The Mathematics of Financial Modelingand Investment Management

12-FinEcon-Model Sel Page 320 Wednesday, February 4, 2004 12:59 PM

320 The Mathematics of Financial Modeling and Investment Management

bound for the variance σ^2

Y of Y. In fact, under mild regularity condi-

tions, it can be demonstrated that

2 1

σY = var Y ≥ -----

In

^2 

I = nE ∂

 ------log f(X θ) 



n

∂θ 

The Cramer-Rao bound can be generalized to the estimates of a k-

vector of parameters θθθθ. In this case, one must consider the Fisher infor-

mation matrix I(θθθθ) (see below) which is defined as the variance-covari-

ance matrix of the vector

∂

------log f(X θ)

∂θ

It can be demonstrated that the difference between the variance-covari-

ance matrix of the vector θθθθand the inverse of the Fisher information

matrix is a nonnegative definite matrix.

This does not mean that the entries of the variance-covariance

matrix of the vector θθθθare systematically bigger than the elements of the

inverse of the Fisher information matrix. However, we can determine a

lower bound for the variance of each parameter θi. In fact, as all the

diagonal elements a nonnegative definite matrix are nonnegative, the

following relationship holds:

σ^2

θi = var θi ≥ {I

– (^1) }
ii,
In other words, the lower bound of the variance of the i-th parameter
θi is the i-th diagonal entry of the inverse of the Fisher information
matrix. Estimators that attain the Cramer-Rao bound are called efficient
estimators. In the following section we will show that the maximum like-
lihood (ML) estimators attain the Cramer-Rao lower bound and are
therefore efficient estimators.
There are various methodologies for determining estimators. An
important methodology is based on the maximum likelihood estimation
(MLE). MLE is a principle of statistical estimation which, given a para-
metric model, prescribes choosing those parameters that maximize the