tstatistic
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In statistics, the tstatistic is the ratio of the departure of the estimated value of a parameter from its hypothesized value to its standard error. It is used in hypothesis testing via Student's ttest. For example, it is used in estimating the population mean from a sampling distribution of sample means if the population standard deviation is unknown.
Contents
Definition and features
Let be an estimator of parameter β in some statistical model. Then a tstatistic for this parameter is any quantity of the form
where β_{0} is a nonrandom, known constant which may or may not match the actual unknown parameter value β, and is the standard error of the estimator for β.
By default, statistical packages report tstatistic with β_{0} = 0 (these tstatistics are used to test the significance of corresponding regressor). However, when tstatistic is needed to test the hypothesis of the form H_{0}: β = β_{0}, then a nonzero β_{0} may be used.
If is an ordinary least squares estimator in the classical linear regression model (that is, with normally distributed and homoscedastic error terms), and if the true value of the parameter β is equal to β_{0}, then the sampling distribution of the tstatistic is the Student's tdistribution with (n − k) degrees of freedom, where n is the number of observations, and k is the number of regressors (including the intercept).
In the majority of models, the estimator is consistent for β and is distributed asymptotically normally. If the true value of the parameter β is equal to β_{0} and the quantity correctly estimates the asymptotic variance of this estimator, then the tstatistic will asymptotically have the standard normal distribution.
In some models the distribution of the tstatistic is different from the normal distribution, even asymptotically. For example, when a time series with a unit root is regressed in the augmented Dickey–Fuller test, the test tstatistic will asymptotically have one of the Dickey–Fuller distributions (depending on the test setting).
Use
Most frequently, t statistics are used in Student's ttests, a form of statistical hypothesis testing, and in the computation of certain confidence intervals.
The key property of the t statistic is that it is a pivotal quantity – while defined in terms of the sample mean, its sampling distribution does not depend on the population parameters, and thus it can be used regardless of what these may be.
One can also divide a residual by the sample standard deviation:
to compute an estimate for the number of standard deviations a given sample is from the mean, as a sample version of a zscore, the zscore requiring the population parameters.
Prediction
Given a normal distribution with unknown mean and variance, the tstatistic of a future observation after one has made n observations, is an ancillary statistic – a pivotal quantity (does not depend on the values of μ and σ^{2}) that is a statistic (computed from observations). This allows one to compute a frequentist prediction interval (a predictive confidence interval), via the following tdistribution:
Solving for yields the prediction distribution
from which one may compute predictive confidence intervals – given a probability p, one may compute intervals such that 100p% of the time, the next observation will fall in that interval.
History
The term "tstatistic" is abbreviated from "hypothesis test statistic",^{[citation needed]} while "Student" was the pen name of William Sealy Gosset, who introduced the tstatistic and ttest in 1908, while working for the Guinness brewery in Dublin, Ireland.
Related concepts
 zscore (standardization): If the population parameters are known, then rather than computing the tstatistic, one can compute the zscore; analogously, rather than using a ttest, one uses a ztest. This is rare outside of standardized testing.
 Studentized residual: In regression analysis, the standard errors of the estimators at different data points vary (compare the middle versus endpoints of a simple linear regression), and thus one must divide the different residuals by different estimates for the error, yielding what are called studentized residuals.
See also
References
External links