Likelihoods That Eliminate Nuisance Parameters
In many cases, the likelihood is a function of more than one parameter but interest focuses on the estimation of only one, or at most a few of them, with the others being considered as nuisance parameters. Several alternative approaches have been developed to eliminate such nuisance parameters so that a likelihood can be written as a function of only the parameter (or parameters) of interest; the main approaches being marginal, conditional and profile likelihoods.
These approaches are useful because standard likelihood methods can become unreliable or fail entirely when there are many nuisance parameters or when the nuisance parameters are high-dimensional. This is particularly true when the nuisance parameters can be considered to be "missing data"; they represent a non-negligible fraction of the number of observations and this fraction does not decrease when the sample size increases. Often these approaches can be used to derive closed-form formulae for statistical tests when direct use of maximum likelihood requires iterative numerical methods. These approaches find application in some specialized topics such as sequential analysis.
Read more about this topic: Likelihood Function
Famous quotes containing the words eliminate, nuisance and/or parameters:
“Out of the element of participation follows the certainty of faith; out of the element of separation follows the doubt in faith. And each is essential for the nature of faith. Sometimes certainty conquers doubt, but it cannot eliminate doubt. The conquered of today may become the conqueror of tomorrow. Sometimes doubt conquers faith, but it still contains faith. Otherwise it would be indifference.”
—Paul Tillich (1886–1965)
“Without infringing on the liberty we so much boast, might we not ask our professional Mayor to call upon the smokers, have them register their names in each ward, and then appoint certain thoroughfares in the city for their use, that those who feel no need of this envelopment of curling vapor, to insure protection may be relieved from a nuisance as disgusting to the olfactories as it is prejudicial to the lungs.”
—Harriot K. Hunt (1805–1875)
“What our children have to fear is not the cars on the highways of tomorrow but our own pleasure in calculating the most elegant parameters of their deaths.”
—J.G. (James Graham)