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.
The concept is defined as follows.
Definition
Let
be a
parameter vector describing the
distribution of a sample
.
Let
be the likelihood function of the sample
,
depending on the parameter
.
Let
be the log-likelihood
function![[eq3]](/images/score-vector__8.png){kind=small}
Then,
the
vector of first derivatives of
with respect to the entries of
,
denoted by
![[eq5]](/images/score-vector__12.png){kind=small}
is
called the score vector.
The symbol
is read nabla and is often used to denote the gradient
of a function.
In the next example, the likelihood depends on a
parameter. As a consequence, the score is a
vector.
Example
Suppose the sample
is a vector of
draws
,
...,
from a normal
distribution with mean
and variance
.
As proved in the lecture on
maximum
likelihood estimation of the parameters of a normal distribution, the
log-likelihood of the sample is
![[eq6]](/images/score-vector__22.png)
The
two parameters (mean and variance) together form a
vector![[eq7]](/images/score-vector__24.png)
The
partial derivative of the log-likelihood with respect to
is
![[eq8]](/images/score-vector__26.png)
and
the partial derivative with respect to the variance
is
![[eq9]](/images/score-vector__28.png)
The
score vector
is![[eq10]](/images/score-vector__29.png)
The maximum likelihood estimator
of the parameter
solves the maximization
problem![[eq11]](/images/score-vector__32.png){kind=small}
Under some regularity conditions, the solution of this problem can be found by
solving the first order
condition![[eq12]](/images/score-vector__33.png){kind=small}
that
is, by equating the score function to
.
The score is often used to construct test statistics and conduct hypothesis tests of model restrictions known as score tests.
More details about the log-likelihood and the score vector can be found in the lecture entitled Maximum likelihood.
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Next entry: Size of a test
Please cite as:
Taboga, Marco (2021). "Score vector", Lectures on probability theory and mathematical statistics. Kindle Direct Publishing. Online appendix. https://www.statlect.com/glossary/score-vector.
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