Likelihood-ratio test

The likelihood-ratio test, or likelihood ratio test is a statistical test which is a typical hypothesis testing in parametric models. Many classical tests such as the F- test for variance ratio or the two- sample t - test can be interpreted as examples of likelihood-ratio tests.

Definition

Formally, one considers the typical parametric test problem: Given a basic set of probability distributions, depending on an unknown parameter which is from a known basic quantity. As a null hypothesis to be tested whether the parameter belongs to a proper subset. So:

The alternative is accordingly:

The complement to in referred.

The observed data are realizations of random variables, each having the (unknown ) distribution, and statistically independent.

The concept of likelihood-ratio tests already suggests that the decision of the tests based on the formation of a quotient. Is followed in such a way that, starting from the data and the belonging to each parameter density functions calculated the following expression:

Heuristically speaking: one determined by the first data parameter of the given base set that provides the greatest likelihood that the data found are realized according to the distribution. The value of the density function with respect to this parameter will be set as a representative of the whole quantity. The numerator is considered as the basic amount the space of the null hypothesis, ie, for the denominator if one considers the entire base set.

It is intuitive to close: The larger the ratio is, the more likely. A value of near unity implies that based on the data is not a big difference between the two sets of parameters, and to identify. The null hypothesis should in such cases, therefore, not be discarded.

Thus, the hypothesis is rejected at the level a likelihood-ratio test, if

Applies. Here, the critical value is to be selected so that the following applies.

The specific provision of this critical value is generally problematic.

Example 1

For independent random variables, each having a normal distribution with known variance and unknown expectation, the result for the test problem to the following likelihood ratio:

With the independent of the specific data constants. Then obtained that is equivalent to the inequality

Is. This gives as a result the well-known Gauss - test; one chooses, the quantile of a standard normal distribution respectively.

Approximating the likelihood-ratio function by χ ² - distribution

Under certain conditions, can the generally difficult to be observed test statistic approximated by χ ² - distributed random variables, so that comparatively easily derive asymptotic tests. In general, this is possible if the null hypothesis can be represented by a linear transformation parameters as a special case of the alternative hypothesis, as in the below example of the coin toss. Precisely formulated is next to the more technical assumptions on the distribution family, the following assumption of a " parameterizability the null hypothesis " fundamental:

Let the parameter space of the alternative and the null hypothesis given both sets are open and it is true. In addition, exist with a twice continuously differentiable mapping whose Jacobian matrix has full rank for each.

Then:

Where the random variables converge in distribution.

The idea of ​​the proof is based on a statement about the existence of maximum likelihood estimators in general parametric families and their convergence to a normal random variable whose variance is the inverse of the Fisher information.

Example 2: coin toss

An example is the comparison whether two coins have the same chance to get head as a result ( null hypothesis ). If the first coin times thrown head throws and the second coin times thrown head tosses, then the Kontigenztabelle results under observation. Under the null hypothesis () and the alternative hypothesis () the probabilities arise as under the alternative hypothesis and the null hypothesis.

Under the null hypothesis, the likelihood function is measured as the

And it follows by means of the log-likelihood estimation.

Under validity of the alternative hypothesis, the likelihood function is measured as the

And it follows by means of the log-likelihood estimates or.

This results in a

And as a test value

Is compared with a predetermined critical value of the distribution. As we have in the alternative hypothesis two parameters (,) and the null hypothesis a parameter (), the number of degrees of freedom results in as.