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Score vector

by , PhD

In the theory of maximum likelihood estimation, the score vector (or score function, or, simply, the score) is the gradient (i.e., the vector of first derivatives) of the log-likelihood function with respect to the parameters being estimated.

Table of Contents


The concept is defined as follows.

Definition Let $	heta $ be a Kx1 parameter vector describing the distribution of a sample $xi $. Let [eq1] be the likelihood function of the sample $xi $, depending on the parameter $	heta $. Let [eq2] be the log-likelihood function[eq3]Then, the Kx1 vector of first derivatives of [eq4] with respect to the entries of $	heta $, denoted by [eq5]is called the score vector.

The symbol $
abla $ is read nabla and is often used to denote the gradient of a function.


In the next example, the likelihood depends on a $2	imes 1$ parameter. As a consequence, the score is a $2	imes 1$ vector.

Example Suppose the sample $xi $ is a vector of n draws $x_{1}$, ..., $x_{n}$ from a normal distribution with mean mu and variance sigma^2. As proved in the lecture on maximum likelihood estimation of the parameters of a normal distribution, the log-likelihood of the sample is [eq6]The two parameters (mean and variance) together form a $2	imes 1$ vector[eq7]The partial derivative of the log-likelihood with respect to mu is [eq8]and the partial derivative with respect to the variance sigma^2 is [eq9]The score vector is[eq10]

How the score is used to find the maximum likelihood estimator

The maximum likelihood estimator $widehat{	heta }$ of the parameter $	heta $ solves the maximization problem[eq11]

Under some regularity conditions, the solution of this problem can be found by solving the first order condition[eq12]that is, by equating the score function to 0.

Usage in hypothesis testing

The score is often used to construct test statistics and conduct hypothesis tests of model restrictions known as score tests.

More details

More details about the log-likelihood and the score vector can be found in the lecture entitled Maximum likelihood.

Keep reading the glossary

Previous entry: Sample variance

Next entry: Size of a test

How to cite

Please cite as:

Taboga, Marco (2021). "Score vector", Lectures on probability theory and mathematical statistics. Kindle Direct Publishing. Online appendix.

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