Comparing the kurtosis measures for symmetric-scale distribution functions considering a new kurtosis
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1 Proceedings of the 8th WSAS International Conference on APPLID MATHMATICS, Tenerife, Sain, December 6-8, 00 (90-9 Comaring the urtosis measures for symmetric-scale distribution functions considering a new urtosis Hedieh Jafarour Rahman arnoosh Deartment of Mathematics Islamic Azad niversity of Shiraz - niversity of Science Technology IRAN htt:/ htt:/ Abstract: In this aer we consider the different inds of urtosis measures. We comare them for symmetric-scale distribution functions. We show the disadvantages of urtosis measures then by introducing a urtosis measure we modify the urtosis measures. inally we discuss the roerties of introduced measure. Keywords: Kurtosis measure, Scale functions, Symmetric distribution. Introduction It is the most oular that the urtosis measure has been comuted by: ( µ ( µ ( where µ is the mean of the rom variable. This is the fourth moment which is divided by suare of the second moment. or a continuous rom variable with distribution function which is symmetric we ca write formula ( by: ( ( V But is not a robust measure, because it is so sensitive to outliers. or eamle by generating 000 data from a normal distribution is comuted, 3.0, if we relace one of the generated value by an outlier value is comuted 3.8, which is far from 3. We now that is eual to 3 for normal distribution. does not measure only eaed ness of a distribution. It measures both eaed ness tail weight of a distribution, so it does not sort the distribution based on the height of distributions each other. If second moment of distribution is infinite, does not eist for the distribution. Bala McGillivray (988 showed that for some distributions which have different shae from normal distribution, is almost 3, so 3 is not a sufficient condition for normality does not measure dearture from normality. is not a good measure for a secial miture distribution function which is introduced by Ali (97. Φ + Φ where Φ is normal distribution function,3. This seuence converges in distribution to the stard normal distribution as 3( +, however. Statistician tried to introduce another urtosis measure which does not have the disadvantages.
2 Proceedings of the 8th WSAS International Conference on APPLID MATHMATICS, Tenerife, Sain, December 6-8, 00 (90-9 Hogss (97 introduced a urtosis measure which measures of tail weight. or 0 < 0. define ( d L ( d where is the distribution function is the th uantile of. ( L ( Q ( L He used Q. or symmetric we have ( Q ( ( 0. ( Q is not a robust measure either, but it is not so sensitive as Ruert (987 construct a robust urtosis measure. or 0 < < η < 0. he defined ( Rη, ( η ( η or symmetric ( Rη, ( η Groeneveld (988 defined a urtosis measure for symmetric distributions. or 0<<0. we have: γ, ( 0.7 Proerties of the urtosis measures Oja (98 called the invariant scale location functional T as a urtosis measure if for two symmetric distribution function G which G has at least as urtosis s as, s G, we can conclude that T ( T ( G. Kurtosis measures have two following roerties. T ( a + b T ( a > 0 s G T ( T ( G Van Zwet (96 introduced for the class of symmetric distributions an ordering defined by s G iff R, G G ( is conve for > m where m is the oint of symmetry of. Van Zwet ordered the following distributions as: niform < lalace < Logistic < Lalace is comuted 0.0, 3,.6 6 resectively. He showed that has the roerties of urtosis measure. We have to note that > 3 does not mean that the density function is higher than normal distribution. It may the density function has much mass in the tails rather than normal density function. 3 Introducing a modified urtosis measure is so sensitive with resect to the outliers, so we offer the following measure: [ ( ( ] I, ( ( I (, [ ] where, ( are th th uantiles of which ~ (.. ( is always finite is symmetric distribution function -. irst of all we show that the introduced measure have two roerties. irst roerty is intuitive. or showing the
3 Proceedings of the 8th WSAS International Conference on APPLID MATHMATICS, Tenerife, Sain, December 6-8, 00 (90-9 second roerty, we show that G ( α, α ( 0, is decreasing ( α R if R( is conve then is a non decreasing function of 0. Without loss of generality we can tae m 0. In this aer we consider symmetric distribution, so the median of distributions are zero. R ( 0 G ( ( 0 G 0 or > 0 we tae the first derivative of R R. We obtain that R. or showing the last euality we note that R ( 0 0. By using the mean value theorem we have R R 0 < < By the conveity of R(, R > 0 hence R is non decreasing R R ( R R So is a non decreasing for >0. A similar roof holds for <0. So G ( ( ( α G ( α α ( 0, ( α ( α is non decreasing. or roving second roerty we have to show that G G G ( It is sufficient ( to show that ( I, I ( (, ( ( G, G I( I G, G ( G G We can tae So. ( I ( (, d( ( ( u d( The last ineuality is euivalent to: ( ( u d u u ( G u d u ( G u By using the mean value theorem the last ineuality is relaced by ( ( u ( ( u 0 < u < u < ( G ( u ( G ( u ( α Because we now that is non G ( α decreasing when s G the above ineuality is hold. Proosition : The urtosis measure ( for, which has been introduced by Ali (97, converges to Φ. Proof: ( ( φ φ where A d φ B ( ( d + A d + B φ d d φ 3
4 Proceedings of the 8th WSAS International Conference on APPLID MATHMATICS, Tenerife, Sain, December 6-8, 00 ( ( Note that 0 lim A lim 6 φ lim d ( ( 0 Similarly we have lim lim ( lim φ d φ d φ. ( d B 0. So φ φ d + A d + B Difference between stardized fourth moment we consider the stardized fourth central moment given by α γ As a new measure of urtosis, where α > 0 are location scale arameters of the distribution. Two urtosis measure are called shae arameter that measures eaed- ness tailed ness of distributions. The denominator of is fourth ower of stard deviation, but the denominator of γ is the fourth ower of scale arameter. If the scale arameter,, the stard deviation, σ, are eual in addition if the location arameter, α, the mean, µ, are eual then γ. or the class of family of normal distributions γ are eual, because α in γ are mean stard deviation. We note that for the family of normal distribution γ 3. Not only for stardized normal distribution but also for any desirable µ σ in normal distribution we have γ 3. Note that location suare of scale arameters are different from mean variance. Stard deviation is one of the estimators of scale arameter. or eamle for stard Lalace distribution function 6 γ. Because f e < < We now that α 0,, µ 0, σ. So, γ ( ( ( 6 ( ( + var( So in general: ( ( µ µ α Concluson In this aer the urtosis measure has been reviewed. The roerties of urtosis measure have been considered. The usual urtosis measure has been modified the roerties of a urtosis measure have been roved for this urtosis measure. The introduced urtosis measure does not have the disadvantages of the other urtosis
5 Proceedings of the 8th WSAS International Conference on APPLID MATHMATICS, Tenerife, Sain, December 6-8, 00 (90-9 measures esecially it wors well for the miture normal distribution function which has been introduced by Ali. The difference of two tyes of urtosis measures γ has been shown. If the mean the stard deviation be the location scale arameter then, otherwise. γ γ References: [] Ali, M.M, Stochastic Ordering Kurtosis Measure, JASA, Vol.69, No.36, 97,.3-. [] Brys, Hubert, M Stryyf, A, Robust measure of tail weight, Comutational Statistics Data Analysis, 00,.-7. [3] Groeneveld, R. A., A class of Quantile measure for urtosis, The American Statistician, Vol., 970,.9-. [] Groeneveld, R. A. Meeden, G., Measuring sewness urtosis, The Statistician, Vol. 33, 98, [] Kevin P. Bala H. L. MacGillivray, Kurtosis: a critical review, JASA, Vol., No., 988,.-9. [6] Oja, H. On location, scale, sewness urtosis of univariate distributions, Sc J Statist, Vol.8, 98,.-68. [7] Ruert, D, What is Kurtosis?, The American Statistician, Vol.,987,.-. [8] Van Zwet, W.R., Conve transformation of rom variables, Mathematical Centrum, Amsterdam, 96.
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