Nonparametric Checks for Single - Index Models

In this paper we study goodness-of-fit testing of single-index models. The large sample behavior of certain score-type test statistics is investigated. As a by-product, we obtain asymptotically distribution-free maximin tests for a large class of local alternatives. Furthermore, characteristic function based goodness-of-fit tests are proposed which are omnibus and able to detect peak alternatives. Simulation results indicate that the approximation through the limit distribution is acceptable already for moderate sample sizes. Applications to two real data sets are illustrated. 1. Introduction. Suppose that a response variable Y depends on a vector X = (x 1 ,. .. , x p) T of covariates, where T denotes transposition. We may then decompose Y into a function m(X) of X and a noise variable ε, which is orthogonal to X, that is, for the conditional expectation of ε given X we have E(ε|X) = 0. When Y is unknown, the optimal predictor of Y given X = x equals m(x). Since in practice the regression function m is unknown, statistical inference about m is an important issue. In a purely parametric framework, m is completely specified up to a parameter. For example, in linear regression m(x) = β T x, where β is an unknown p-vector which needs to be estimated from the available data. Slightly more generally we may consider m(x) = Φ(β T x), where the link-function Φ may be nonlinear but is again specified. This is the so-called generalized linear model. When Φ remains unspecified, we arrive at a semiparametric model which is more flexible on the one hand and, on the other hand, avoids the curse of dimensionality one faces in fully nonparametric models. The estimator of β, as well as of the link function Φ, in this so-called single-index model