Matrix Differential Calculus with Applications in Statistics and Econometrics, 3rd Edition

Chapter 15Maximum likelihood estimation

1 INTRODUCTION

The method of maximum likelihood (ML) estimation has great intuitive appeal and generates estimators with desirable asymptotic properties. The estimators are obtained by maximization of the likelihood function, and the asymptotic precision of the estimators is measured by the inverse of the information matrix. Thus, both the first and the second differential of the likelihood function need to be found and this provides an excellent example of the use of our techniques.

2 THE METHOD OF MAXIMUM LIKELIHOOD (ML)

Let {y₁, y₂, …} be a sequence of random variables, not necessarily independent or identically distributed. The joint density function of y = (y₁, … , y_n) ∈ ℝⁿ is denoted by h_n(·; γ₀) and is known except for γ₀, the true value of the parameter vector to be estimated. We assume that γ₀ ∈ Γ, where Γ (the parameter space) is a subset of a finite‐dimensional Euclidean space. For every (fixed) y ∈ ℝⁿ, the real‐valued function

is called the likelihood function, and its logarithm

is called the loglikelihood function.

For fixed y ∈ ℝⁿ every value with

(1)

is called an ML estimate of γ₀. In general, there is no guarantee that an ML ...