Response to Alexander Shen's note

W.Mebane

Abstract

Response to Alexander Shen's note "On the likelyhood for finite mixture models and Kirill Kalinin’s paper “Validation of the Finite Mixture Model Using Quasi-Experimental Data and Geography”"


Alexander Shen writes, "the expression being maximized, considered as a function of \(f_0, f_i, f_e\), is a linear function on the triangle \(f_0 + f_i + f_e = 1, f_0; f_i; f_e \geq 0\)." The expression being maximized (from [5]) is not a function of \(f_0\), \(f_i\) and \(f_e\). These "probabilities" are functions of the likelihood and so depend on all the other parameter estimates. For example, when both "incremental" and "extreme" frauds are included the R code that implements the method, which follows [2, 1, 6, 4], iteratively evaluates the following until stable, maximizing values for the likelihood are found:

\(F = (1-f_{\mathrm{i}}-f_{\mathrm{e}})F_0 + f_{\mathrm{i}}F_I + f_{\mathrm{e}}F_E \)

\(h_0 = (1-f_{\mathrm{i}}-f_{\mathrm{e}})F_0/F \)

\( h_I = f_{\mathrm{i}}F_I/F \)

\(h_E = f_{\mathrm{e}}F_E/F \)

\(f_{\mathrm{i}} = \text{mean}(h_I) \)

\(f_{\mathrm{e}} = \text{mean}(h_E)\)

where \(F_0\), \(F_I\) and \(F_E\) are vectors of length \(n\) (the number of observations) that have the observation-specific likelihoods as elements. \(h_0\), \(h_I\) and \(h_E\) are also vectors of length \(n\), and \(F_0/F\), \(F_I/F\) and \(F_E/F\) are evaluated elementwise. The likelihood value actually maximized is \(\sum_{i=1}^n(\log(h_0F_0 + h_IF_I + h_EF_E))\) where \(h_0F_0\), \(h_IF_I\) and \(h_EF_E\) are elementwise products. Shen's "triangle" argument does not apply.

Results from the model of [5] di ffer from results produced by the algorithm of [3] in part because [3] describes a Monte Carlo simulation method not a statistical estimation method based on any kind of likelihood or probability specification.

Received 02.07.2018.


References

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