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Jensen's inequality generalizes the statement that a secant line of a convex function lies above its graph. Visualizing convexity and Jensen's inequality. In mathematics, Jensen's inequality, named after the Danish mathematician Johan Jensen, relates the value of a convex function of an integral to the integral of the convex function.
The following theorem presents a strengthened version of the Bernoulli inequality, incorporating additional terms to refine the estimate under specific conditions. Let the expoent r {\displaystyle r} be a nonnegative integer and let x {\displaystyle x} be a real number with x ≥ − 2 {\displaystyle x\geq -2} if r {\displaystyle r} is odd and ...
An inequality is said to be sharp if it cannot be relaxed and still be valid in general. Formally, a universally quantified inequality φ is called sharp if, for every valid universally quantified inequality ψ, if ψ ⇒ φ holds, then ψ ⇔ φ also holds. For instance, the inequality ∀a ∈ R. a 2 ≥ 0 is sharp, whereas the inequality ∀ ...
Using Blinder–Oaxaca decomposition one can distinguish between "change of mean" contribution (purple) and "change of effect" contribution. The Blinder–Oaxaca decomposition (/ ˈ b l aɪ n d ər w ɑː ˈ h ɑː k ɑː /) or Kitagawa decomposition, is a statistical method that explains the difference in the means of a dependent variable between two groups by decomposing the gap into within ...
The Atkinson index is defined as: (, …,) = {(=) / (=) / = (,...,) = +where is individual income (i = 1, 2, ..., N) and is the mean income.. In other words, the Atkinson index is the complement to 1 of the ratio of the Hölder generalized mean of exponent 1−ε to the arithmetic mean of the incomes (where as usual the generalized mean of exponent 0 is interpreted as the geometric mean).
Microsoft Excel is a spreadsheet editor developed by Microsoft for Windows, macOS, Android, iOS and iPadOS.It features calculation or computation capabilities, graphing tools, pivot tables, and a macro programming language called Visual Basic for Applications (VBA).
For a fixed length n, the Hamming distance is a metric on the set of the words of length n (also known as a Hamming space), as it fulfills the conditions of non-negativity, symmetry, the Hamming distance of two words is 0 if and only if the two words are identical, and it satisfies the triangle inequality as well: [2] Indeed, if we fix three words a, b and c, then whenever there is a ...
[2] Whether a space supports a Poincaré inequality has turned out to have deep connections to the geometry and analysis of the space. For example, Cheeger has shown that a doubling space satisfying a Poincaré inequality admits a notion of differentiation. [3] Such spaces include sub-Riemannian manifolds and Laakso spaces.