Abstract For a multivariate normal distribution with independent components, the observed vector is an inadmissible estimator of the mean vector. The James–Stein estimator defines a statistic with lower squared error loss than the observed vector. It has led to research into families of minimax estimators, some of which are summarized.

Abstract Particle filtering is a (sequential) Monte Carlo technique for simulation‐based inference in intractable general state space hidden Markov models for which such inference is not analytically tractable. It is an approach to characterize the distribution of an indirectly observed Markov chain as observations become available, which involves simulating the evolution of a system of particles, which can be used to approximate estimators of interest.

Abstract Psychometric reliability, ρ XX ′ , is the degree to which test scores are precise and free from measurement error. Classical test theory defines reliability as the ratio of true score variance to observed score variance: ρ XX ′ . This article provides a comprehensive definition of reliability and then describes the methods by which reliability is commonly estimated, including their limitations. The article then discusses several closely related topics: reliability of ratings; the standard error of measurement; reliability in subpopulations; the Spearman–Brown prophesy formula; true‐score correlations; effect of reliability upon validity; reliability of speeded tests; problems of correlated errors; and the relationship between classical reliability and item response theory information.

Manifold‐based learning combines elements of geometry, computer science, and statistics and is a major technique in dimensionality reduction. It is frequently a promising approach in nonparametric dimension reduction and has been widely adopted in data mining and statistical learning. The objective of manifold learning is to recover the low‐dimensional structure that is embedded in a higher dimensional space. We survey the nonlinear methods that have been proposed in manifold learning in the unsupervised learning setting. Other settings including supervised and semisupervised learning are briefly discussed. In particular, we present a unified framework for nonlinear manifold‐based learning methods, so that readers can see the intrinsic connections and commonalities across different methods. Our survey emphasizes their linear algebra formulations. Statistical and other theoretical considerations and extensions are discussed in the Conclusion. Numerical examples are presented abundant in the literature and are omitted in this article due to space limit.

Abstract Manifold‐based learning is a major tool in dimensionality reduction, which is widely used for feature extraction and feature selection in data mining and statistical learning. It combines techniques from various fields, such as geometry, algorithm, and statistics. The objective of manifold‐based learning is to recover the low‐dimensional structure that is embedded in a higher dimensional space. The focus in this article is on linear methods, where the underlying embedded structure is a linear subspace. We mainly survey the algorithms for manifold‐based learning methods under the unsupervised learning setting. Other nonlinear methods for manifold‐based learning are briefly discussed. We will provide the readers more references for more details on nonlinear manifold‐based learning methods. Our survey emphasizes their linear algebra formulations. Numerical examples are presented to illustrate the learning methods. More applications can be found in the literature provided in this article.

Abstract A unifying information‐theoretic approach for the evaluation of surrogate endpoints is discussed. This methodological framework offers an intuitive yet rigorous definition of surrogacy and provides a general metric that can be applied across different scenarios to quantify the validity of a surrogate, within both the meta‐analytic and causal‐inference paradigms.

Abstract Originally, the Bornhuetter–Ferguson was merely a claim‐reserving procedure, without a statistical model as foundation. This procedure is described. The more recent literature contains a number of models compatible with the procedure. Some of these are nonparametric models, and others are Bayesian. Those most prominent in the literature are described. Comment is added on the estimation of the prediction error associated with some of these models.

Abstract The measurement in socioeconomic context is achieved as a consequence of a definition confirmed through the relationship observed between observations and the concept to be measured. Almost all the measures developed in social sciences belong to this group. The measures developed in this context are defined indicators.

Abstract Item response theory (IRT) models are tools developed in psychometrics and applied primarily to validate, analyze, and score tests and questionnaires measuring abilities and attitudes. However, their potentials extend well beyond the education and psychometric fields. In fact, recent developments of these models allow us to pursue two key objectives of the social sciences, that is, verifying and retaining the multidimensionality of the data, either with an explorative or a confirmatory approach, and classifying units into homogeneous groups as regard to the identified relevant dimensions. Concurrently accounting for multidimensionality and classification instances is a step forward compared to statistical methods of data complexity reduction concerned either, on one side, with the structure of variables, or with the structure of units on the other, as it allows us to explain and qualify the differences between clusters of units alongside the distinctive latent dimensions underlying the data at issue.

Abstract Missing data occurs commonly in longitudinal studies. When data are missing at random, full‐likelihood approaches may be used for analysis. However, when the missing observations are nonignorable and nonmonotone, a full‐likelihood approach may yield inconsistent estimates. To correct for any bias that may arise in longitudinal studies with nonignorable missing data, various modeling strategies have been introduced, including selection, pattern‐mixture, and shared‐parameter models. However, because these classes of joint models explicitly model the nonignorable missingness mechanism, they require strong and unverifiable assumptions to guarantee model identifiability. To avoid this, the protective estimator has been proposed as an approach to analyze longitudinal data subject to nonignorable, nonmonotone missingness. The protective estimator avoids estimating the nonignorable missing data by utilizing a key assumption: the probability that a response is missing at any instance depends on the value of the variable alone in an unspecified way. In this article, we introduce the protective estimator and present its mathematical framework. We elaborate on advantages and disadvantages of the modeling approach and review examples of the protective estimator used in practice across the literature.