China
Africa
Explore chapters and articles related to this topic
Convergence And Approximation Of The Pareto Model
Published in Graham V. Weinberg, Radar Detection Theory of Sliding Window Processes, 2017
Since (13.8) measures the information lost in the approximation of X by Y, it can be used to assess the convergence of these dsistributions. It can be shown that DKL(X||Y) ≥ 0, a result known as Gibb’s Inequality, which follows from Jensen’s Inequality (Arndt, 2004). It is clear that if the two random variables X and Y coincide then DKL(X||Y) = 0. The converse of this can also be demonstrated to be true. However, it is clear from (13.8) that the Kullback‐Leibler divergence is not symmetric, nor satisfies a triangle inequality. Consequently it is not a metric in the usual analysis sense but is a pseudo‐metric. Its merit in assessing convergence in distribution follows from the Pinsker‐Csiszár Inequality (Pinsker, 1964), (Csiszár, 1967), (Kullback, 1967). Suppose that for the two random variables X and Y their distribution functions are FX(t) and Fy (t) respectively, with support the nonnegative real line. Then this inequality states that ‖FX-FY‖∞=supt≥0|FX(t)-FY(t)|≤2DKL(X‖Y), $$ \|{F_X} - {F_Y}\|_{\infty}= {\rm{sup}}_{t\ge 0}|F_X(t)-F_Y(t)|\le \sqrt{2D_{KL}(X\|Y)}, $$
Multivariate robust second-order stochastic dominance and resulting risk-averse optimization
Published in Optimization, 2019
Pseudo-metric is a generalization of the metric because the distance between two distinct points may be zero. Here we adopt a class of pseudo-metrics which is also known as ζ-structure metrics. Pseudo-metric has been widely used in quantitative and qualitative stability analyses [48]. Let the set of functions be The pseudo-metric is defined as We define the pseudo-metric between two sets by and
Segmented pseudometrics and four-point Fermat-Torricelli problems
Published in Optimization, 2023
Frank Plastria, Francisco Guevara
We call such a pseudometric -like by analogy to the well-known metric (also called Manhattan, rectilinear, rectangular or taxicab distance). is a metric if and only if all the functions are injective.