Theory and Methods of Statistics

10.4 The M-Estimators

The method of maximum likelihood is based on the fact that subject to identifiability of the family $\left\{f\left(\cdot ,t\right)\right\}$, if X has pdf $f\left(x,\theta \right)$ , then the function ${\text{E}}_{\theta }\left[logf(X,t)\right]$ has t = θ as its unique maximizer. The MLE of θ based on a random sample X₁, …, X_n is the maximizer of $n^{-1}{\sum }_{i=1}^{n}logf\left(X_i;t\right)$ which is a natural estimate of ${\text{E}}_{\theta }\left[logf(X,t)\right]$. Equivalently, ...