Z statistic
Encyclopedia
In statistics
, the Vuong closeness test is likelihood-ratio
-based test for model selection
using the Kullback-Leibler information criterion. This statistic makes probabilistic statements about two models. It tests the null hypothesis, that two models (nested, non-nested or overlapping) are as close to the actual model against the alternative that one model is closer. It cannot make any decision whether the "closer" model is the true model.
With non-nested models and iid exogenous variables, model 1 (2) is preferred with significance level α, if the z statistic
with
exceeds the positive (falls below the negative) (1 − α)-quantile of the standard normal distribution. Here K1 and K2 are the numbers of parameters in models 1 and 2 respectively.
The numerator is the difference between the maximum likelihoods of the two models, corrected for the number of coefficients analogous to the BIC, the term in the denominator,
, is defined by setting 
equal to either the mean of the squares of the pointwise log-likelihood ratios 
, or to the sample variance of these values, where
For nested or overlapping models the statistic
has to be compared to critical values from a weighted sum of chi squared distributions. This can be approximated by a gamma distribution:
with
and
is a vector of eigenvalues of a matrix
of conditional expectations
. The computation is quite difficult, so that in the overlapping and nested case many authors only derive statements from a subjective evaluation of the Z statistic (is it subjectively "big enough" to accept my hypothesis?).
Statistics
Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....
, the Vuong closeness test is likelihood-ratio
Likelihood-ratio test
In statistics, a likelihood ratio test is a statistical test used to compare the fit of two models, one of which is a special case of the other . The test is based on the likelihood ratio, which expresses how many times more likely the data are under one model than the other...
-based test for model selection
Model selection
Model selection is the task of selecting a statistical model from a set of candidate models, given data. In the simplest cases, a pre-existing set of data is considered...
using the Kullback-Leibler information criterion. This statistic makes probabilistic statements about two models. It tests the null hypothesis, that two models (nested, non-nested or overlapping) are as close to the actual model against the alternative that one model is closer. It cannot make any decision whether the "closer" model is the true model.
With non-nested models and iid exogenous variables, model 1 (2) is preferred with significance level α, if the z statistic
Z statistic
In statistics, the Vuong closeness test is likelihood-ratio-based test for model selection using the Kullback-Leibler information criterion. This statistic makes probabilistic statements about two models. It tests the null hypothesis, that two models are as close to the actual model against the...
with
exceeds the positive (falls below the negative) (1 − α)-quantile of the standard normal distribution. Here K1 and K2 are the numbers of parameters in models 1 and 2 respectively.
The numerator is the difference between the maximum likelihoods of the two models, corrected for the number of coefficients analogous to the BIC, the term in the denominator,
, is defined by setting equal to either the mean of the squares of the pointwise log-likelihood ratios , or to the sample variance of these values, whereFor nested or overlapping models the statistic
has to be compared to critical values from a weighted sum of chi squared distributions. This can be approximated by a gamma distribution:
with
and
is a vector of eigenvalues of a matrixMatrix (mathematics)
In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...
of conditional expectations
Expected value
In probability theory, the expected value of a random variable is the weighted average of all possible values that this random variable can take on...
. The computation is quite difficult, so that in the overlapping and nested case many authors only derive statements from a subjective evaluation of the Z statistic (is it subjectively "big enough" to accept my hypothesis?).