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 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. McWilliam and Loh (2009) consider the characterisation of a hyper-cuboid normal mixture 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. Spectral method Some problems in mixture model estimation can be solved using spectral methods. In particular it becomes useful if data points xi are points in high-dimensional real space, and the hidden distributions are known to be log-concave (such as Gaussian distribution or Exponential distribution). Spectral methods of learning mixture models are based on the use of Singular Value Decomposition of a matrix which contains data points. The idea is to consider the top k singular vectors, where k is the number of distributions to be learned. The projection of each data point to a linear subspace spanned by those vectors groups points originating from the same distribution very close together, while points from different distributions stay far apart. One distinctive feature of the spectral method is that it allows us to prove that if distributions satisfy certain separation condition (e.g., not too close), then the estimated mixture will be very close to the true one with high probability. Graphical Methods Tarter and Lock describe a graphical approach to mixture identification in which a kernel function is applied to an empirical frequency plot so to reduce intra-component variance. In this way one may more readily identify components having differing means. While this λ-method does not require prior knowledge of the number or functional form of the components its success does rely on the choice of the kernel parameters which to some extent implicitly embeds assumptions about the component structure. Other methods Some of them can even probably learn mixtures of heavy-tailed distributions including those with infinite variance (see links to papers below). In this setting, EM based methods would not work, since the Expectation step would diverge due to presence