M-estimator, method of moments, generalised method of moments क्या है?

An M-estimator maximises (or zeroes the derivative of) a sample average $n^{-1}\sum_im(X_i; \theta)$; maximum likelihood takes $m = \ln f$, least squares $m = -(y - x^\top\theta)^2$. The method of moments solves $n^{-1}\sum_ig(X_i) = \E_\theta[g(X)]$ for as many moments as parameters. The generalised method of moments (GMM) minimises $\bar g_n(\theta)^\top W\bar g_n(\theta)$ for $\bar g_n(\theta) = n^{-1}\sum_ig(X_i; \theta)$ with more moment conditions than parameters and a weight matrix $W$.