Quasimaximum

Quasimaximum is a term used in mathematics and statistics to denote an approximate maximum value of a function or estimator that is approached but not necessarily attained within a given domain. It is often encountered when the true maximum is not achieved because the domain is not closed, the function is not continuous, or the optimization is only carried out to finite precision.

In mathematics, a quasimaximum can refer to the limit of a maximizing sequence. If f: D → R has a supremum on D but no x in D satisfies f(x) = sup_D f, then a sequence {x_n} in D with f(x_n) → sup_D f is a sequence of quasi-maximizers, and any limit point of {x_n} may be called a quasimaximum (or a quasimaximizing point). The value sup_D f is the quasimaximum value, even though it is not attained within D.

In optimization practice, quasimaximum can describe a solution obtained when incremental improvements fall below a predefined tolerance, yielding a quasi-optimal solution rather than a true optimum.

In statistics, quasi-maximum likelihood estimation (QMLE) is a distinct concept in which parameters are estimated using a likelihood-like objective that may be misspecified. The phrase quasi-maximum is sometimes used informally to refer to the estimator obtained from such a quasi-likelihood, but this usage is separate from the mathematical idea of a quasimaximum.

Example: maximize f(x) = x on the interval [0,1). The supremum is 1, but it is not attained. The sequence x_n = 1 - 1/n yields f(x_n) → 1; the limit point x* = 1 can be viewed as a quasimaximum location with quasimaximum value 1.

See also: quasi-maximum likelihood, maximum, supremum, maximizing sequence.