This article introduces a parametric mixed-tail quantile model which flexibly combines Gumbel, Fréchet and Weibull tail behaviours through weighted quantile functions. Unlike classical extreme value models, our approach simultaneously captures multiple tail types, enhancing finite-sample adaptability and modelling accuracy. The extension of the approach for the context of regression for extremes is here termed mixed-tail quantile regression. Parameters are estimated using a computationally efficient least squares approach based on the expected order statistics. We establish asymptotic properties and address inference challenges via bootstrap methods. Theoretical results, including tail dominance and adaptive bias–variance decomposition, guide practical quantile estimation. Simulations and an application on global ice extents demonstrate improved performance in modelling extreme quantiles.
Generalized additive smooth models, including those for location scale and shape, would sometimes benefit from the imposition of shape constraints on some smooth components. When these components are represented using spline basis expansions, many constraints, such as monotonicity or convexity, can be expressed as linear inequality constraints on the basis coefficients. Given smoothing parameters, model fitting by Newton's method then becomes a sequential quadratic programming problem, while the equally important task of finding initial coefficients satisfying the constraints is a simple linear programming problem. Smoothing parameters can then be estimated by an adaptation of the extended Fellner-Schall method. This article brings together and reviews the necessary methods required to provide a general purpose framework for shape constrained smooth additive modelling, covering the widely applicable case in which not all smooth terms are constrained. As well as providing examples of shape constrained generalized additive models, it is shown how the same approach facilitates non-parametric density estimation. The article is accompanied by a new function scasm in R package mgcv.
We introduce a novel extension of Benford’s Law that shifts attention from leading digits to the joint distribution of final digit pairs in numerical data. Traditional Benford-based tests rely on first-digit frequencies and require observations to span several orders of magnitude, a condition frequently violated in precinct-level election returns. To overcome this limitation, we develop a formal probabilistic framework for analyzing the behaviour of the last two digits under Benfordian assumptions, which we denote L2-BL 10. Within this framework, we derive closed-form expressions for digit-pair probabilities and establish convergence bounds toward their limiting uniform distribution, providing new theoretical results that extend the statistical foundations of Benford analysis beyond its classical first-digit form. We then show that precinct-level vote counts are well approximated by a discrete Weibull distribution, linking this theoretical development to realistic data-generating processes encountered in elections. Monte Carlo simulations and empirical illustrations using data from nine states in the 2016 and 2020 US presidential elections demonstrate how the L2-BL 10 framework can be operationalized as a practical diagnostic for identifying anomalous digit behaviour. The result is a method that is both theoretically grounded and directly applicable to election forensics in settings where standard Benford tests are not appropriate.
Longitudinal studies have known widespread use in the last years in several fields of research, as they allow to distinguish between different sources of variation. We may observe differences at the beginning of the study that stay persistent through time, and changes in the response that are due to temporal dynamics in the observed covariates. Individual-specific, time-constant, effects are often included in the linear predictor to allow for unobserved individual-specific, time constant, heterogeneity motivated by omitted individual features. The random effect approach to estimation is based on considering such effects as random variables, usually with a specific parametric distribution. This approach has been frequently criticized, as it is often employed not considering correlation between observed (i.e., covariates) and unobserved (i.e., random effects) terms. To solve this issue, we may explicitly account for correlation between observed and unobserved heterogeneity, using the so-called correlated effects approach. In this article, we show that a more general solution may be developed by estimating the random effect conditional distribution non-parametrically via a discrete probability distribution on a finite number of locations. The approach we propose is assessed via a large-scale simulation study and illustrated by the analysis of a benchmark dataset.
