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 functionThen, the vector of first derivatives of with respect to the entries of , denoted by 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 The two parameters (mean and variance) together form a vectorThe partial derivative of the log-likelihood with respect to is and the partial derivative with respect to the variance is The score vector is

The maximum likelihood estimator of the parameter solves the maximization problem

Under some regularity conditions, the solution of this problem can be found by solving the first order conditionthat 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.

Previous entry: Sample variance

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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