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| {{Expert-subject|statistics|date=November 2008}}
| | 42 year old Marine Biologist Courtney from Bedford, spends time with pastimes like gardening, property developers in new ec launch singapore - [http://www.emdriachicago.net/activity/p/69870/ use this link] - and digital photography. At all times enjoys visiting destinations like Historic Centre of Riga. |
| In [[statistics]], the '''Vuong closeness test''' is [[likelihood-ratio test|likelihood-ratio]]-based test for [[model selection]] using the [[Kullback-Leibler divergence|Kullback-Leibler information criterion]]. This statistic makes probabilistic statements about two models. They can be nested, non-nested or overlapping. The statistic tests the null hypothesis that the two models are equally 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.
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| With non-nested models and [[iid]] exogenous variables, model 1 (2) is preferred with significance level α, if the [[z statistic]]
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| :<math>Z=\frac{LR_N(\beta_{ML,1},\beta_{ML,2})} {\sqrt{N}\omega_N}</math>
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| with
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| : <math> {LR_N(\beta_{ML,1},\beta_{ML,2})} = L^1_N-L^2_N-\frac{K_1-K_2} {2} \log N</math>
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| exceeds the positive (falls below the negative) (1 − α)-quantile of the [[standard normal distribution]]. Here ''K''<sub>1</sub> and ''K''<sub>2</sub> are the numbers of parameters in models 1 and 2 respectively.
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| The numerator is the difference between the maximum likelihoods of the two models, corrected for the number of coefficients analogous to the [[Bayesian information criterion|BIC]], the term in the denominator of the expression for ''Z'', <math>\omega_N \,</math>, is defined by setting <math>\omega_N^2</math> equal to either the mean of the squares of the pointwise log-likelihood ratios <math>\ell_i\,</math>, or to the sample variance of these values, where
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| :<math>\ell_i = \log\frac{f_1(y_i|x_i,\beta_{ML,1})}{f_2(y_i|x_i,\beta_{ML,2})}.</math> | |
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| For nested or overlapping models the statistic
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| : <math>2LR_N(\beta_{ML,1},\beta_{ML,2})\,</math>
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| has to be compared to critical values from a weighted sum of [[chi squared distribution]]s. This can be approximated by a [[gamma distribution]]:
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| :<math>M_m(.,\bold\lambda)\sim \Gamma(b,p)\,</math>
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| with
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| : <math> \bold\lambda=(\lambda_1, \lambda_2, \dots, \lambda_m),\,</math>
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| : <math>m=K_1+K_2,\ b=\frac 1 2 \frac {\sum\lambda_i} {\sum\lambda_i^2}</math>
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| and
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| : <math>p=\frac 1 2 \frac {{(\sum\lambda i)}^2} {\sum\lambda_i^2}.</math>
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| <math>\bold\lambda</math> is a vector of [[eigenvalue]]s of a [[Matrix (mathematics)|matrix]] of conditional [[expected value|expectations]]. The computation is quite difficult, so that in the overlapping and nested case many authors{{Who|date=February 2011}} only derive statements from a subjective evaluation of the Z statistic (is it subjectively "big enough" to accept my hypothesis?).
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| ==References==
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| *{{cite journal
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| | last1 = Vuong
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| | first1 = Quang H.
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| | year = 1989
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| | title = Likelihood Ratio Tests for Model Selection and non-nested Hypotheses
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| | journal = [[Econometrica]]
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| | volume = 57
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| | issue = 2
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| | pages = 307–333
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| | jstor = 1912557
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| }}
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| *{{cite journal
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| | last1 = Genius| first1 = Margarita
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| |last2=Strazzera|first2=Elisabetta
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| | year = 2002
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| | title = A note about model selection and tests for non-nested contingent valuation models
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| | journal = Economics Letters| volume = 74
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| | issue = 3
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| | pages = 363–370
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| | doi = 10.1016/S0165-1765(01)00566-3
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| }}
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| [[Category:Statistical tests]]
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| [[Category:Econometrics]]
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42 year old Marine Biologist Courtney from Bedford, spends time with pastimes like gardening, property developers in new ec launch singapore - use this link - and digital photography. At all times enjoys visiting destinations like Historic Centre of Riga.