ࡱ > n p m ~ ~ 4 bjbj > h h , ( ( k k k k k 8 5 G < < < < /5 15 15 15 15 15 15 D7 9 15 k ! @ < ! ! 15 k k y F5 ' ' ' ! k k /5 ' ! /5 ' ' r 2 T 3 @rs $ _2 5 \5 0 5 m2 r: % ~ r: 3 r: k 3 < 0 ' j < < < 15 15 ;' < < < 5 ! ! ! ! r: < < < < < < < < < ( 1 : 1905 2005 Kurtosis is a property of distributions related to the heaviness of the tails and the peakedness of the distribution. Ethimology: The word kurtosis comes from the Greek kurtos (curvature) and had been previously used to indicate curvature in Medicine and Mathematics. Timeline of Kurtosis 1905 Pearson defines Kurtosis, EMBED Equation.3 = EMBED Equation.3 , as a measure of departure from normality in a paper published in Biometrika. EMBED Equation.3 =3 for the normal distribution and the terms leptokurtic ( EMBED Equation.3 >3), mesokurtic ( EMBED Equation.3 =3), platikurtic ( EMBED Equation.3 >3) are introduced. 1906-1910 Articles appear in Biometrika comparing frequency distributions of generally very large data sets (anthropometric measurements, indicators of severity of smallpox, size of paramecium Chilomona, pig fertility) to the normal distribution using EMBED Equation.3 and symmetry1920s.. the kurtosis statistic is mentioned in all statistics textbooks, even introductory ones, together with location, spread and symmetry 1943aArticles start to appear pointing to some misconceptions about kurtosis in introductory textbooks. Some times people tended to think that if f(0) was higher for one density function than for another, necessarily that distribution had higher kurtosis. 1943 bThe density crossing sufficient condition for one distribution to have a higher value of EMBED Equation.3 than another one appears (Dyson, 1943, Finucan 1963)). If two density functions(with common variance) cross twice at each side of 0, one has higher EMBED Equation.3 than the other. The graph at the right shows that the Laplace and Normal distributions with equal variances cross twice,.the Laplace distribution has higher kurtosis than the Normal. On the other hand, from the comparison of the density functions we could not assure that the Triangular has higher EMBED Equation.3 than the Normal because they cross 3 times.1964 Van Zwet defines kurtosis as an ordering of symmetric distributions and says that we should not be representing it by a single measure. F has less kurtosis than G iff EMBED Equation.3 is convex when x>mF ( mF is the common point of symmetry). Not all symmetric distributions are ordered (see graph that shows that the Normal and Laplace are comparable but Laplace and t(6) are not). Van Zwet proved that the following distributions are ordered U-shaped EMBED Equation.3 Uniform EMBED Equation.3 Normal EMBED Equation.3 Logstic EMBED Equation.3 Laplace Later more distributions were proved to be ordered. The graphs below show that: the Laplace distribution has more kurtosis than the Normal distribution the Laplace and the t(6) distributions are not really kurtosis comparable . The graph below compares the normal and triangular distribution, EMBED Equation.3 is not convex for low values of x, therefore the distributions are not kurtosis comparable according to Van Zwets criteria. EMBED MSPhotoEd.3 From 1964 on, two lines of work develop in Kurtosis. One works with ordering of distributions without using measures. The other one defines kurtosis measures and studies the properties of their sample estimators. However when a new measure is defined, to be considered as a valid measure of kurtosis it has to respect Van Zwets ordering. 1970The discussion if kurtosis measures should detect bimodality opens (currently the general understanding is that they do not have to). Here the graph shows the classic example of the double gamma and the normal distribution (Hildebrandt). We have added the plot of EMBED Equation.3 that shows that from the point of view of Van Zwets criteria, the two distributions are not comparable in terms of Kurtosis (the line is part convex part concave) 1982A robust kurtosis statistic is defined (Stavig) as EMBED Equation.3 1987Using the influence function Ruppert addresses the on going discussion if kurtosis is related to peak or to tails (it is related to both) 1988Moors defines a measure of kurtosis based on the octiles EMBED Equation.3 1988Balanda y MacGillivray give a more flexible definition of kurtosis as the movement of mass, adjusted for mean and dispersion, from the shoulders of the distribution to th center and the tail, being possible of quantify it of many different ways.1990Balanda & MacGillivray extend Van Zwets criteria to non-symmetric distributions by defining the spread function EMBED Equation.3 = EMBED Equation.3 , 00. See display of spread function and spread-spread plot for the Uniform and EMBED Equation.3 distributions. EMBED MtbGraph.Document 1990 L-kurtosis is defined (Hosking) as EMBED Equation.3 , EMBED Equation.3 and EMBED Equation.3 . L-kurtosis becomes popular in water resources research1998Groeneveld defines quantile kurtosis for symmetric distributions with mean 0 EMBED Equation.3 , 0