Statistical tests in the partially linear additive regression models

In the present paper, we are mainly concerned with statistical tests in the partially linear additive model defined by Yi=Zi⊤β+∑ℓ=1dmℓ(Xi,ℓ)+εi,1≤i≤n, where Zi=(Zi,1,…,Zip)⊤ and Xi=(Xi,1,…,Xid)⊤ are vectors of explanatory variables, β=(β1,…,βp)⊤ is a vector of unknown parameters, m1,…,mdm1,…,md are unknown univariate real functions, and ε1,…,εnε1,…,εn are independent random errors with mean zero and finite variances σε2. More precisely, we first consider the problem of testing the null hypothesis β=β0. The second aim of this paper is to propose a test for the null hypothesis H0σ:σε2=σ02, in the partially linear additive regression models. Under the null hypotheses, the limiting distributions of the proposed test statistics are shown to be standard chi-squared distributions. Finally, simulation results are provided to illustrate the finite sample performance of the proposed statistical tests.