Finite mixture models are widely used in scientific investigations. Due to their non-regularity, there are many technical challenges concerning inference problems on various aspects of the finite mixture models. After decades of effort by statisticians, substantial progresses are recorded recently in characterising large sample properties of some classical inference methods when applied to finite mixture models, providing effective numerical solutions for mixture model-based data analysis, and the invention of novel inference approaches. This paper aims to provide a comprehensive summary on large sample properties of some classical statistical methods and recently developed modified likelihood ratio test and EM-test for the order of the finite mixture model. The presentation de-emphasises the rigour in order to gain some insights behind some complex technical issues. The paper wishes to recommend the EM-test as the most promising approach to data analysis problems from all models with mixture structures.

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