Likelihood-ratio Test - Definition (likelihood Ratio Test For Composite Hypotheses)

Definition (likelihood Ratio Test For Composite Hypotheses)

A null hypothesis is often stated by saying the parameter is in a specified subset of the parameter space .

\begin{align} H_0 &:& \theta \in \Theta_0\\ H_1 &:& \theta \in \Theta_0^{\complement} \end{align}

The likelihood function is (with being the pdf or pmf) is a function of the parameter with held fixed at the value that was actually observed, i.e., the data. The likelihood ratio test statistic is

Here, the notation refers to the Supremum function.

A likelihood ratio test is any test with critical region (or rejection region) of the form where is any number satisfying . Many common test statistics such as the Z-test, the F-test, Pearson's chi-squared test and the G-test are tests for nested models and can be phrased as log-likelihood ratios or approximations thereof.

Read more about this topic:  Likelihood-ratio Test

Famous quotes containing the words definition, ratio, test and/or composite:

    “According to our social pyramid, all men who feel displaced racially, culturally, and/or because of economic hardships will turn on those whom they feel they can order and humiliate, usually women, children, and animals—just as they have been ordered and humiliated by those privileged few who are in power. However, this definition does not explain why there are privileged men who behave this way toward women.”
    —Ana Castillo (b. 1953)

    “People are lucky and unlucky not according to what they get absolutely, but according to the ratio between what they get and what they have been led to expect.”
    —Samuel Butler (1835–1902)

    “The test of literature is, I suppose, whether we ourselves live more intensely for the reading of it.”
    —Elizabeth Drew (1887–1965)

    “A positive learning climate in a school for young children is a composite of many things. It is an attitude that respects children. It is a place where children receive guidance and encouragement from the responsible adults around them. It is an environment where children can experiment and try out new ideas without fear of failure. It is an atmosphere that builds children’s self-confidence so they dare to take risks. It is an environment that nurtures a love of learning.”
    —Carol B. Hillman (20th century)