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The research described in this paper is motivated by model checking forparametric single-index models with diverging number of predictors. Toconstruct a test statistic, we first study the asymptotic property of theestimators of involved parameters of interest under the null and alternativehypothesis when the dimension is divergent to infinity as the sample size goesto infinity. For the testing problem, we study an adaptive-to-modelresidual-marked empirical process as the basis for constructing a teststatistic. By modifying the approach in the literature to suit the divergingdimension settings, we construct a martingale transformation. Under the null,local and global alternative hypothesis, the weak limits of the empiricalprocess are derived and then the asymptotic properties of the test statisticare investigated. Simulation studies are carried out to examine the performanceof the test.

Estimation and adaptive-to-model testing for regressions with diverging number of predictors