Score (statistics)

scorescore functionscoringscore equationscoresscoring system
In statistics, the score (or informant) is the gradient of the log-likelihood function with respect to the parameter vector.wikipedia
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Score test

Lagrange multiplier testLagrange multiplierLagrange multiplier (LM) test
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.
In statistics, the score test assesses constraints on statistical parameters based on the gradient of the likelihood function—known as the score—evaluated at the hypothesized parameter value under the null hypothesis.

Fisher information

Fisher information matrixinformation matrixinformation
:The latter is known as the Fisher information and is written.
Formally, it is the variance of the score, or the expected value of the observed information.

Likelihood function

likelihoodlikelihood ratiolog-likelihood
In statistics, the score (or informant) is the gradient of the log-likelihood function with respect to the parameter vector. 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 \theta.
This ensures that the score has a finite variance.

Maximum likelihood estimation

maximum likelihoodmaximum likelihood estimatormaximum likelihood estimate
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.
where is the score and is the inverse of the Hessian matrix of the log-likelihood function, both evaluated the r th iteration.

Support curve

The function being plotted is used in the computation of the score and Fisher information, and the graph has a direct interpretation in the context of maximum likelihood estimation and likelihood-ratio tests.

Scoring algorithm

Fisher scoringFisher's scoringFisher scoring algorithm
First, suppose we have a starting point for our algorithm \theta_0, and consider a Taylor expansion of the score function, V(\theta), about \theta_0:

Statistics

statisticalstatistical analysisstatistician
In statistics, the score (or informant) is the gradient of the log-likelihood function with respect to the parameter vector.

Gradient

gradientsgradient vectorvector gradient
In statistics, the score (or informant) is the gradient of the log-likelihood function with respect to the parameter vector. 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 \theta.

Statistical parameter

parametersparameterparametrization
In statistics, the score (or informant) is the gradient of the log-likelihood function with respect to the parameter vector.

Slope

gradientslopesgradients
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.

Infinitesimal

infinitesimalsinfinitely closeinfinitesimally
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.

Continuous function

continuouscontinuitycontinuous map
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.

Parameter space

weight space
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.

Zero of a function

rootrootszeros
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.

Maxima and minima

maximumminimumlocal maximum
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.

Realization (probability)

realizationrealizationsobserved data
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.

Sampling error

sampling variabilitysampling variationless reliable
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.

Test statistic

Common test statisticst''-test of test statistics
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.

Integral

integrationintegral calculusdefinite integral
Further, the ratio of two likelihood functions evaluated at two distinct parameter values can be understood as a definite integral of the score function.

Partial derivative

partial derivativespartial differentiationpartial differential
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 \theta.

Natural logarithm

lnnatural logarithmsnatural log
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 \theta.

Expected value

expectationexpectedmean
While the score is a function of \theta, 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.

Sample space

event spacespacerepresented by points
While the score is a function of \theta, 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. To see this rewrite the likelihood function \mathcal L as a probability density function, and denote the sample space \mathcal{X}.

Probability density function

probability densitydensity functiondensity
To see this rewrite the likelihood function \mathcal L as a probability density function, and denote the sample space \mathcal{X}.

Leibniz integral rule

differentiation under the integral signLeibniz's rulebasic form
The assumed regularity conditions allow the interchange of derivative and integral (see Leibniz integral rule), hence the above expression may be rewritten as