Likelihood Ratio Statistic

The likelihood ratio statistic evaluates the relative plausibility of two competing hypotheses on the basis of a collection of sample data. The favored hypothesis is determined by whether the ratio is greater than or less than one.

To introduce the likelihood ratio, suppose that yoBS denotes a vector of observed data. Assume that a parametric joint density is postulated for the random vector Y corresponding to the realization yoBS. Let f(y; θ) represent this density, with parameter vector θ. The likelihood of θ based on the data yoBS is defined as the joint density:

Although the likelihood and the density are the same function, they are viewed differently: The density f(y; θ) assigns probabilities to various outcomes for the random vector Y based on a fixed value of θ, whereas the likelihood L(θ; yoBs) reflects the plausibility of various values for θ based on the observed data yoBS.

In formulating the likelihood, multiplicative factors that do not depend on θ are routinely omitted, and the function is redefined based on the remaining terms, which comprise the kernel. For instance, when considering a binomial experiment based on n trails with success probability θ, the density for the success count Y is [Page 713]whereas the corresponding likelihood for θ based on the observed success count yoBS is often defined as

Now, consider two possible values for the parameter vector θ, say θ0 and θ1. The likelihood ratio statistic L(θ0;yoBs)/L(θ1;yoBs) might be used to determine which of these two candidate values is more “likely” (i.e., which is better supported by the data yoBs). If the ratio is less than one, θ1 is favored; if the ratio is greater than one, θ0 is preferred.

As an example, in a classroom experiment to illustrate simple Mendelian genetic traits, suppose that a student is provided with 20 seedlings that might flower either white or red. She is told to plant these seedlings and to record the colors after germination. Let θ denote the probability of a seedling flowering white. If Y denotes the number of seedlings among the 20 planted that flower white, then Y might be viewed as arising from a binomial distribution with density

The student is told that θ is either θ0 = 0:75 or θ1 = 0:50; she must use the outcome of her experiment to determine the correct probability. After planting the 20 seedlings, she observes that yoBS = 13 flower white. In this setting, the likelihood ratio statistic

equals 1.52. Thus, the likelihood ratio implies that the value θ0 = 0:75 is the more plausible value for the probability θ. Based on the ratio, the student should choose the value θ0 = 0:75.

The likelihood ratio might also be used to test formally two competing point hypotheses H0: θ = θ0 versus H1: θ = θ1. In fact, the Ney-man—Pearson Lemma establishes that the power of such a test will be at least as high as the power of any alternative test, assuming that the tests are conducted using the same levels of significance.

A generalization of the preceding test allows one to evaluate two competing composite hypotheses H0: θ∊Θ0 versus H1: θ∊ Θ1. Here, Θ0 and Θ1 refer to disjoint parameter spaces where the parameter vector θ might lie. The conventional test statistic, which is often called the generalized likelihood ratio statistic, is given by

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