The asymptotic distribution of robust maximum likelihood estimator with Huber function for the mixed spatial autoregressive model with outliers

There is a wide range of outliers in spatial data, and these potential outliers will have a great impact on parameter estimation and corresponding statistical inference. Relying on the framework of maximum likelihood estimation (MLE), we investigate the asymptotic distribution of robust ML estimator under the mixed spatial autoregressive models with outliers and compare it with that of the ML estimator. Furthermore, based on the asymptotic theoretical result, we conduct the confidence interval of robust MLE and MLE. Similar to the results of MLE, we construct the second-order-corrected robust confidence interval using the parametric and semi-parametric bootstrap method. Simulation studies using Monte Carlo show that the robust estimator with the Huber loss function is more accurate and outperforms the MLE in most sample settings when data is contaminated by outliers. Then the use of the method is demonstrated in the analysis of the Neighborhood Crimes Data and the Boston Housing Price Data. The results further support the eligibility of the robust method in practical situations.