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Concept
Informant (statistics)
Summary
In statistics, the informant (or score) is the gradient of the log-likelihood function with respect to the parameter vector. Evaluated at a particular point of the parameter vector, the score indicates the steepness of the log-likelihood function and thereby the sensitivity to infinitesimal changes to the parameter values. If the log-likelihood function is continuous over the parameter space, the score will vanish at a local maximum or minimum; this fact is used in maximum likelihood estimation to find the parameter values that maximize the likelihood function.
Since the score is a function of the observations that are subject to sampling error, it lends itself to a test statistic known as score test in which the parameter is held at a particular value. Further, the ratio of two likelihood functions evaluated at two distinct parameter values can be understood as a definite integral of the score function.
The score is the gradient (the vector of partial derivatives) of , the natural logarithm of the likelihood function, with respect to an m-dimensional parameter vector .
This differentiation yields a row vector, and indicates the sensitivity of the likelihood (its derivative normalized by its value).
In older literature, "linear score" may refer to the score with respect to infinitesimal translation of a given density. This convention arises from a time when the primary parameter of interest was the mean or median of a distribution. In this case, the likelihood of an observation is given by a density of the form . The "linear score" is then defined as
While the score is a function of , it also depends on the observations at which the likelihood function is evaluated, and in view of the random character of sampling one may take its expected value over the sample space. Under certain regularity conditions on the density functions of the random variables, the expected value of the score, evaluated at the true parameter value , is zero. To see this, rewrite the likelihood function as a probability density function , and denote the sample space .
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