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.