Editing Mixture model (section)

=== Moment matching ===
The [[Method of moments (statistics)|method of moment matching]] is one of the oldest techniques for determining the mixture parameters dating back to Karl Pearson's seminal work of 1894.
In this approach the parameters of the mixture are determined such that the composite distribution has moments matching some given value.  In many instances extraction of solutions to the moment equations may present non-trivial algebraic or computational problems.  Moreover, numerical analysis by Day<ref name=day>{{Cite journal | last1 = Day | first1 = N. E. | title = Estimating the Components of a Mixture of Normal Distributions | journal = Biometrika | volume = 56 | issue = 3 | pages = 463–474 | doi = 10.2307/2334652 | jstor = 2334652| year = 1969 }}</ref> has indicated that such methods may be inefficient compared to EM. Nonetheless, there has been renewed interest in this method, e.g., Craigmile and Titterington (1998) and Wang.<ref name=wang>{{citation 
|title=Generating daily changes in market variables using a multivariate mixture of normal distributions
|first = J. | last = Wang
|date = 2001
|journal = Proceedings of the 33rd Winter Conference on Simulation |pages   =283–289
}}</ref>

McWilliam and Loh (2009) consider the characterisation of a hyper-cuboid normal mixture [[copula (statistics)|copula]] in large dimensional systems for which EM would be computationally prohibitive.  Here a pattern analysis routine is used to generate multivariate tail-dependencies consistent with a set of univariate and (in some sense) bivariate moments.  The performance of this method is then evaluated using equity log-return data with [[Kolmogorov–Smirnov]] test statistics suggesting a good descriptive fit.