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For example, x ∗ is a strict global maximum point if for all x in X with x ≠ x ∗, we have f(x ∗) > f(x), and x ∗ is a strict local maximum point if there exists some ε > 0 such that, for all x in X within distance ε of x ∗ with x ≠ x ∗, we have f(x ∗) > f(x). Note that a point is a strict global maximum point if and only if ...
Near the maximum likelihood estimate, low Fisher information therefore indicates that the maximum appears "blunt", that is, the maximum is shallow and there are many nearby values with a similar log-likelihood. Conversely, high Fisher information indicates that the maximum is sharp.
As an example, both unnormalised and normalised sinc functions above have of {0} because both attain their global maximum value of 1 at x = 0. The unnormalised sinc function (red) has arg min of {−4.49, 4.49}, approximately, because it has 2 global minimum values of approximately −0.217 at x = ±4.49.
The probability of having a specific range value, t, can be determined by adding the probabilities of having two samples differing by t, and every other sample having a value between the two extremes. The probability of one sample having a value of x is (). The probability of another having a value t greater than x is:
For example, if one takes the function () that is equal to zero everywhere except at = where () =, then the supremum of the function equals one. However, its essential supremum is zero since (under the Lebesgue measure ) one can ignore what the function does at the single point where f {\displaystyle f} is peculiar.
At points of discontinuity, a Fourier series converges to a value that is the average of its limits on the left and the right, unlike the floor, ceiling and fractional part functions: for y fixed and x a multiple of y the Fourier series given converges to y/2, rather than to x mod y = 0. At points of continuity the series converges to the true ...
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value theorem was originally proven by Bernard Bolzano in the 1830s in a work Function Theory but the work remained unpublished until 1930.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.
For a sample set, the maximum function is non-smooth and thus non-differentiable. For optimization problems that occur in statistics it often needs to be approximated by a smooth function that is close to the maximum of the set. A smooth maximum, for example, g(x 1, x 2, …, x n) = log( exp(x 1) + exp(x 2) + … + exp(x n) )