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:

    “The physicians say, they are not materialists; but they are:MSpirit is matter reduced to an extreme thinness: O so thin!—But the definition of spiritual should be, that which is its own evidence. What notions do they attach to love! what to religion! One would not willingly pronounce these words in their hearing, and give them the occasion to profane them.”
    —Ralph Waldo Emerson (1803–1882)

    “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 an adventure is that when you’re in the middle of it, you say to yourself, “Oh, now I’ve got myself into an awful mess; I wish I were sitting quietly at home.” And the sign that something’s wrong with you is when you sit quietly at home wishing you were out having lots of adventure.”
    —Thornton Wilder (1897–1975)

    “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)