Statistical Inference based on the Empirical Identity Process

The empirical identity process (Enrico Bibbona, Giovanni Pistone, and Mauro Gasparini. The Empirical Identity Process: asymptotics and applications. The Canadian Journal of Statistics, 2017) gives rise to a test statistic $d_n$ which can be used for statistical inference. In this thesis, a new statistic $o_n$ based on the process is devised, and two inference settings are studied; goodness-of-fit and parameter estimation. For the latter, minimum-distance estimators are constructed. In the goodness-of-fit setting, we show that the new statistics are powerful in uniform settings with alternatives containing clusters. However, they are outperformed in other cases if the parameters of the null distribution are estimated from the data. In the parameter estimation setting, the minimum-distance estimator based on $d_n$ is shown to have excellent performance, beating the maximum-likelihood estimator with the expectation-maximization algorithm in normal mixtures with high component overlap. Moreover, our minimum-distance estimators have excellent robustness properties, especially compared with the maximum-likelihood estimator which is shown not to be robust. However, the maximum-likelihood estimator is less sensitive to initialization.