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The total variation distance (or half the norm) arises as the optimal transportation cost, when the cost function is (,) =, that is, ‖ ‖ = (,) = {(): =, =} = [], where the expectation is taken with respect to the probability measure on the space where (,) lives, and the infimum is taken over all such with marginals and , respectively.
In addition to these several statistics have been developed with nominal data in mind. A number have been summarized and devised by Wilcox (Wilcox 1967), (Wilcox 1973), who requires the following standardization properties to be satisfied: Variation varies between 0 and 1. Variation is 0 if and only if all cases belong to a single category.
In statistics, dispersion (also called variability, scatter, or spread) is the extent to which a distribution is stretched or squeezed. [1] Common examples of measures of statistical dispersion are the variance, standard deviation, and interquartile range. For instance, when the variance of data in a set is large, the data is widely scattered.
An advantage of variance as a measure of dispersion is that it is more amenable to algebraic manipulation than other measures of dispersion such as the expected absolute deviation; for example, the variance of a sum of uncorrelated random variables is equal to the sum of their variances. A disadvantage of the variance for practical applications ...
In probability theory and statistics, the coefficient of variation (CV), also known as normalized root-mean-square deviation (NRMSD), percent RMS, and relative standard deviation (RSD), is a standardized measure of dispersion of a probability distribution or frequency distribution.
The f-divergences are probably the best-known way to measure dissimilarity of probability distributions. It has been shown [ 5 ] : sec. 2 that the only functions which are both IPMs and f -divergences are of the form c TV ( P , Q ) {\displaystyle c\,\operatorname {TV} (P,Q)} , where c ∈ [ 0 , ∞ ] {\displaystyle c\in [0,\infty ]} and TV ...
In probability theory and statistics, the index of dispersion, [1] dispersion index, coefficient of dispersion, relative variance, or variance-to-mean ratio (VMR), like the coefficient of variation, is a normalized measure of the dispersion of a probability distribution: it is a measure used to quantify whether a set of observed occurrences are clustered or dispersed compared to a standard ...
The variance function is a measure of heteroscedasticity and plays a large role in many settings of statistical modelling. It is a main ingredient in the generalized linear model framework and a tool used in non-parametric regression , [ 1 ] semiparametric regression [ 1 ] and functional data analysis . [ 2 ]