Minimum Likelihood Distance Estimation For A Semiparameter Location-shifted Mixture Model

Mixture model has been one of the focus of study in statistics during the last few decades,especially finite mixture model.It provides an non-uniform representation for finite classes.That is to say,it is a method for constructing statistical models by mixing(or weighting)other distributions.Many statistics pay more attention to finite mixture model and they have been applied in many fields including education,economics,genetics,medicine,chemistry,psychology and other fields.Semiparameter location-shifted mixture model has many great properties,for example,two components semiparameter location-shifted mixture model is identifiable.Two components semiparameter location-shifted mixture model is a great statistical model with practical value and practical significance.So many experts and scholars have discussed the theory and application of this model.One of the focus of study is to estimate parameters.The common methods is minimum distance estimation,one of the most classic is minimum Hellinger distance estimation.Based on the analysis of previous work,we estimated parameters of two components semiparameter location-shifted mixture model by the minimum distance estimation.Minimum distance estimation has many representations depending on the distance forms.We mainly considers the distance derived from likelihood distance.First,we studied K-L distance and symmetry K-L distance.The minimum adjusted likelihood distance estimation and minimum symmetry adjusted likelihood distance estimation are constructed based on the minimum K-L distance estimation.Then we discussed the algorithm of minimum K-L distance estimation,minimum symmetry K-L distance estimation,minimum adjusted likelihood distance estimation and minimum symmetry adjusted likelihood distance estimation.Under some certain conditions,estimators based on four minimum likelihood distance estimations are proved to be consistent.The simulation analysis about Comparing four minimum distance estimations with minimum Hellinger distance estimation and minimum Pearson's chi-square distance estimation is given.The simulation result show that the estimators based on four minimum likelihood distance estimations have consistency;Under Laplace distribution the effect of minimum adjusted likelihood distance estimation and minimum symmetry adjusted likelihood distance estimation are better than minimum Hellinger distance estimation and minimum Pearson's chi-square distance estimation.Furthermore,the computing time of minimum adjusted likelihood distance estimation and minimum symmetry adjusted likelihood distance estimation are less than others.Final,The proposed method is applied to the waiting time of Old Faithful Geysar data set and Guangxi Normal University students' score data set.Numerical studies and real data examples show the practicability of proposed method.