# An Optimal Test for the Additive Model with Discrete or Categorical Predictors

## TL;DR

This paper introduces an asymptotically optimal test for additive models with discrete or categorical predictors, applicable to models with continuous covariates, and demonstrates its effectiveness through simulations and real data application.

## Contribution

The paper develops a new goodness-of-fit test for additive models with discrete or categorical predictors, extending existing methods to handle mixed predictor types.

## Key findings

- Test is asymptotically optimal with a $n^{-1/2}$ detection rate.
- Simulation studies confirm theoretical properties.
- Applied successfully to real data on diamond pricing.

## 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 is presented to support the theoretical results derived in this paper. Finally, the method is applied to a real data to model the diamond price based on its quality attributes and physical measurements.