Statistics — Page 51
Explore statistical methods, their assumptions, interpretation and limitations.
Research field
Method family
UNIDIFF Mobility Model
The uniform difference (UNIDIFF) model, equivalently the log-multiplicative layer effect model, compares the strength of origin–destination association across several mobility tables — countries, time periods, or birth cohorts. It assumes t...
Unfolding Model
The Unfolding Model is a geometric approach to preference analysis that represents both individuals and choice objects (stimuli) as points in a shared low-dimensional space. Originating with Clyde Coombs's foundational 1950 work on preferen...
Universal Kriging
Universal kriging generalizes ordinary kriging to data whose mean varies systematically across space — a spatial trend or 'drift'. It models the mean as a function of the coordinates (or covariates) and krigs the residuals, so it can interp...
Urban Density Gradient Model
The urban density gradient model is the broad family of functional relationships that describe how population density varies with distance from a city's centre. Its canonical member is Colin Clark's 1951 negative-exponential form, but the f...
Urban Scaling Laws
Urban scaling laws describe how the aggregate properties of cities — wealth, innovation, infrastructure, crime — change systematically with population size, following power laws rather than growing in simple proportion. Building on the 2007...
VAR Model
Vector Autoregression is a multivariate time-series model that treats several interdependent series symmetrically, letting each variable depend on its own past values and the past values of all the others. It is the standard tool for captur...
VECM
The Vector Error Correction Model is a multivariate time-series model for cointegrated series that captures both their short-run dynamics and their long-run equilibrium relationship. It was introduced by Engle and Granger in 1987 as part of...
VaR Backtesting
VaR backtesting is a family of statistical tests that validate a risk model by comparing its Value-at-Risk forecasts against realised losses. It builds on Kupiec's (1995) unconditional coverage test, Christoffersen's (1998) conditional cove...
Value-Added Teacher Evaluation
Value-added teacher evaluation uses longitudinal student test scores to estimate how much individual teachers contribute to their students' achievement growth, net of what students brought into the classroom. Statistically it applies value-...
Value-Belief-Norm Model (VBN)
The value-belief-norm (VBN) model explains pro-environmental behavior as the end of a causal chain that runs from basic human values, through environmental beliefs, to a felt moral obligation that activates action. Paul Stern, Thomas Dietz,...
Van der Waerden Test
The Van der Waerden test is a nonparametric k-sample hypothesis test that converts observations into normal scores — the quantiles of a standard normal distribution — before comparing groups. Introduced by Bartel Leendert van der Waerden in...
Variance Inflation Factor
The Variance Inflation Factor (VIF) is a scalar diagnostic statistic proposed by Donald Marquardt (1970) that quantifies how much the variance of an estimated regression coefficient increases due to linear dependence—multicollinearity—among...
Variational Inference
Variational inference (VI) is a family of techniques that turn Bayesian posterior computation into an optimisation problem. Instead of drawing samples from the exact posterior — as Markov chain Monte Carlo does — VI posits a simpler, tracta...
Variational Inference with Measurement Error
Variational inference with measurement error is a scalable Bayesian approach that simultaneously estimates model parameters and latent true covariates when observed variables are contaminated by noise. Rather than sampling the posterior via...
Variational Inference with Missing Data
Variational inference with missing data is a scalable Bayesian approach that simultaneously approximates the posterior over latent variables and model parameters while imputing missing observations. Instead of integrating over all possible...
Vector Autoregression
Vector Autoregression is a multivariate time-series model in which each variable is regressed on its own lags and the lags of all other variables in the system. Originally proposed by Sims (1980) as a data-driven alternative to large struct...
Vector Error Correction Model
The Vector Error Correction Model extends the Vector Autoregression (VAR) framework to a system of variables that share one or more long-run equilibrium relationships. It jointly models short-run dynamics and the speed at which each variabl...
Von Thünen Land-Use Model
The von Thünen model is the founding theory of agricultural land use, explaining how the pattern of farming around a market emerges from transport costs and land rent. Set out by Johann Heinrich von Thünen in his 1826 work Der isolierte Sta...
Vote Buying Analysis
Vote buying analysis is the measurement-and-modeling toolkit for studying the distribution of private goods or cash in exchange for electoral support. Two problems define the field. First, who is being paid: Simeon Nichter's 2008 article sh...
W-Estimator
The W-estimator is a family of robust M-estimator variants for linear regression that use the Tukey bisquare and Welsch weight functions, introduced in the line of work going back to Beaton and Tukey (1974). Because its weights fall rapidly...
Weber Industrial Location Model
Weber's industrial location model is the classic least-cost theory of where a manufacturing plant should locate. Developed by Alfred Weber in 1909, it finds the site that minimizes total transport cost between the sources of raw materials a...
Weibull Regression
Weibull regression is a fully parametric survival model, formalised by Kalbfleisch and Prentice, that assumes survival times follow a Weibull distribution. A shape parameter controls whether the hazard increases, decreases, or remains const...
Weighted Least Squares
Weighted Least Squares is a generalization of Ordinary Least Squares (OLS) regression that assigns each observation a weight inversely proportional to its error variance, thereby down-weighting high-variance data points and up-weighting pre...
Welch ANOVA
Welch ANOVA is a parametric hypothesis test that compares the means of three or more independent groups when their variances are not equal. Introduced by B. L. Welch in 1951, it replaces classic one-way ANOVA whenever the homogeneity-of-var...