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Energy distance is a statistical distance between probability distributions. If X and Y are independent random vectors in Rd with cumulative distribution functions (cdf) F and G respectively, then the energy distance between the distributions F and G is defined to be the square root of For a simple proof of this equivalence, see Székely (2002). In higher dimensions, however, the two distances are different because the energy distance is rotation invariant while Cramér's distance is not. (Notice that Cramér's distance is not the same as the distribution-free Cramér–von Mises criterion.)

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  • Distancia de energia (es)
  • Energy distance (en)
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  • Energy distance is a statistical distance between probability distributions. If X and Y are independent random vectors in Rd with cumulative distribution functions (cdf) F and G respectively, then the energy distance between the distributions F and G is defined to be the square root of For a simple proof of this equivalence, see Székely (2002). In higher dimensions, however, the two distances are different because the energy distance is rotation invariant while Cramér's distance is not. (Notice that Cramér's distance is not the same as the distribution-free Cramér–von Mises criterion.) (en)
  • La distancia de energía es una distancia estadística entre distribuciones de probabilidad. Si X y Y son dos vectores aleatorios independientes en Rd con funciones de distribución acumulada (cdf) F y G respectivamente, la distancia de energía entre las distribuciones F y G se define mediante la la raíz cuadrada de que muestra la equivalencia con la distancia de Harald Cramér. Para una prueba elemental de esta equivalencia, véase por ejemplo Székely (2002).​ (es)
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  • Energy distance is a statistical distance between probability distributions. If X and Y are independent random vectors in Rd with cumulative distribution functions (cdf) F and G respectively, then the energy distance between the distributions F and G is defined to be the square root of where (X, X', Y, Y') are independent, the cdf of X and X' is F, the cdf of Y and Y' is G, is the expected value, and || . || denotes the length of a vector. Energy distance satisfies all axioms of a metric thus energy distance characterizes the equality of distributions: D(F,G) = 0 if and only if F = G.Energy distance for statistical applications was introduced in 1985 by Gábor J. Székely, who proved that for real-valued random variables is exactly twice Harald Cramér's distance: For a simple proof of this equivalence, see Székely (2002). In higher dimensions, however, the two distances are different because the energy distance is rotation invariant while Cramér's distance is not. (Notice that Cramér's distance is not the same as the distribution-free Cramér–von Mises criterion.) (en)
  • La distancia de energía es una distancia estadística entre distribuciones de probabilidad. Si X y Y son dos vectores aleatorios independientes en Rd con funciones de distribución acumulada (cdf) F y G respectivamente, la distancia de energía entre las distribuciones F y G se define mediante la la raíz cuadrada de donde (X, X', Y, Y') son variables aleatorias independientes, siendo X y X' distribuidas de acuerdo a F, Y e Y' según G, es el operador esperanza, y || . || denota la norma euclidiana usual. La distancia de energía satisface todos los axiomas de una distancia, por tanto, la distancia de energía caracteriza la igualdad de distribuciones, esto es, D(F,G) = 0 si y sólo si F = G. La noción de distancia de fue introducida inicialmente en 1985 por Gábor J. Székely, quién demostró para el caso unidimensional la siguiente relación: que muestra la equivalencia con la distancia de Harald Cramér. Para una prueba elemental de esta equivalencia, véase por ejemplo Székely (2002).​ En dimensiones más altas (p>1), como la distancia de energía es invariante ante rotaciónes, mientras la distancia de Cramér no lo es, el test estadístico asociado al problema de testear la igual de distribución entre dos muestras no es de distribución libre. (es)
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