Mathematik, Informatik und Statistik - Open Access LMU - Teil 03/03

In regression models for ordinal response, each covariate can be equipped with either a simple, global effect or a more flexible and complex effect which is specific to the response categories. Instead of a priori assuming one of these effect types, as is done in the majority of the literature, we argue in this paper that effect type selection shall be data-based. For this purpose, we propose a novel and general penalty framework that allows for an automatic, data-driven selection between global and category-specific effects in all types of ordinal regression models. Optimality conditions and an estimation algorithm for the resulting penalized estimator are given. We show that our approach is asymptotically consistent in both effect type and variable selection and possesses the oracle property. A detailed application further illustrates the workings of our method and demonstrates the advantages of effect type selection on real data.

In regression models for ordinal response, each covariate can be equipped with either a simple, global effect or a more flexible and complex effect which is specific to the response categories. Instead of a priori assuming one of these effect types, as is done in the majority of the literature, we argue in this paper that effect type selection shall be data-based. For this purpose, we propose a novel and general penalty framework that allows for an automatic, data-driven selection between global and category-specific effects in all types of ordinal regression models. Optimality conditions and an estimation algorithm for the resulting penalized estimator are given. We show that our approach is asymptotically consistent in both effect type and variable selection and possesses the oracle property. A detailed application further illustrates the workings of our method and demonstrates the advantages of effect type selection on real data.

Regression models with functional responses and covariates constitute a powerful and increasingly important model class. However, regression with functional data poses well known and challenging problems of non-identifiability. This non-identifiability can manifest itself in arbitrarily large errors for coefficient surface estimates despite accurate predictions of the responses, thus invalidating substantial interpretations of the fitted models. We offer an accessible rephrasing of these identifiability issues in realistic applications of penalized linear function-on-function-regression and delimit the set of circumstances under which they are likely to occur in practice. Specifically, non-identifiability that persists under smoothness assumptions on the coefficient surface can occur if the functional covariate's empirical covariance has a kernel which overlaps that of the roughness penalty of the spline estimator. Extensive simulation studies validate the theoretical insights, explore the extent of the problem and allow us to evaluate their practical consequences under varying assumptions about the data generating processes. A case study illustrates the practical significance of the problem. Based on theoretical considerations and our empirical evaluation, we provide immediately applicable diagnostics for lack of identifiability and give recommendations for avoiding estimation artifacts in practice.

Regression models with functional covariates for functional responses constitute a powerful and increasingly important model class. However, regression with functional data poses challenging problems of non-identifiability. We describe these identifiability issues in realistic applications of penalized linear function-on-function-regression and delimit the set of circumstances under which they arise. Specifically, functional covariates whose empirical covariance has lower effective rank than the number of marginal basis function used to represent the coefficient surface can lead to unidentifiability. Extensive simulation studies validate the theoretical insights, explore the extent of the problem and allow us to evaluate its practical consequences under varying assumptions about the data generating processes. Based on theoretical considerations and our empirical evaluation, we provide easily verifiable criteria for lack of identifiability and provide actionable advice for avoiding spurious estimation artifacts. Applicability of our strategy for mitigating non-identifiability is demonstrated in a case study on the Canadian Weather data set.

Evaluation of a new k-means approach for exploratory clustering of items