# Score test

Lagrange multiplier testLagrange multiplierLagrange multiplier (LM) testLagrange multiplier statistics
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.wikipedia
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### Score (statistics)

scorescore functionscoring
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.
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.

### Samuel D. Silvey

S. D. Silvey
The equivalence of these two approaches was first shown by S. D. Silvey in 1959, which led to the name Lagrange multiplier test that has become more commonly used, particularly in econometrics, since Breusch and Pagan's much-cited 1980 paper.
Among his contributions are the Lagrange multiplier test, and the use of eigenvalues of the moment matrix for the detection of multicollinearity.

### Wald test

WaldWald estimatorWald statistic
The main advantage of the score test over the Wald test and likelihood-ratio test is that the LM test only requires the computation of the restricted estimator.
Together with the Lagrange multiplier and the likelihood-ratio test, the Wald test is one of three classical approaches to hypothesis testing.

### Likelihood-ratio test

likelihood ratio testlikelihood ratiolikelihood-ratio
The main advantage of the score test over the Wald test and likelihood-ratio test is that the LM test only requires the computation of the restricted estimator.
The likelihood-ratio test is the oldest of the three classical approaches to hypothesis testing, together with the Lagrange multiplier test and the Wald test.

A. R. PaganPagan
The equivalence of these two approaches was first shown by S. D. Silvey in 1959, which led to the name Lagrange multiplier test that has become more commonly used, particularly in econometrics, since Breusch and Pagan's much-cited 1980 paper.
Pagan is known for work in time-series econometrics and hypothesis testing, notably including the Breusch–Pagan test for heteroscedasticity and other applications of the Lagrange multiplier test.

### C. R. Rao

While the finite sample distributions of score tests are generally unknown, it has an asymptotic χ 2 -distribution under the null hypothesis as first proved by C. R. Rao in 1948, a fact that can be used to determine statistical significance.

### Lagrange multiplier

Lagrange multipliersLagrangianLagrangian multiplier
Since function maximization subject to equality constraints is most conveniently done using a Lagrangean expression of the problem, the score test can be equivalently understood as a test of the magnitude of the Lagrange multipliers associated with the constraints where, again, if the constraints are non-binding at the maximum likelihood, the vector of Lagrange multipliers should not differ from zero by more than sampling error.

### Maximum likelihood estimation

maximum likelihoodmaximum likelihood estimatormaximum likelihood estimate
Suppose that is the maximum likelihood estimate of \theta under the null hypothesis H_0 while U and I are respectively, the score and the Fisher information matrices under the alternative hypothesis.
This in turn allows for a statistical test of the "validity" of the constraint, known as the Lagrange multiplier test.

### Structural break

Sup-LR testStructural break testSup-LM test
For cases 1 and 2, the sup-Wald (i.e., the supremum of a set of Wald statistics), sup-LM (i.e., the supremum of a set of Lagrange multiplier statistics), and sup-LR (i.e., the supremum of a set of likelihood ratio statistics) tests developed by Andrews (1993, 2003) may be used to test for parameter instability when the number and location of structural breaks are unknown.

### Logrank test

Log-rank test
When the data consists of failure time data in two groups, the score statistic for the Cox partial likelihood is the same as the log-rank statistic in the log-rank test.

### Statistics

statisticalstatistical analysisstatistician
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.

### Constraint (mathematics)

constraintconstraintsconstrained
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.

### Statistical parameter

parametersparameterparametrization
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.

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.

### Likelihood function

likelihoodlikelihood ratiolog-likelihood
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.

### Null hypothesis

nullnull hypotheseshypothesis
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.

### Maxima and minima

maximumminimumlocal maximum
Intuitively, if the restricted estimator is near the maximum of the likelihood function, the score should not differ from zero by more than sampling error.

### Sampling error

sampling variabilitysampling variationless reliable
Intuitively, if the restricted estimator is near the maximum of the likelihood function, the score should not differ from zero by more than sampling error.

### Sampling distribution

finite sample distributiondistributionsampling
While the finite sample distributions of score tests are generally unknown, it has an asymptotic χ 2 -distribution under the null hypothesis as first proved by C. R. Rao in 1948, a fact that can be used to determine statistical significance.

### Chi-squared distribution

chi-squaredchi-square distributionchi square distribution
While the finite sample distributions of score tests are generally unknown, it has an asymptotic χ 2 -distribution under the null hypothesis as first proved by C. R. Rao in 1948, a fact that can be used to determine statistical significance.

### Statistical significance

statistically significantsignificantsignificance level
While the finite sample distributions of score tests are generally unknown, it has an asymptotic χ 2 -distribution under the null hypothesis as first proved by C. R. Rao in 1948, a fact that can be used to determine statistical significance.

### Magnitude (mathematics)

magnitudemagnitudesmagnitude of the vector
Since function maximization subject to equality constraints is most conveniently done using a Lagrangean expression of the problem, the score test can be equivalently understood as a test of the magnitude of the Lagrange multipliers associated with the constraints where, again, if the constraints are non-binding at the maximum likelihood, the vector of Lagrange multipliers should not differ from zero by more than sampling error.

### Trevor S. Breusch

BreuschT. S. BreuschTrevor Breusch
The equivalence of these two approaches was first shown by S. D. Silvey in 1959, which led to the name Lagrange multiplier test that has become more commonly used, particularly in econometrics, since Breusch and Pagan's much-cited 1980 paper.

### Boundary (topology)

boundaryboundariesboundary point
This makes testing feasible when the unconstrained maximum likelihood estimate is a boundary point in the parameter space.

### Parameter space

weight space
This makes testing feasible when the unconstrained maximum likelihood estimate is a boundary point in the parameter space.