On the single-index model estimate of the conditional density function: Consistency and implementation

- A new approach to estimate the unknown single-index parameter is proposed.
- We propose a highly efficient smooth minimum distance estimation method for β0.
- A statistic to test if the true parameter satisfies some assumptions is proposed.
- Estimation method and asymptotic properties for the estimator are obtained.
- Simulated and real data studies show the advantages of our methods.

We consider the estimation for the unknown single-index parameter in the conditional density function. Firstly, estimation method and asymptotic properties for the estimator are obtained. Secondly, to test a hypothesis on the single-index parameter, a test statistic based on the difference between the minimization criteria under the null and alternative hypotheses is proposed. We show that the limiting distribution for the test statistics is a weighted sum of independent standard chi-squared distributions. Besides, a local alternative hypothesis that converges to the null hypothesis at an n−1/2 rate is also considered. A bootstrap procedure is proposed to calculate critical values. Finally, simulation studies are conducted to demonstrate the performance of the proposed procedure and a real example is analyzed as an illustration.