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Furthermore, a global maximum (or minimum) either must be a local maximum (or minimum) in the interior of the domain, or must lie on the boundary of the domain. So a method of finding a global maximum (or minimum) is to look at all the local maxima (or minima) in the interior, and also look at the maxima (or minima) of the points on the ...
The maximum of a subset of a preordered set is an element of which is greater than or equal to any other element of , and the minimum of is again defined dually. In the particular case of a partially ordered set , while there can be at most one maximum and at most one minimum there may be multiple maximal or minimal elements.
If it does, it is a minimum or least element of . Similarly, if the supremum of belongs to , it is a maximum or greatest element of . For example, consider the set of negative real numbers (excluding zero).
The extreme value theorem was originally proven by Bernard Bolzano in the 1830s in a work Function Theory but the work remained unpublished until 1930. Bolzano's proof consisted of showing that a continuous function on a closed interval was bounded, and then showing that the function attained a maximum and a minimum value.
The sample maximum and minimum are the least robust statistics: they are maximally sensitive to outliers.. This can either be an advantage or a drawback: if extreme values are real (not measurement errors), and of real consequence, as in applications of extreme value theory such as building dikes or financial loss, then outliers (as reflected in sample extrema) are important.
In a totally ordered set the maximal element and the greatest element coincide; and it is also called maximum; in the case of function values it is also called the absolute maximum, to avoid confusion with a local maximum. [1] The dual terms are minimum and absolute minimum. Together they are called the absolute extrema. Similar conclusions ...
In mathematics, a smooth maximum of an indexed family x 1, ..., x n of numbers is a smooth approximation to the maximum function (, …,), meaning a parametric family of functions (, …,) such that for every α, the function is smooth, and the family converges to the maximum function as .
In statistics, the mid-range or mid-extreme is a measure of central tendency of a sample defined as the arithmetic mean of the maximum and minimum values of the data set: [1] M = max x + min x 2 . {\displaystyle M={\frac {\max x+\min x}{2}}.}