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The L2E method

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Abstract Estimation theory and practice is generally focused on maximum likelihood methodology, which boasts of claims of efficiency and widespread availability in software. Likelihood methods occasionally encounter problems with small sample sizes, or if the data are contaminated with outliers. The first problem can be addressed by regularization methods such as Bayesian estimation; the latter problem can be solved by using robust methodology such as the M‐estimator, for example. is a particular example of an M‐estimator. This article motivates its special properties and provides detailed examples that take advantage of those properties. Copyright © 2009 John Wiley & Sons, Inc. This article is categorized under: Statistical and Graphical Methods of Data Analysis > Density Estimation

Kernel estimate of 1000 points from a mixture; see text for details. The true mixture (red dotted line) and single normal MLE fit (blue dotted line) are also depicted.

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The two three‐component normal fits with a kernel estimate.

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Two slices of the three‐parameter L2E criterion; see text for details.

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Five‐parameter L2E fit (blue) superimposed on kernel estimate (black dotted) and true mixture (red dotted).

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Two‐parameter L2E fit (red) superimposed on kernel estimate. The blue dotted line is the one‐parameter L2E fit shown in Figure 3.

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L2E surface for the two parameters θ = (µ, σ); here, .

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Two L2E fits superimposed on a kernel estimate.

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L2E criterion for mixture data with local minima at and 3.79.

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