An optimal test for the additive model with discrete or categorical predictors

Abstract

In multivariate nonparametric regression, the additive models are very useful when a suitable parametric model is difficult to find. The backfitting algorithm is a powerful tool to estimate the additive components. However, due to complexity of the estimators, the asymptotic p value of the associated test is difficult to calculate without a Monte Carlo simulation. Moreover, the conventional tests assume that the predictor variables are strictly continuous. In this paper, a new test is introduced for the additive components with discrete or categorical predictors, where the model may contain continuous covariates. This method is also applied to the semiparametric regression to test the goodness of fit of the model. These tests are asymptotically optimal in terms of the rate of convergence, as they can detect a specific class of contiguous alternatives at a rate of $n^{-1/2}$. An extensive simulation study and a real data example are presented to support the theoretical results.

Appendix A: Regularity conditions

To derive the asymptotic distribution of the GLR test statistic, we need the following assumptions:

• (C1) Suppose $c_{pj} = P(X_p=x_{pj})$, then $c_{pj} \in (0,1)$ for all $j=1,2, \ldots , k_p$ and $p=1,2, \ldots , P$, where $\sum _{j=1}^{k_p} c_{pj} =1$.

• (C2) The kernel function K(z) is bounded and Lipschitz continuous with a bounded support.

• (C3) If $Z_q$ is continuous, then the density $f_q$ of $Z_q$ is Lipschitz continuous and bounded away from 0 and has bounded supports $\Omega _q$ for $q \in \{1,2, \ldots , Q\}$.

• (C4) If both $Z_q$ and $Z_{q'}$ are continuous, then the joint density $f_{qq'}$ of $Z_q$ and $Z_{q'}$ is Lipschitz continuous on its support $\Omega _q \times \Omega _{q'}$ for $q \ne q' \in \{1,2, \ldots , Q\}$.

• (C5) $nh_q / \log (n) \rightarrow \infty$ as $n \rightarrow \infty$ and $h_q \rightarrow 0$ for $q = 1,2, \ldots , Q$.

• (C6) If $Z_q$ is continuous and $d_q$ is the degree of the polynomial used for smoothing of $Z_q$, then the $( d_q + 1)$th derivative of $m_{P+q}$, for $q \in \{1,2, \ldots , Q\}$, exists and is bounded and continuous.

• (C7) $\sigma ^2 = \text{ Var }[\epsilon ] = E[\epsilon ^2 ] < \infty$.

• (C8) $E[ m_q (Z_q)| X_p = x_{pj} ] = 0$ for all $j=1,2,\ldots , k_p$, $p = 1,2, \ldots , P$ and $q = 1,2, \ldots , Q$.