A popular instrumental variable approach is to use a two-step procedure and estimate equation ( 2 ) first and then use the estimates of this first step to estimate equation ( 1 ) in a second step.
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Though originally defined for least squares, lasso regularization is easily extended to a wide variety of statistical models including generalized linear models, generalized estimating equations, proportional hazards models, and M-estimators, in a straightforward fashion.
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This still leaves the question of how to obtain estimators in a given situation and carry the computation, several methods have been proposed : the method of moments, the maximum likelihood method, the least squares method and the more recent method of estimating equations.
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From the condition that at a maximum, the partial derivative with respect to the shape parameter equals zero, we obtain the following system of coupled maximum likelihood estimate equations ( for the average log-likelihoods ) that needs to be inverted to obtain the ( unknown ) shape parameter estimates \ hat { \ alpha }, \ hat { \ beta } in terms of the ( known ) average of logarithms of the samples " X " 1, . . ., " X N ":
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