singleindex

Singleindex refers to a class of statistical models known as the single-index model. In these models, the conditional expectation of a response variable Y given a vector of covariates X = (X1, ..., Xd) depends on X only through a single linear combination X^T beta, where beta is a parameter vector. The common form is Y = g(X^T beta) + epsilon, where g is an unknown smooth link function and epsilon is a random error with E[epsilon|X] = 0. This structure reduces high-dimensional regression to estimating a univariate function g while allowing flexible nonlinear relationships with respect to the index.

Identifiability and variants are important considerations. To identify beta and g, one typically fixes a scale and sometimes the sign, for example by enforcing ||beta|| = 1. Variants include generalized single-index models where the conditional mean E[Y|X] = h(X^T beta) uses a known or parametric link h (such as logistic or probit), or semi-parametric forms where g is estimated nonparametrically while beta is estimated jointly or in a profiling step.

Estimation in single-index models combines dimension reduction with nonparametric or semi-parametric smoothing. Methods include average derivative estimation (ADE) and sliced inverse regression (SIR) to obtain beta, followed by smoothing approaches (kernel, local polynomial) to estimate g. Alternative approaches use profile likelihoods or iterative procedures in semi-parametric setups. Practical use requires adequate sample size to support nonparametric estimation of g and to stabilize the index estimation.

Applications and extensions cover econometrics, finance, biostatistics, and social sciences, where nonlinear relationships with many predictors are plausible but fully nonparametric models would be impractical. Extensions include multi-index models with several linear indices, additive single-index models, and generalized single-index models for different outcome distributions.