presence of outliers if modeling is done with light-tailed distributions such as the normal distribution. In this paper, we study robustness to outliers in location-scale parameter models using both the Bayesian and frequentist approaches. We find sufficient conditions (e.g., on tail behavior of the model) to obtain whole robustness to outliers, in the sense that the impact of the outliers gradually decreases to nothing as the conflict grows infinitely. To this end, we introduce the family of log-Pareto-tailed symmetric distributions that belongs to the larger family of log-regularly varying distributions.

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