Penalized regression methods have for quite some time been a popular choice for addressing challenges in high dimensional data analysis. Despite their popularity, their application to time series data has been limited. This paper concerns bridge penalized methods in a linear regression time series model. We first prove consistency, sparsity and asymptotic normality of bridge estimators under a general mixing model. Next, as a special case of mixing errors, we consider bridge regression with autoregressive and moving average (ARMA) error models and develop a computational algorithm that can simultaneously select important predictors and the orders of ARMA models. Simulated and real data examples demonstrate the effective performance of the proposed algorithm and the improvement over ordinary bridge regression.

Estimating a curve nonparametrically from data measured with error is a difficult problem that has been studied by many authors. Constructing a consistent estimator in this context can sometimes be quite challenging, and in this paper we review some of the tools that have been developed in the literature for kernel-based approaches, founded on the Fourier transform and a more general unbiased score technique. We use those tools to rederive some of the existing nonparametric density and regression estimators for data contaminated by classical or Berkson errors, and discuss how to compute these estimators in practice. We also review some mistakes made by those working in the area, and highlight a number of problems with an existing R package decon.

In this paper we present methods for inference on data selected by a complex sampling design for a class of statistical models for the analysis of ordinal variables. Specifically, assuming that the sampling scheme is not ignorable, we derive for the class of cub models (Combination of discrete Uniform and shifted Binomial distributions) variance estimates for a complex two stage stratified sample. Both Taylor linearization and repeated replication variance estimators are presented. We also provide design-based test diagnostics and goodness-of-fit measures. We illustrate by means of real data analysis the differences between survey-weighted and unweighted point estimates and inferences for cub model parameters.

The different constituents of physical mixtures such as coloured paint, cocktails, geological and other samples can be represented by *d*-dimensional vectors called compositions with non-negative components that sum to one. Data in which the observations are compositions are called compositional data. There are a number of different ways of thinking about and consequently analysing compositional data. The log-ratio methods proposed by Aitchison in the 1980s have become the dominant methods in the field. One reason for this is the development of normative arguments converting the properties of log-ratio methods to ‘essential requirements’ or Principles for any method of analysis to satisfy. We discuss different ways of thinking about compositional data and interpret the development of the Principles in terms of these different viewpoints. We illustrate the properties on which the Principles are based, focussing particularly on the key subcompositional coherence property. We show that this Principle is based on implicit assumptions and beliefs that do not always hold. Moreover, it is applied selectively because it is not actually satisfied by the log-ratio methods it is intended to justify. This implies that a more open statistical approach to compositional data analysis should be adopted.

A Bayes linear space is a linear space of equivalence classes of proportional σ-finite measures, including probability measures. Measures are identified with their density functions. Addition is given by Bayes' rule and substraction by Radon–Nikodym derivatives. The present contribution shows the subspace of square-log-integrable densities to be a Hilbert space, which can include probability and infinite measures, measures on the whole real line or discrete measures. It extends the ideas from the Hilbert space of densities on a finite support towards Hilbert spaces on general measure spaces. It is also a generalisation of the Euclidean structure of the simplex, the sample space of random compositions. In this framework, basic notions of mathematical statistics get a simple algebraic interpretation. A key tool is the centred-log-ratio transformation, a generalization of that used in compositional data analysis, which maps the Hilbert space of measures into a subspace of square-integrable functions. As a consequence of this structure, distances between densities, orthonormal bases, and Fourier series representing measures become available. As an application, Fourier series of normal distributions and distances between them are derived, and an example related to grain size distributions is presented. The geometry of the sample space of random compositions, known as Aitchison geometry of the simplex, is obtained as a particular case of the Hilbert space when the measures have discrete and finite support.