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Characterizing distributions by quantile measures

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CBM R 7626 1992 578 ~~~ ~j, ~`'~~ JQ`~~~o~ ,,,,,~,~a ~~ ~~ ~~O ~`~0 ~~Q~ ~~ .h,~~ ~~~ c..~~O (.0~~. C1~~ , II III IIII~ IIIII I I II n I I IVIkII IIII IVI III IIII r F !7 6 . .. ti. . .-; - ~;t 6~,~~~..:-,~... ~~~ CHARACTERIZING DISTRIBUTIONS BY QUANTILE MEASURES R.Th.A. Wagemakers, J.J.A. Moors, M.J.B.T. Janssens ~ 578 Communicated by Dr. R.M.J. Heuts 1 CHARACTERIZING DISTRIBUTIONS BY QUANTILE MEASURES R.Th.A. Wagemakers~ J.J.A. Moors~ M.J.B.T. Janssens~ Abstract. Modelling an empirical distribution by means of a simple theore- tical distribution is an interesting issue in applied statistics. A rea- sonable first step in this modelling process is to demand that measures for location, dispersion, skewness and kurtosis for the two distributions coincide. Up to now, the four measures used hereby were based on moments. In this paper measures are considered which are based on quan- tiles. Of course the four values of these quantile measures do not unique- ly determine the modelling distribution. They do, however, within specific systems of distributions, like Pearson's or Johnson's. This opens the possibility of modelling - within a specific system - an empirical distribution by means of quantile measures. Since moment- based measures are sensitive for outliers, this approach may lead to a better fit. ~ Tilburg University, P.O. Box 90153, 5000 LE Tilburg, Netherlands. 2 1. A quantile measure for kurtosis Consider a random variable x with mean u- E(x) and central moments iui - E(x-u) , i - 2.3. .. The (very familiar) moment-based measures for location, dispersion, skew- ness and kurtosis now are . the mean y, . the variance u2 . the third standardized moment p - u ~u3~21 3 2 . the fourth standardized moment g- u ~yt22 4 2 They all exist provided E(x4) ~ m. For the first three measures quantile-based alternatives are well- known. Defining quartiles Qi by P(x ~ Qi) s i~4 , P(x ) Qi) s 1- i~4 for i- 1,2,3, they are given by . the median Q - Q2 . the

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