score function

Given a statistical model {f𝐗(𝒙𝜽):𝜽Θ} with log-likelihood functionMathworldPlanetmath (𝜽𝒙), the score functionMathworldPlanetmath U is defined to be the gradient of :


Since the score function U is also a function of the random vector 𝒙, U is itself a random vector. By setting U to 0, we have a system of k equation(s), otherwise known as the likelihood equation(s):


If 𝜽=θ is one-dimensional, then the score function is simply referred to as the score of θ.

The maximum likelihood estimate (MLE) 𝜽^ of the parameter vector 𝜽 can usually be found by finding the solutions of the likelihood equations. The likelihood equations may also be formed by setting the gradient of the plain likelihood function to zero. The use of the log function often facilitates the algebra as many distributionsPlanetmathPlanetmath are exponentialPlanetmathPlanetmath in nature. For some distributions it may also be necessary to test that the solution to the likelihood equations is really a minimum as opposed to a point of inflection.

Example. n independent observations are made from a random variableMathworldPlanetmath X with a Poisson distributionMathworldPlanetmath with parameter λ. The observed values are x1,,xn. The log-likelihood of the joint pdf is


and so the score function is


where nx¯=xi. To find the MLE of λ, we set U=0 and solve for λ. So the MLE λ^ of λ=x¯.

Title score function
Canonical name ScoreFunction
Date of creation 2013-03-22 14:28:02
Last modified on 2013-03-22 14:28:02
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 13
Author CWoo (3771)
Entry type Definition
Classification msc 62A01
Synonym score
Synonym score statistic
Defines likelihood equation