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.