Iterative Process of $M_{\mathrm{split}(q)}$ Estimation

${\mathit{M}}_{\mathrm{split}(\mathit{q})}$ estimation allows us to estimate competitive parameters, namely different versions of the parameter vector within the split functional model. In the univariate model, such parameters can be regarded as location parameters for different observation aggregations. The whole observation set might be an unrecognized mixture of observations that belong to such aggregations. There are two main variants of ${\mathit{M}}_{\mathrm{split}(\mathit{q})}$ estimation: the squared and absolute ${\mathit{M}}_{\mathrm{split}(\mathit{q})}$ estimations, which differ from each other in objective functions. The estimation process is always an iterative one, irrespective of the estimation variant. This paper addresses the main practical problem in such a context, namely the choice of the starting point and its possible influence on the estimation results. The paper shows that this issue is important; it also proposes the best choice that guarantees the correct solutions of the optimization problem. The authors also consider two types of iterative processes and conclude that the traditional iterative process is recommended for squared ${\mathit{M}}_{\mathrm{split}(\mathit{q})}$ estimation, whereas the parallel process is suitable for absolute ${\mathit{M}}_{\mathrm{split}(\mathit{q})}$ estimation.