Mathematik, Informatik und Statistik - Open Access LMU - Teil 03/03 Ludwig-Maximilians-Universität München

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

For simulation studies on the exploratory factor analysis (EFA), usually rather simple population models are used without model errors. In the present study, real data characteristics are used for Monte Carlo simulation studies. Real large data sets are examined and the results of EFA on them are taken as the population models. First we apply a resampling technique on these data sets with sub samples of different sizes. Then, a Monte Carlo study is conducted based on the parameters of the population model and with some variations of them. Two data sets are analyzed as an illustration. Results suggest that outcomes of simulation studies are always highly influenced by particular specification of the model and its violations. Once small residual correlations appeared in the data for example, the ranking of our methods changed completely. The analysis of real data set characteristics is therefore important to understand the performance of different methods.