In probability theory and statistics , kurtosis from the Greek word , kyrtos or kurtos , meaning ... 920613 9 ref In a similar way to the concept of skewness , kurtosis is a descriptor of the shape of a probability .... One common measure of kurtosis, originating with Karl Pearson , is based on a scaled version of the fourth ... proc.hlp a002473332.htm SAS Elementary Statistics Procedures , SAS Institute section on Kurtosis ref For this measure, higher kurtosis means more of the variance is the result of infrequent extreme Deviation ..., the L kurtosis is a scaled version of the fourth L moment . Image KurtosisChanges.png thumb 200px ... of kurtosis in older works, but is not the definition used here. Kurtosis is more commonly ... 4 3 math which is also known as excess kurtosis . The minus 3 at the end of this formula is often explained as a correction to make the kurtosis of the normal distribution equal to zero. Another reason can be seen by looking at the formula for the kurtosis of the sum of random variables. Suppose ... This formula would be much more complicated if kurtosis were defined just as sub 4 sub ... whereas this identity would not hold if the definition did not include the subtraction of 3. Kurtosis ... kurtosis must be &minus 2 or more. This lower bound is realized by the Bernoulli distribution with p , or coin toss . There is no upper limit to the excess kurtosis and it may be infinite. Terminology and examples A high kurtosis distribution has a sharper peak and longer, fatter tails , while a low kurtosis distribution has a more rounded peak and shorter, thinner tails. Distributions with zero excess kurtosis are called mesokurtic , or mesokurtotic. The most prominent example of a mesokurtic ... positive excess kurtosis is called leptokurtic , or leptokurtotic. Lepto means slender http medical ... excess kurtosis is called platykurtic , or platykurtotic. Platy means broad http www.yourdictionary.com ... one obtains heads when flipping a coin once, a coin toss , for which the excess kurtosis is &minus 2 ... more details
Kurtosis risk in statistics and decision theory denotes the fact that observations are spread in a wider fashion than the normal distribution entails. In other words, fewer observations cluster near the average, and more observations populate the extremes either far above or far below the average compared to the Normal distribution bell curve shape of the normal distribution. Kurtosis risk applies to any kurtosis related quantitative model that relies on the normal distribution for certain of its independent variables when the latter may have kurtosis much greater than the normal distribution. Kurtosis risk is commonly referred to as fat tail risk. The fat tail metaphor explicitly describes the situation of having more observations at either extreme than the tails of the normal distribution would suggest therefore, the tails are fatter . Ignoring kurtosis risk will cause any model to understate the risk of variables with high kurtosis. For instance, Long Term Capital Management , a hedge fund cofounded by Myron Scholes , ignored kurtosis risk to its detriment. After four successful years, this hedge fund had to be bailed out by major investment banks in the late 90s because it understated the kurtosis of many Security finance financial securities underlying the fund s own trading positions. ref cite news title Rashomon in Connecticut last Krugman first Paul date 2 October 1998 url http www.slate.com id 1908 publisher Slate Magazine language English accessdate 2008 05 16 ref Beno t Mandelbrot , a French mathematician, extensively researched this issue. He felt that the extensive reliance on the normal distribution for much of the body of modern finance and investment theory is a serious flaw of any related models including the Black Scholes option model developed by Myron ... of Markets A Fractal View of Risk, Ruin, and Reward . See also Skewness risk Kurtosis Taleb distribution ... and Kurtosis Risk on the Swedish Stock Market , Masters Thesis, Department of Economics, Lund ... more details
In probability theory and statistics , a shape parameter is a kind of numerical parameter of a parametric family of probability distribution s. ref Everitt B.S. 2002 Cambridge Dictionary of Statistics. 2nd Edition. CUP. ISBN 0 521 81099 x ref Definition A shape parameter is any parameter of a probability distribution that is neither a location parameter nor a scale parameter nor a function of either or both of these only, such as a rate parameter . Such a parameter must affect the shape of a distribution rather than simply shifting it as a location parameter does or stretching shrinking it as a scale parameter does . Examples The following continuous probability distributions have a shape parameter Beta distribution Burr distribution Erlang distribution ExGaussian distribution Exponential power distribution Gamma distribution Generalized extreme value distribution Log logistic distribution Inverse gamma distribution Pareto distribution Pearson distribution Tukey lambda distribution Weibull distribution Student t distribution Student s t distribution By contrast, the following continuous distributions do not have a shape parameter, so their shape is fixed and only their location or their scale or both can change. It follows that where they exist the skewness and kurtosis of these distribution are constants, as skewness and kurtosis are independent of location and scale parameters. Exponential distribution Cauchy distribution Logistic distribution Normal distribution Raised cosine distribution Uniform distribution Wigner semicircle distribution See also skewness kurtosis References references Category Theory of probability distributions Category Statistical terminology es Par metro de forma fa fr Param tre de forme pl Parametr kszta tu sl Parameter oblike ... more details
math These quantities consistent estimator consistently estimate the theoretical skewness and kurtosis ..., then the exact finite sample distributions of the skewness and kurtosis can themselves be analysed ... and a kurtosis of nowrap 0, SD 0.15 , where SD indicates the standard deviation. cn date January 2012 Transformed sample skewness and kurtosis The sample skewness g sub 1 sub and kurtosis g sub 2 sub ... the sample kurtosis g sub 2 sub has both the skewness and the kurtosis of approximately 0.3, which ... of g sub 1 sub , and sub 2 sub sub 2 sub g sub 1 sub is the kurtosis the expressions ... skewness or kurtosis harv D Agostino Belanger D Agostino 1990 math K 2 Z 1 g 1 2 Z 2 g 2 2 , math ... journal title Distribution of the kurtosis statistic b sub 2 sub for normal statistics first1 F.J. last1 ... more details
Unreferenced date December 2009 In probability theory and statistics , the k th standardized moment of a probability distribution is math frac mu k sigma k math where math mu k math is the k th moment about the mean and is the standard deviation . It is the normalization statistics normalization of the k th moment with respect to standard deviation . The power of k is because moments scale as math x k math , meaning that math mu k lambda X lambda k mu k X math they are homogeneous polynomial s of degree k , thus the standardized moment is scale invariant . This can also be understood as being because moments have dimension in the above ratio defining standardized moments, the dimensions cancel, so they are dimensionless number s. The first standardized moment is zero, because the first moment about the mean is zero The second standardized moment is one, because the second moment about the mean is equal to the variance the square of the standard deviation The third standardized moment is the skewness The fourth standardized moment is the kurtosis Note that for skewness and kurtosis alternative definitions exist, which are based on the third and fourth cumulant respectively. Other normalizations Details Normalization statistics Another scale invariant, dimensionless measure for characteristics of a distribution is the coefficient of variation , math frac sigma mu math . However, this is not a standardized moment, firstly because it is a reciprocal, and secondly because math mu math is the first moment about zero the mean , not the first moment about the mean which is zero . See Normalization statistics for further normalizing ratios. See also Coefficient of variation Moment mathematics Standard score Other normalizations Standard score Other normalizations Statistics DEFAULTSORT Standardized Moment Category Statistical deviation and dispersion Category Statistical ratios Category Theory of probability distributions cs Standardizovan moment es Momento est ndar sl St ... more details
Unreferenced date December 2009 Expert subject Mathematics date November 2008 In mathematics, in the area of statistical analysis , the trispectrum is a statistic used to search for nonlinear interaction s. The Fourier transform of the second order cumulant , i.e., the autocorrelation function , is the traditional power spectrum . The Fourier transform of C4 t1, t2, t3 fourth order cumulant generating function is called the trispectrum or trispectral density. The trispectrum T f1,f2,f3 falls into the category of higher order spectra, or polyspectra , and provides supplementary information to the power spectrum. The trispectrum is a three dimensional construct. The symmetry symmetries of the trispectrum allow a much reduced support set to be defined, contained within the following verticies, where 1 is the Nyquist frequency . 0,0,0 1 2,1 2, 1 2 1 3,1 3,0 1 2,0,0 1 4,1 4,1 4 . The plane containing the points 1 6,1 6,1 6 1 4,1 4,0 1 2,0,0 divides this volume into an inner and an outer region. A stationary signal will have zero strength statistically in the outer region. The trispectrum support is divide into regions by the plane identified above, and by the f1,f2 plane. Each region has different requirements in terms of the bandwidth of signal required for non zero values. In the same way that the bispectrum identifies contributions to a signal s skewness as a function of frequency triples, the trispectrum identifies contributions to a signal s kurtosis as a function of frequency quadruplets. The trispectrum has been used to investigate the domains of applicability of maximum kurtosis phase estimation used in the deconvolution of seismic data to find layer structure. The trispectrum is the non zero stationary support for the four dimensional non stationary trispectrum. Category Time series analysis Category Nonlinear time series analysis ... more details
Unreferenced stub auto yes date December 2009 In finance , volatility clustering refers to the observation, as noted by Beno t Mandelbrot Mandelbrot 1963 , that large changes tend to be followed by large changes, of either sign, and small changes tend to be followed by small changes. A quantitative manifestation of this fact is that, while returns themselves are uncorrelated, absolute returns rt or their squares display a positive, significant and slowly decaying autocorrelation function corr rt , rt 0 for ranging from a few minutes to several weeks. Observations of this type in financial time series have led to the use of GARCH models in financial forecasting and Derivative finance derivatives pricing. The ARCH Robert F. Engle Engle , 1982 and GARCH Tim Bollerslev Bollerslev , 1986 models aim to more accurately describe the phenomenon of volatility clustering and related effects such as kurtosis . The main idea behind these two widely used models is that volatility is dependent upon past realizations of the asset process and related volatility process. This is a more precise formulation of the intuition that asset Volatility finance volatility tends to revert to some mean rather than remaining constant or moving in monotonic fashion over time. See also Stochastic volatility Volatility DEFAULTSORT Volatility Clustering Category Derivatives finance Category Technical analysis Econ stub ... more details
Expert subject Statistics date February 2009 Higher order statistics HOS are descriptive measures of, among other things, qualities of probability distribution s and sample distribution s, and are, themselves, extensions of first and second order measures such as the mean , variance , autocorrelation function , and power spectrum to higher orders. Skewness and kurtosis are examples of this. Arguably, the estimates of coefficients and associated significance test s, in a regression analysis , of terms for effects higher than quadratic could be included here, too. In statistical theory , one long established approach to higher order statistics, for univariate and multivariate distributions is through the use of cumulant s and cumulant joint cumulants . ref Kendall, MG., Stuart, A. 1969 The Advanced Theory of Statistics, Volume 1 Distribution Theory, 3rd Edition , Griffin. ISBN 0852641419 Chapter 3 ref In time series analysis , the extension of these is to higher order spectra, for example the bispectrum and trispectrum . An alternative to generalising the use of moments is to consider L moment s which allow higher order statistics to be based on linear combinations of order statistic s. References reflist External links http www.maths.leeds.ac.uk Applied news.dir issue2 hos intro.html http lpce.cnrs orleans.fr ddwit lalonde lalonde presentations horbury2.pdf http www.ics.uci.edu welling publications papers RobCum aistats.pdf Category Summary statistics statistics stub ... more details
unreferenced date March 2011 Probability distribution name Benktander distribution of the first kind type density pdf image No image available cdf image No image available parameters math a 0 math real number real br math b 0 math real number real support math x 1 math pdf math e b mathrm Log x 2 x 2 a left tfrac 2b a left 1 a 2b mathrm Log x right left 1 tfrac 2b mathrm Log x a right right math cdf math 1 e b mathrm Log x 2 x 1 a left 1 tfrac 2b mathrm Log x a right math mean math 1 tfrac 1 a math median mode variance math frac sqrt b a e tfrac 1 a 2 4b sqrt pi operatorname erfc left tfrac 1 a 2 sqrt b right a 2 sqrt b math skewness kurtosis entropy mgf math math char math math Benktander distribution of the first kind Related distributions math lim b to 0 mathrm Benktander a,b sim mathrm Pareto 1,a 1 math ProbDistributions continuous semi infinite Category Continuous distributions ... more details
In statistics , the Jarque Bera test is a goodness of fit test of whether sample data have the skewness and kurtosis matching a normal distribution . The test is named after Carlos Jarque and Anil K. Bera . The test statistic JB is defined as math mathit JB frac n 6 left S 2 frac14 K 3 2 right math where n is the number of observations or degrees of freedom in general S is the sample skewness , and K is the sample kurtosis math begin align & S frac hat mu 3 hat sigma 3 frac frac1n sum i 1 n x i bar x 3 left frac1n sum i 1 n x i bar x 2 right 3 2 & K frac hat mu 4 hat sigma 4 frac frac1n sum i 1 n x i bar x 4 left frac1n sum i 1 n x i bar x 2 right 2 , end align math where math hat mu 3 math and math hat mu 4 math are the estimates of third and fourth central moment s, respectively, math bar x math is the sample mean , and math hat sigma 2 math is the estimate of the second central moment, the variance . If the data come from a normal distribution, the JB statistic asymptotic analysis asymptotically has a chi squared distribution with two degrees of freedom statistics degrees of freedom , so the statistic can be used to statistical hypothesis testing test the hypothesis that the data are from a normal distribution . The null hypothesis is a joint hypothesis of the skewness being zero and the excess kurtosis being zero. Samples from a normal distribution have an expected skewness of 0 and an expected excess kurtosis of 0 which is the same as a kurtosis of 3 . As the definition of JB shows, any deviation from this increases the JB statistic. For small samples the chi squared approximation is overly sensitive, often rejecting the null hypothesis when it is in fact true. Furthermore, the distribution of p values departs from a uniform distribution and becomes a right skewed uni modal distribution, especially for small p values. This leads to a large Type I error rate. The table below shows some p values approximated by a chi squared distribution that differ from their tr ... more details
McCullagh and Nelder 1989 , Section 4.2.2. ref . The kurtosis goes to infinity for high and low values of p , but for math p 1 2 math the Bernoulli distribution has a lower kurtosis than any ... more details
Probability distribution name Type 2 Gumbel type density pdf image cdf image parameters math a math real number real br math b math shape real support pdf math a b x a 1 e b x a math cdf math e b x a math mean median mode variance skewness kurtosis entropy mgf char In probability theory , the Type 2 Gumbel probability density function is math f x a,b a b x a 1 e b x a , math for math 0 x infty math . This implies that it is similar to the Weibull distribution s, substituting math b lambda k math and math a k math . Note however that a positive k as in the Weibull distribution would yield a negative a , which is not allowed here as it would yield a negative probability density. For math 0 a le 1 math the mean is infinite. For math 0 a le 2 math the variance is infinite. The cumulative distribution function is math F x a,b 1 e b x a , math The moments math E X k , math exist for math k a , math The special case b 1 yields the Fr chet distribution Based on http www.gnu.org software gsl manual html node The Type 002d2 Gumbel Distribution.html The GNU Scientific Library , used under GFDL. See also Extreme value theory Gumbel distribution Type 1 Gumbel distribution ProbDistributions continuous semi infinite Category Continuous distributions sl Gumbelova porazdelitev 2. tipa ... more details
Unreferenced date December 2009 EDITORS Please see Wikipedia WikiProject Probability Standards for a discussion of standards used for probability distribution articles such as this one. Probability distribution name Rademacher type mass pdf image cdf image parameters support math k in 1,1 , math pdf math f k begin cases 1 2, & k 1 1 2, & k 1 end cases math cdf math F k begin cases 0, & k 1 1 2, & 1 leq k 1 1, & k geq 1 end cases math mean math 0 , math median math 0 , math mode N A variance math 1 4 , math skewness math 0 , math kurtosis math 2 , math entropy math ln 2 , math mgf math cosh t , math char math cos t , math In probability theory and statistics , the Rademacher distribution named after Hans Rademacher is a discrete probability distribution discrete probability distribution which has a 50 chance for either 1 or 1. The probability mass function of this distribution is math f k left begin matrix 1 2 & mbox if k 1, 1 2 & mbox if k 1, 0 & mbox otherwise. end matrix right. math it can be also written, in term of the Dirac delta function, as math f k frac 1 2 left delta left k 1 right delta left k 1 right right math The Rademacher distribution has been used in Bootstrapping statistics bootstrapping . Related distributions Bernoulli distribution If X has a Rademacher distribution then math frac X 1 2 math has a Bernoulli 1 2 distribution. ProbDistributions discrete finite DEFAULTSORT Rademacher Distribution Category Discrete distributions fa it Distribuzione discreta uniforme Altre distribuzioni sl Rademacherjeva porazdelitev tr Rademacher da l m ... more details
magnitude of deviations in the sample data. This test is useful in cases where one faces kurtosis ... and kurtosis were some of the earliest tests for normality. Mardia s test Mardia s multivariate skewness and kurtosis tests generalize the moment tests to the multivariate case. ref Mardia, K. V. 1970 . Measures of multivariate skewness and kurtosis with applications. Biometrika 57, 519 530 ... more details
In statistics , the concept of the shape of the distribution refers to the shape of a probability distribution and it most often arises in questions of finding an appropriate distribution to use to model the statistical properties of a population, given a sample from that population. The shape of a distribution may be considered either descriptively, using terms such as J shaped , or numerically, using quantitative measures such as skewness and kurtosis . Considerations of the shape of a distribution arise in statistical data analysis , where simple quantitative descriptive statistics and plotting techniques such as histograms can lead on to the selection of a particular family of distributions for modelling purposes. File Standard deviation diagram.svg right thumb 350px The Normal distribution , often called the bell curve Image Exponential distribution pdf.png thumb Exponential distribution Descriptions of shape The shape of a distribution will fall somewhere in a continuum where a flat distribution might be considered central and where types of departure from this include mounded or unimodal , U shaped, J shaped, reverse J shaped and multi modal. ref Yule & Kendall 1950 Chapter 4 &mdash Frequency Distributions ref A bimodal distribution would have two high points rather than one. The shape of a distribution is sometimes characterised by the behaviours of the tails as in a long or short tail . For example a flat distribution can be said either to have no tails, or to have short tails. A normal distribution is usually regarded as having short tails, while an exponential distribution has exponential tails and a Pareto distribution has long tails. Even in the relatively simple case of a mounded distribution, the distribution may be skewed to the left or skewed to the right, with symmetric corresponding to no skew. See also Shape parameter List of probability distributions Notes Reflist References Yule, G.U., Kendall, M.G. 1950 An Introduction to the Theory of Statist ... more details
Orphan date October 2008 The Binder parameter ref K. Binder, Z. Phys. B 43, 119 1981 ref in statistical physics , also known as the fourth order culumant math U L 1 frac langle s 4 rangle L 3 langle s 2 rangle 2 L math in Ising model ref K. Binder & D. W. Heermann, Monte Carlo Simulation in Statistical Physics An Introduction, Ed. 4, Spring ref , is used to identify phase transition points in numerical simulations. It is defined as the kurtosis of the order parameter. For example in spin glass es one defines the Binder as math B frac 1 2 left 3 frac overline langle q 4 rangle overline langle q 2 rangle 2 right math where math langle cdot rangle math stands for Boltzmann average, math overline cdot math for average over the disorder and math q math is the overlap between two identical replicas of the system. The phase transition point is usually identified comparing the behavior of math B math as a function of the temperature for different values of the system size math L math . The transition temperature is the unique point where the different curves cross. This is based on finite size scaling hypothesis, according to which, in the critical region math T approx T c math the Binder behaves as math B T,L b epsilon L 1 nu math , where math epsilon frac T T c T math . References references Category Statistical mechanics condensedmatter stub ... more details
Test for population mean, for standard deviation, for skewness and for Excess kurtosis excess kurtosis . Normality test using Jarque Bera test , Shapiro Wilk test , and Chi square test methods ... Calculate the Excess kurtosis excess kurtosis of the standardized Exponential power distribution GED ... more details
Empirica Capital a tail hedging firm started by Nassim Nicholas Taleb in 1999 carrying an insurance style strategy now called Black Swan Protection aiming to protect investors from large adverse events ref cite web last Harrington first Shannon D. url http www.bloomberg.com news 2010 07 20 pimco sells black swan protection as wall street profits from selling fear.html title Pimco Sells Black Swan Protection as Wall Street Markets Fear publisher Bloomberg date accessdate 2010 10 01 ref ref name Pat cite Mr. Volatility and the Swan cite , The Wall Street Journal , July 13, 2007 ref The investment strategy of the fund has been explained in a New Yorker article. ref http www.gladwell.com 2002 2002 04 29 a blowingup.htm Gladwell.com ref One of Empirica s funds, Empirica Kurtosis LLC, was reported to have made a 60 return in 2000 followed by losses in 2001, 2002, and single digit gains in 2003 and 2004. Taleb claims that the fund was closed so that he could become a writer and scholar . ref name Pat ref name WSJ08 cite http online.wsj.com article SB122567265138591705.html October Pain Was Black Swan Gain cite , The Wall Street Journal , November 4, 2008 ref In 2007, Mark Spitznagel , Taleb s former partner at Empirica, founded the firm Universa Investments L.P., to which Taleb is an adviser, and which runs portfolio hedging strategies similar to Empirica s. ref name WSJ08 ref name WSJ09 cite http online.wsj.com article SB124519615631521063.html Black Swan Trader Bets Reputation on Inflation cite , The Wall Street Journal , June 17, 2009 ref References reflist Category Hedge fund firms of the United States ... more details
saved book title Economic Statistics subtitle cover image The Normal Distribution.svg cover color Orange Economic Statistics Introduction Data collection Statistics History of statistics Descriptive statistics Statistical inference Descriptive Statistics Average Mean Median Mode statistics Mode Statistical dispersion Variance Standard deviation Range statistics Range Percentile Interquartile range Shape of the distribution Skewness Kurtosis Moment mathematics Moment Grouped data Frequency distribution Contingency table Bar chart Box plot Histogram Scatter plot Radar chart Probability Theory Probability Probability theory Random variable Probability distribution Discrete probability distribution Probability mass function Uniform distribution discrete Uniform distribution discrete Bernoulli distribution Poisson distribution Binomial distribution Continuous probability distribution Uniform distribution continuous Uniform distribution continuous Normal distribution Chi squared distribution Student s t distribution Probability density function Cumulative distribution function Law of large numbers Central limit theorem Estimation Theory Estimator Estimation theory Statistical Inference Frequentist inference Bayesian inference Statistical hypothesis testing Sampling distribution Null hypothesis Alternative hypothesis P value Statistical power Fisher s exact test Pearson s chi squared test Type I and type II errors Z test Student s t test Chi squared test F test Correlation and Regression Analysis Correlation and dependence Regression analysis Linear regression Simple linear regression Ordinary least squares Analysis of variance Analysis of covariance ... more details
Probability distribution name Benini type density pdf image No image available cdf image No image available parameters math alpha 0 math shape parameter shape real number real br math beta 0 math shape parameter shape real number real br math sigma 0 math scale parameter scale real number real support math x sigma math pdf math e alpha log frac x sigma beta log left frac x sigma right 2 left frac alpha x frac 2 beta log frac x sigma x right math cdf math 1 e alpha log frac x sigma beta log frac x sigma 2 math mean math sigma tfrac sigma sqrt 2 beta H 1 left tfrac 1 alpha sqrt 2 beta right math br where math H n x math is the probabilists Hermite polynomial s median math sigma left e frac alpha sqrt alpha 2 beta log 16 2 beta right math mode variance math left sigma 2 tfrac 2 sigma 2 sqrt 2 beta H 1 left tfrac 2 alpha sqrt 2 beta right right mu 2 math skewness math math kurtosis entropy mgf math math char math math Context date March 2011 Related distributions If math X sim mathrm Benini alpha,0, sigma , math , then X has a Pareto distribution with math x mathrm m sigma math If math X sim mathrm Benini 0, tfrac 1 2 sigma 2 ,1 , math then math X sim e U math where math U sim mathrm Rayleigh sigma math Unreferenced date March 2011 ProbDistributions continuous semi infinite Category Probability theory probability stub ... more details
context date March 2011 Probability distribution name Benktander distribution of the second kind type density pdf image No image available cdf image No image available parameters math a 0 math real number real br math 0 b 1 math real number real support nowrap x 1, pdf math e tfrac a 1 x b b x 2 b 1 b a x b math cdf math 1 e tfrac a 1 x b b x 1 b math mean math 1 tfrac 1 a math median math begin cases 1 frac Log 2 a & text if b 1 left frac 1 b a W left frac 2 frac b 1 b a e frac a 1 b 1 b right right tfrac 1 b & text otherwise end cases math br Where math W x math is the Lambert W function mode 1 variance math frac 1 frac 2a e tfrac a b rm E 1 tfrac 1 b , tfrac a b b a 2 math br Where math rm E n,x math is shorthand for math rm E n x math the generalized Exponential integral skewness kurtosis entropy mgf char Benktander distribution of the second kind unreferenced date March 2011 refs need checking References Benktander, G. and Seherdahl, C.O. 1960 On the analytical representation of claim distributions with special reference to excess of loss reinsurance . Trans. 16 th Intern. Congress Actuaries , 626 646. Embrechts, P., Kl uppelberg, C. and Mikosch, T. 1997 . Modelling Extremal Events , Springer Verlag, New York. ProbDistributions continuous semi infinite Category Continuous distributions ... more details
facts, Or date July 2009 where sub 2 sub is the coefficient of excess kurtosis , defined as sub 2 sub sub 4 sub sub 2 sub &minus 3. class wikitable Kurtosis sub 2 sub Most efficient ... more details
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