Statistica Sinica 14(2004), 1-39

A GOODNESS-OF-FIT TEST FOR SINGLE-INDEX MODELS

Yingcun Xia$^1$, W. K. Li$^2$, Howell Tong$^{2,3}$ and Dixin Zhang$^4$

$^1$National University of Singapore, $^2$University of Hong Kong,

$^3$London School of Economics and $^4$Nanjing University

Abstract: The single-index model with an unknown link function is a generalized linear model that has been intensively investigated. This article considers a goodness-of-fit test for this model. Cramér-von Mises tests are constructed and the bootstrap method is used to provide $p-$values. The problem of bias in nonparametric estimation is tackled by the bootstrap method. Therefore, we do not need to undersmooth or oversmooth the link function. Some simulations are reported and some data are used for illustration.

Key words and phrases: Bias correction, bootstrap, Cramér-von Mises test, goodness of fit, local linear smoother, single-index model.