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The original ridge estimator of the unknown p ×1 vector of β -coefficients in a linear model used a single scalar, k , to determine a point on a shrinkage path of finite length that extends from the Ordinary Least Squares estimator, ^ β 0 , to the shrinkage terminus (usually ^ β ≡ 0). Generalized ridge estimators use two or more parameters to determine not only the shape of their shrinkage path but also a specific point on that path. The efficient generalized ridge regression path proposed here is a continuous two-piece linear function that (1) starts at ^ β 0 , the Best Linear Unbiased Estimator, (2) has its interior knot at the ^ β -estimator with Maximum Likelihood of achieving overall minimum MSE risk under normal distribution-theory, and (3) ends at the shrinkage terminus. This new path is efficient in the senses that it is the shortest path and, at least when p > 2, essentially the only known shrinkage path that always contains the ^ β -vector that is most likely to be optimally biased . Functions in R-packages freely distributed via CRAN perform the calculations and produce the graphics used here to illustrate shrinkage concepts by interpreting ridge Trace diagnostic plots. These new concepts and visual tools provide invaluable data-analytic insights and improved self-confidence to applied researchers and data scientists fitting linear models when p , the number of non-constant X -predictor variables in the model, is strictly less than the number of observations available.

Variational Auto-Encoder (VAE) is a handy and computationally friendly Bayesian tool for Inverse Reinforcement Learning (IRL) problems, with the native setting of absent reward functions in a Markov Decision Process (MDP). However, recent works mainly deal with single reward, which turn out to be insufficient for complex dynamic environments with multiple demonstrators of various characteristics (hence multiple reward functions). This paper extends the dimensionality of reward (from ℝ to ℝ K ) by incorporating a latent embedding and clustering step on top of a scalable Bayesian IRL model, which enhances her applicability to multi-reward scenarios. We introduce our method, Clustered Autoencoded Variational Inverse Reinforcement Learning (CAVIRL), which is able to approximate multiple posterior reward functions and learn the corresponding policies for experts of various characteristics and skills. As a by-product, the proposed model also thrives to determine the number of clusters K on her own, as opposed to the competing multi-reward imitation learning models that require K to be prespecified. We trained the proposed model within a grid world with multiple types of players, where we achieved 100% correctness in determining the number of players’ types and 80%-83.9% match between the model-learned policies and the players’ demonstrations from the data.

In the ongoing discussion on improving quality and reproducibility of research, the data pipeline had been identified as one crucial element. Everything happening with the data already prior to the final analysis may have a highly relevant impact on the final analysis itself and its interpretation. Consequently, there is an increased focus on these processes. Sound practices for initial data analyses (IDA) have been described and the rigorous use of IDA has been advocated. However, IDA can only have a positive impact on research quality if findings from a systematic data screening lead to reasonable changes in the final analysis or its interpretation. In this communication some general challenges in finding adequate responses to findings from data screening are identified. They reflect a lack of knowledge and consensus. Consequently, rigorous future use of IDA can only be advocated if guidance for responding to data screening findings is further developed. With the knowledge present today, IDA may do more harm than good.

In this paper, we study the asymptotic properties (bias, variance, mean squared error) of Bernstein estimators for cumulative distribution functions and density functions near and on the boundary of the d- dimensional simplex. Our results generalize those found by Leblanc (2012b), who treated the case d = 1, and complement the results from Ouimet (2021a) in the interior of the simplex. Since the “edges” of the d- dimensional simplex have dimensions going from 0 (vertices) up to d − 1 (facets) and our kernel function is multinomial, the asymptotic expressions for the bias, variance and mean squared error are not straightforward extensions of one-dimensional asymptotics as they would be for product-type estimators studied by almost all past authors in the context of Bernstein estimators or asymmetric kernel estimators. This point makes the mathematical analysis much more interesting.

This article methodologically complements the R package nparMD , and it presents the follow-up to a previous article that had built the foundation for the procedure implemented initially within the package. We consider non-parametric rank-based inference methods for multivariate samples in completely randomized factorial designs. Assumption on the data are minimal: the variables need to be at least ordinal (including binary), and the multivariate observation vectors from different experimental units are assumed independent although there may be dependencies within the vectors. Different endpoints may be measured on different scales. That is, multivariate observation vectors with a count variable, a continuous metric response, and ordinal endpoints, are allowed for. Two different asymptotic settings are considered: (1) sample sizes are large while the number of factor levels is small and bounded, (2) small samples while the number of factor levels is large. Special attention is paid to deriving inference methods for the interaction effect, as here the variance-covariance matrix does not simplify under the null. In addition, we discuss some finite sample approximations and demonstrate application of the methods using real data from a psychological study on Seasonal Affective Disorder (SAD), also commonly referred to as Winter depression