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 A statistical model is a mathematical model that embodies a set of statistical assumptions concerning the generation of Sample (statistics), sample data (and similar data from a larger Statistical population, population). A statistical model repre ...

that is formed by maximizing a function that is related to the logarithm of the

likelihood function A likelihood function (often simply called the likelihood) measures how well a statistical model explains observed data by calculating the probability of seeing that data under different parameter values of the model. It is constructed from the ...

, 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 stati ...

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 In statistics, a nuisance parameter is any parameter which is unspecified but which must be accounted for in the hypothesis testing of the parameters which are of interest. The classic example of a nuisance parameter comes from the normal distri ...

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 deductive logic, a consistent theory is one that does not lead to a logical contradiction. A theory T is consistent if there is no formula \varphi such that both \varphi and its negation \lnot\varphi are elements of the set of consequences ...

and asymptotically normal. 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