Response to Alexander Shen's note



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.


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