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Score vector

by , PhD

In the theory of maximum likelihood estimation, the score vector (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.

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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 vector to 0.

More details

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

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