Abstract Different types of omnibus goodness-of-fit tests for a binomial null hypothesis are developed, which are based on the counts’ probability generating function. For each statistic, closed-form asymptotics are derived and used to implement the tests in practice without the need for a bootstrap procedure. The finite-sample performance of the tests is analyzed by means of a comprehensive simulation study, and practical recommendations for the choice of the test statistic are given. The practical application of the tests is illustrated by a couple of real-world data examples.

Abstract Consider a model with parameter $$\theta $$ and likelihood function $$L(\theta )$$ . Suppose that $$\theta $$ can be written $$\theta = (\psi, \lambda )$$ , where $$\psi $$ is a real-valued parameter-of-interest and $$\lambda $$ is a nuisance parameter and that our goal is likelihood-based inference regarding $$\psi $$ . In some cases, it may be beneficial to reparameterize the model, keeping the parameter-of-interest unchanged but modifying the nuisance parameter of the model. The purpose of this paper is to present some results regarding a specific nuisance parameter, known as the zero-score expectation (ZSE) parameter, that has been shown to be useful in integrated likelihood inference. These results include a more straightforward motivation for the ZSE parameter, an approximation useful for calculating the corresponding likelihood function, and a new interpretation of an integrated likelihood constructed using the ZSE parameter.

Abstract Despite the increasing prevalence of vector observations, computation of optimal experimental design for multi-response models has received limited attention. To address this problem within the framework of approximate designs, we introduce mREX, an algorithm that generalizes the randomized exchange algorithm REX (J Am Stat Assoc 115:529, 2020), originally specialized for single-response models. The mREX algorithm incorporates several improvements: a novel method for computing efficient sparse initial designs, an extension to all differentiable Kiefer’s optimality criteria, and an efficient method for performing optimal exchanges of weights. For the most commonly used D-optimality criterion, we propose a technique for optimal weight exchanges based on the characteristic polynomial of a matrix. The mREX algorithm is applicable to linear, nonlinear, and generalized linear models, and scales well to large problems. It typically converges to optimal designs faster than available alternative methods, although it does not require advanced mathematical programming solvers. We demonstrate the usefulness of mREX on bivariate dose-response Emax models for clinical trials, both without and with the inclusion of covariates.

Skewness and kurtosis are natural characteristics of a distribution. While it has long been recognized that they are more intrinsically entangled than other characteristics like location and dispersion, this has recently been made more explicit by Eberl and Klar (Stat Papers 65:415–433, 2024) with regard to orders of kurtosis. In this paper, we analyze the implications of this entanglement on kurtosis measures in general and for several specific examples. As a key finding, we show that kurtosis measures that are defined in the classical order-based way, which is analogous to measures of location, dispersion and skewness, do not exist. This raises serious doubts about the frequent application of such measures to skewed data. We then consider old and new proposals for kurtosis measures and evaluate under which additional conditions they measure kurtosis in a meaningful way. Some measures also allow more specific representations of the influence of skewness on the measurement of kurtosis than are available in a general setting. This works particularly well for a family of newly introduced density-based kurtosis measures.