Studentized residual

studentized residualsexternallystudentization of residualsTau-distribution
In statistics, a studentized residual is the quotient resulting from the division of a residual by an estimate of its standard deviation.wikipedia
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Errors and residuals

residualserror termresidual
In statistics, a studentized residual is the quotient resulting from the division of a residual by an estimate of its standard deviation. Typically the standard deviations of residuals in a sample vary greatly from one data point to another even when the errors all have the same standard deviation, particularly in regression analysis; thus it does not make sense to compare residuals at different data points without first studentizing.
One can standardize statistical errors (especially of a normal distribution) in a z-score (or "standard score"), and standardize residuals in a t-statistic, or more generally studentized residuals.

William Sealy Gosset

StudentGossetGosset, William Sealy
It is among several named in honor of William Sealey Gosset, who wrote under the pseudonym Student.
Although introduced by others, Studentized residuals are named in Student's honour because, like the problem that led to Student's t-distribution, the idea of adjusting for estimated standard deviations is central to that concept.

T-statistic

Student's t-statistict''-statisticStudent's ''t''-statistic
It is a form of a Student's t-statistic, with the estimate of error varying between points.

Normalization (statistics)

normalizednormalisedaverage
Dividing a statistic by a sample standard deviation is called studentizing, in analogy with standardizing and normalizing.

Outlier

outliersstatistical outliersconservative estimate
This is an important technique in the detection of outliers.

Leverage (statistics)

leveragehigh-leverage pointleverage score
The leverage h ii is the ith diagonal entry in the hat matrix.
The corresponding studentized residual—the residual adjusted for its observation-specific estimated residual variance—is then

Projection matrix

hat matrixannihilator matrixobservation matrix
and the hat matrix H is the matrix of the orthogonal projection onto the column space of the design matrix:

Student's t-distribution

Student's ''t''-distributiont-distributiont''-distribution
If the errors are independent and normally distributed with expected value 0 and variance σ 2, then the probability distribution of the ith externally studentized residual t_{i(i)} is a Student's t-distribution with n − m − 1 degrees of freedom, and can range from to.

Statistics

statisticalstatistical analysisstatistician
In statistics, a studentized residual is the quotient resulting from the division of a residual by an estimate of its standard deviation.

Unit of observation

data pointdata pointsobservation
Typically the standard deviations of residuals in a sample vary greatly from one data point to another even when the errors all have the same standard deviation, particularly in regression analysis; thus it does not make sense to compare residuals at different data points without first studentizing.

Regression analysis

regressionmultiple regressionregression model
Typically the standard deviations of residuals in a sample vary greatly from one data point to another even when the errors all have the same standard deviation, particularly in regression analysis; thus it does not make sense to compare residuals at different data points without first studentizing. The key reason for studentizing is that, in regression analysis of a multivariate distribution, the variances of the residuals at different input variable values may differ, even if the variances of the errors at these different input variable values are equal.

Student

studentscollege studentpupils
It is among several named in honor of William Sealey Gosset, who wrote under the pseudonym Student.

Standard deviation

standard deviationssample standard deviationSD
In statistics, a studentized residual is the quotient resulting from the division of a residual by an estimate of its standard deviation. Dividing a statistic by a sample standard deviation is called studentizing, in analogy with standardizing and normalizing.

Standard score

normalizednormalisedz-score
Dividing a statistic by a sample standard deviation is called studentizing, in analogy with standardizing and normalizing.

Joint probability distribution

joint distributionjoint probabilitymultivariate distribution
The key reason for studentizing is that, in regression analysis of a multivariate distribution, the variances of the residuals at different input variable values may differ, even if the variances of the errors at these different input variable values are equal.

Independence (probability theory)

independentstatistically independentindependence
where the errors, are independent and all have the same variance \sigma^2.

Robust statistics

robustbreakdown pointrobustness
It is also reflected in the influence functions of various data points on the regression coefficients: endpoints have more influence.

Linear regression

regression coefficientmultiple linear regressionregression
It is also reflected in the influence functions of various data points on the regression coefficients: endpoints have more influence. Consider the simple linear regression model

Estimator

estimatorsestimateestimates
In statistics, a studentized residual is the quotient resulting from the division of a residual by an estimate of its standard deviation. It is not simply a matter of the population parameters (mean and standard deviation) being unknown – it is that regressions yield different residual distributions at different data points, unlike point estimators of univariate distributions, which share a common distribution for residuals.

Univariate distribution

univariateuni
It is not simply a matter of the population parameters (mean and standard deviation) being unknown – it is that regressions yield different residual distributions at different data points, unlike point estimators of univariate distributions, which share a common distribution for residuals.

Design matrix

data matrixdesign matricesdata matrices
For this simple model, the design matrix is

Projection (linear algebra)

orthogonal projectionprojectionprojection operator
and the hat matrix H is the matrix of the orthogonal projection onto the column space of the design matrix:

Normal distribution

normally distributedGaussian distributionnormal
If the errors are independent and normally distributed with expected value 0 and variance σ 2, then the probability distribution of the ith externally studentized residual t_{i(i)} is a Student's t-distribution with n − m − 1 degrees of freedom, and can range from to.

Expected value

expectationexpectedmean
If the errors are independent and normally distributed with expected value 0 and variance σ 2, then the probability distribution of the ith externally studentized residual t_{i(i)} is a Student's t-distribution with n − m − 1 degrees of freedom, and can range from to.

Probability distribution

distributioncontinuous probability distributiondiscrete probability distribution
If the errors are independent and normally distributed with expected value 0 and variance σ 2, then the probability distribution of the ith externally studentized residual t_{i(i)} is a Student's t-distribution with n − m − 1 degrees of freedom, and can range from to.