The Frontit Model: A Stochastic Frontier for Dichotomic Random Variables

Summary

This paper is aimed to generalize the Stochastic Frontier Model to the case of dichotomic response variables. The proposed generalization is called Frontit model.

Be Y* and R* two random variables and R^D, Y^D be the dichotomic random variables so defined:

$${R^{D}} = \left\{ {\begin{array}{*{20}{c}} 1 \hfill & {{\text{if}}\:{R^{ * }} > 0} \hfill \\ 0 \hfill & {{\text{if}}\:{R^{ * }} \leqslant 0} \hfill \\ \end{array} } \right.,{Y^{D}} = \left\{ {\begin{array}{*{20}{c}} 1 \hfill & {{\text{ if}}\:{Y^{ * }} > 0} \hfill \\ 0 \hfill & {{\text{if}}\:{Y^{ * }} \leqslant 0} \hfill \\ \end{array} } \right.$$

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In this paper R^D * Y* is the random variable so defined:

$${R^{D}}*{Y^{ * }} = \left\{ {\begin{array}{*{20}{c}} {{\text{ }}{Y^{ * }}\;\;{\text{if }}{R^{ * }} > 0} \hfill \\ {{\text{ not defined}}\;\;{\text{if }}{R^{ * }} \leqslant 0} \hfill \\ \end{array} } \right.$$

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R^D *Y* is called random variable Y* truncated by R*.

In the same way we define the truncated dichotomic random variable:

$${{\text{R}}^{D}} * {{\text{Y}}^{D}} = \left\{ {\begin{array}{*{20}{c}} {{\text{ }}{{\text{Y}}^{D}}\;\;{\text{if }}{{\text{R}}^{ * }} > 0} \hfill \\ {{\text{ not defined}}\;\;{\text{if }}{{\text{R}}^{ * }} \leqslant 0} \hfill \\ \end{array} } \right.$$

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