This paper studies a class of Spatial Dynamic Panel Data (SDPD) models in which spatial units are grouped into clusters that have the same parameters within clusters but different across clusters. This specific formulation of the model is called the Clustered SDPD model. When exogenous components are included, efficient estimation of their associated parameters becomes crucial for accurate cluster identification. To address this, we propose a two-step estimation procedure that enhances the efficiency of parameter estimation in a Heterogeneous SDPD framework, where each location has its own parameters, and provides a direct method for estimating fixed effects. This leads to increased precision in cluster detection. Theoretical results establish the consistency and asymptotic properties of the proposed estimators. Monte Carlo simulations and a real data application demonstrate substantial efficiency gains and superior clustering performance compared to standard methods.
This work discusses the article Independent component analysis by robust distance correlation appeared within this same issue of this journal. Particularly, it focuses into the novel bowl transform, a proposal embedded into an Independent Component Analysis strategy to obtain solutions which are resistant to the presence of contamination in the data. The proposal is discussed, and an alternative solution is suggested. This latter exploits data depth contours to embed a multivariate winsorization step within the same proposed strategy.
We are grateful to the editors of Advances in Data Analysis and Classification for bringing together this excellent collection of comments from many esteemed colleagues. We also sincerely thank the discussants for the considerable effort they invested in their thoughtful and stimulating remarks. Their contributions span a wide range of perspectives, from broader reflections on the philosophy of robust ICA to concrete suggestions regarding methodological building blocks and promising directions for future research. In particular, they propose alternative robust whitening procedures, different robustness transformations, and further theoretical developments that enrich the scope of our work. In what follows, we address the discussants’ comments by grouping them according to the main topics that were raised.
This discussion addresses PICARD, a robust approach to Independent Component Analysis (ICA) recently introduced by Leyder et al. My comments center on the distinction between two levels of robustness: that of the local dependence criterion — in particular the newly introduced bowl transform — and that of the full unmixing functional, the composite pipeline mapping the data distribution to the estimated sources. To clarify the mechanisms and trade-offs behind this robustness, I propose targeted empirical diagnostics, notably ablation studies and q-stability paths, where q is the scale parameter of the bowl transform. I further argue for a unified framework in which the dependence contrast, whitening functional, optimization scheme, and perturbation model are treated jointly.
In the correspondence analysis literature, advocates over the past 45 years (or so) have discussed the advantages of studying the square root of the cell counts of a contingency table, rather than the cell counts themselves. Doing so means that the Freeman-Tukey statistic is used as an alternative measure of association to Pearson’s chi-squared statistic. While there are advantages to using the classic Freeman-Tukey statistic, it is not always beneficial since it does not always behave like a chi-squared random variable. To address this issue, various improvements to the classic Freeman-Tukey statistic have been proposed, including a class of Freeman-Tukey statistics introduced in the early 1990s. This paper shows how correspondence analysis can be performed using two special cases of this class. By using these two special cases, we discuss the construction and properties of a low-dimensional space that visually represents the structure of a statistically significant association. We also quantify and interpret the distance between two row (or column) points in this low-dimensional space in terms of the link between their Hellinger and chi-squared differences.
