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Likelihood-ratio test

In statistics, the likelihood-ratio test assesses the goodness of fit of two competing statistical models based on the ratio of their likelihoods, specifically one found by maximization over the entire parameter space and another found after imposing some constraint. If the constraint (i.e., the null hypothesis) is supported by the observed data, the two likelihoods should not differ by more than sampling error. Thus the likelihood-ratio test tests whether this ratio is significantly different from one, or equivalently whether its natural logarithm is significantly different from zero. In statistics, the likelihood-ratio test assesses the goodness of fit of two competing statistical models based on the ratio of their likelihoods, specifically one found by maximization over the entire parameter space and another found after imposing some constraint. If the constraint (i.e., the null hypothesis) is supported by the observed data, the two likelihoods should not differ by more than sampling error. Thus the likelihood-ratio test tests whether this ratio is significantly different from one, or equivalently whether its natural logarithm is significantly different from zero. 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. In fact, the latter two can be conceptualized as approximations to the likelihood-ratio test, and are asymptotically equivalent. In the case of comparing two models each of which has no unknown parameters, use of the likelihood-ratio test can be justified by the Neyman–Pearson lemma. The lemma demonstrates that the test has the highest power among all competitors. Suppose that we have a statistical model with parameter space Θ {displaystyle Theta } . A null hypothesis is often stated by saying that the parameter θ {displaystyle heta } is in a specified subset Θ 0 {displaystyle Theta _{0}} of Θ {displaystyle Theta } . The alternative hypothesis is thus that θ {displaystyle heta } is in the complement of Θ 0 {displaystyle Theta _{0}} , i.e. in Θ ∖ Θ 0 {displaystyle Theta smallsetminus Theta _{0}} , which is denoted by Θ 0 ∁ {displaystyle Theta _{0}^{complement }} . The likelihood ratio test statistic for the null hypothesis H 0 : θ ∈ Θ 0 {displaystyle H_{0},:, heta in Theta _{0}} is given by: where the quantity inside the brackets is called the likelihood ratio. Here, the sup {displaystyle sup } notation refers to the supremum function. As all likelihoods are positive, and as the constrained maximum cannot exceed the unconstrained maximum, the likelihood ratio is bounded between zero and one. Often the likelihood-ratio test statistic is expressed as a difference between the log-likelihoods, L R = − 2 [ ℓ ( θ 0 ) − ℓ ( θ ^ ) ] {displaystyle LR=-2left} , where θ 0 ∈ Θ 0 {displaystyle heta _{0}in Theta _{0}} and θ ^ ∈ Θ {displaystyle {hat { heta }}in Theta } denote the respective arguments of the maxima. The reason for multiplying by negative two is mathematical so that, by Wilks' theorem, L R {displaystyle LR} has an asymptotic χ2-distribution under the null hypothesis. The finite sample distributions of likelihood-ratio tests are generally unknown. The likelihood-ratio test requires that the models be nested—i.e. the more complex model can be transformed into the simpler model by imposing constraints on the former's parameters. Many common test statistics are tests for nested models and can be phrased as log-likelihood ratios or approximations thereof: e.g. the Z-test, the F-test, the G-test, and Pearson's chi-squared test; for an illustration with the one-sample t-test, see below. If the models are not nested, then instead of the likelihood-ratio test, there is a generalization of the test that can usually be used: for details, see relative likelihood. A simple-vs.-simple hypothesis test has completely specified models under both the null hypothesis and the alternative hypothesis, which for convenience are written in terms of fixed values of a notional parameter θ {displaystyle heta } : In this case, under either hypothesis, the distribution of the data is fully specified: there are no unknown parameters to estimate. For this case, a variant of the likelihood-ratio test is available:

[ "Algorithm", "Statistics", "Econometrics", "composite hypothesis testing", "likelihood ratio statistic", "Vuong's closeness test", "Score test" ]
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