In statistics a quasi-maximum likelihood estimate (QMLE), also known as a pseudo-likelihood estimate or a composite likelihood estimate, is an estimate of a

parameter A parameter (), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when ...

''θ'' in a statistical model that is formed by maximizing a function that is related to the logarithm of the

likelihood function The likelihood function (often simply called the likelihood) represents the probability of random variable realizations conditional on particular values of the statistical parameters. Thus, when evaluated on a given sample, the likelihood funct ...

, but in discussing the consistency and (asymptotic) variance-covariance matrix, we assume some parts of the distribution may be mis-specified. In contrast, the

maximum likelihood In statistics, maximum likelihood estimation (MLE) is a method of estimating the parameters of an assumed probability distribution, given some observed data. This is achieved by maximizing a likelihood function so that, under the assumed stat ...

estimate maximizes the actual log likelihood function for the data and model. The function that is maximized to form a QMLE is often a simplified form of the actual log likelihood function. A common way to form such a simplified function is to use the log-likelihood function of a misspecified model that treats certain data values as being independent, even when in actuality they may not be. This removes any parameters from the model that are used to characterize these dependencies. Doing this only makes sense if the dependency structure is a

nuisance parameter Nuisance (from archaic ''nocence'', through Fr. ''noisance'', ''nuisance'', from Lat. ''nocere'', "to hurt") is a common law tort. It means that which causes offence, annoyance, trouble or injury. A nuisance can be either public (also "common") ...

with respect to the goals of the analysis. As long as the quasi-likelihood function that is maximized is not oversimplified, the QMLE (or composite likelihood estimate) is

consistent In classical deductive logic, a consistent theory is one that does not lead to a logical contradiction. The lack of contradiction can be defined in either semantic or syntactic terms. The semantic definition states that a theory is consistent ...

and asymptotically

normal Normal(s) or The Normal(s) may refer to: Film and television * ''Normal'' (2003 film), starring Jessica Lange and Tom Wilkinson * ''Normal'' (2007 film), starring Carrie-Anne Moss, Kevin Zegers, Callum Keith Rennie, and Andrew Airlie * ''Norma ...

. It is less efficient than the maximum likelihood estimate, but may only be slightly less efficient if the quasi-likelihood is constructed so as to minimize the loss of information relative to the actual likelihood. Standard approaches to statistical inference that are used with maximum likelihood estimates, such as the formation of confidence intervals, and statistics for model comparison,{{cite journal , last=Varin , first=Cristiano , author2=Vidoni, Paolo , journal=Biometrika , year=2005 , volume=92 , issue=3 , pages=519–528 , doi=10.1093/biomet/92.3.519 , title=A note on composite likelihood inference and model selection, url=http://paduaresearch.cab.unipd.it/7054/1/2004_4.pdf can be generalized to the quasi-maximum likelihood setting.

References

Maximum likelihood estimation

See also

References