The Mathematics of Financial Modelingand Investment Management

12-FinEcon-Model Sel Page 322 Wednesday, February 4, 2004 12:59 PM

322 The Mathematics of Financial Modeling and Investment Management

n

log L(y θ) = ∑log fi(yi θ)

i = 1

The Fisher score function u is defined as the k-vector of the first deriva-

tives of the log-likelihood:

u ()θθθθ = [uj()θθθθ]

∂

uj()θθθθ = -------log L(y θ), j = 1,2,...,k

∂θj

The ML estimator θθθθˆ of the true parameter θθθθis obtained equating

the score to zero: u ()θθθθˆ = 0. It can be demonstrated that the mean of the

score evaluated at the true parameter value vanishes: E[u(θθθθ)] = 0. The

variance-covariance matrix of the score is called the Fisher information

matrix:

var/cov[u ()θθθθ]= E[u ()θθθθu ()]= I θθθθ

T

θθθθ ()

Under mild regularity conditions it can be demonstrated that the follow-

ing relationship holds:

I θθθθ

∂^2 log L ()θθθθ

()= –E --------------------------

∂θi∂θj

The matrix of the second derivatives on the right side is called the

observed information matrix. The classical theory of ML estimators

states that, in large samples, the distribution of the ML estimator θθθθˆ of θθθθ

is approximately normal with parameters [θθθθ, I–1(θθθθ)], that is, the follow-

ing relationship holds:

θθˆ θθ∼N[θθθθ,I θθθθ

• 1
  ()]

This relationship tells us that ML estimators are efficient estimators as

their variance attains the Cramer-Rao bound. The asymptotic joint nor-

mality of the ML estimators can be used to construct a number of tests

and confidence intervals.