Question

Relating M-estimation and Maximum Likelihood Estimation

1 point possible (graded)

Let (E,{Pθ}θ∈Θ) denote a discrete statistical model and let X1,…,Xn∼iidPθ∗ denote the associated statistical experiment, where θ∗ is the true, unknown parameter. Suppose that Pθ has a probability mass function given by pθ. Let θˆMLEn denote the maximum likelihood estimator for θ∗.

The maximum likelihood estimator can be expressed as an M-estimator– that is,

θˆMLEn=argminθ∈Θ1n∑i=1nρ(Xi,θ)

for some function ρ.

Which of the following represents the correct choice of the function ρ so that the equation above is satisfied?

Answer 1

Maximum Likelihood Estimators can be achieved through maximizing the likelihood/log-likelihood of given data, i.e., minimizing the negative log-likelihood of given data.

$\widehat{\theta} = \arg\min_{\displaystyle\theta}{ \left( \sum_{i=1}^n -\log{( f(x_i, \theta) ) }\right) }\\ \rho\left(x_i,\theta \right )=-\log{( f(x_i, \theta) ) }$