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Topic: Leptokurtic distribution

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	NationMaster - Encyclopedia: Leptokurtic distribution (Site not responding. Last check: 2007-11-03)
		In terms of shape, a leptokurtic dsitribution has a more acute "peak" around the mean (that is, a higher probability than a normally distributed variable of values near the mean) and "fat tails" (that is, a higher probability than a normally distributed variable of extreme values).
		In terms of shape, a leptokurtic distribution has a more acute "peak" around the mean (that is, a higher probability than a normally distributed variable of values near the mean) and "fat tails" (that is, a higher probability than a normally distributed variable of extreme values).
		In terms of shape, a platykurtic distribution has a smaller "peak" around the mean (that is, a lower probability than a normally distributed variable of values near the mean) and "thin tails" (that is, a lower probability than a normally distributed variable of extreme values).
	www.nationmaster.com /encyclopedia/Leptokurtic-distribution (257 words)

	Kurtosis - Wikipedia, the free encyclopedia
		The most prominent example of a mesokurtic distribution is the normal distribution family, regardless of the values of its parameters.
		In terms of shape, a leptokurtic distribution has a more acute "peak" around the mean (that is, a higher probability than a normally distributed variable of values near the mean) and "fat tails" (that is, a higher probability than a normally distributed variable of extreme values).
		In terms of shape, a platykurtic distribution has a smaller "peak" around the mean (that is, a lower probability than a normally distributed variable of values near the mean) and "thin tails" (that is, a lower probability than a normally distributed variable of extreme values).
	en.wikipedia.org /wiki/Kurtosis (690 words)

	Hyperbolic secant distribution (Site not responding. Last check: 2007-11-03)
		In probability theory and statistics, the hyperbolic secant distribution is a continuous probability distribution whose probability density function and characteristic function are proportional to the hyperbolic secant function.
		The hyperbolic secant distribution shares many properties with the standard normal distribution: it is symmetric with unit variance and zero mean, median and mode, and its pdf is proportional to its characteristic function.
		However, the hyperbolic secant distribution is leptokurtic, that is, it has a more acute peak near its mean, compared with the standard normal distribution.
	hyperbolic-secant-distribution.mindbit.com (214 words)

	Ch2 Distributions Pt1 (Site not responding. Last check: 2007-11-03)
		In a platykurtic distribution the individual measures are spread out fairly uniformly across their range, whereas in a leptokurtic distribution they tend to cluster compactly at some particular point in the range.
		In the mesokurtic distribution illustrated by Section C the clustering is more moderate than in the leptokurtic distribution, and the curve as it falls away from the peak is more tapering than in the platykurtic distribution.
		In skewed distributions the mean, median, and mode will tend to be separated from one another, with the mean falling toward the tail of the skew, the mode falling away from the tail, at the peak, and the median falling somewhere in-between.
	vassun.vassar.edu /~lowry/ch2pt1.html (3381 words)

	t Distribution (Site not responding. Last check: 2007-11-03)
		In the introduction to normal distributions it was shown that 95% of the area of a normal distribution is within 1.96 standard deviations of the mean.
		The t distribution is very similar to the normal distribution when the estimate of variance is based on many degrees of freedom but has relatively more scores in its tails when there are fewer degrees of freedom.
		Since the t distribution is leptokurtic, the percentage of the distribution within 1.96 standard deviations of the mean is less than the 95% for the normal distribution.
	psych.rice.edu /online_stat/chapter8/t_distribution.html (737 words)

	PlanetMath: moment
		The skewness measures how “symmetrical”, or rather, how “skewed”, a distribution is with respect to its mode.
		means that the distribution has a longer positive tail.
		The kurtosis measures how “peaked” a distribution is compared to the standard normal distribution.
	planetmath.org /encyclopedia/Leptokurtic.html (183 words)

	Stable Paretian Distribution (Fat Tailed Distributions)
		Their common distribution is not normal, but its mean and standard deviation exist.
		If we assume that returns have a certain distribution over a day, we would like them to have the same distribution (perhaps with a different mean and standard deviation) over a month.
		Cauchy distribution A bell-shaped distribution that is more peaked and has fatter tails than the normal distribution.
	www.riskglossary.com /link/stable_paretian_distributions.htm (1045 words)

	Pareto-Levy Stable Distributions
		For non-normal distribution ν has a value but it is not the same as the standard deviation, which for non-normal stable distributions is infinite.
		For a normal distribution α=2, β=0, ν is equal to the standard deviation and δ is equal to the mean.
		If the distribution is a fat-tailed distribution then that fact would account for the unexpected extreme changes in a variables, the sort of occurrences associated with catastrophes.
	www.sjsu.edu /faculty/watkins/stable.htm (509 words)

	Fat--tails and VaR estimation using power EWMA models. - Journal of Academy of Business and Economics - HighBeam ... (Site not responding. Last check: 2007-11-03)
		An important feature of the normal distribution is that the tail decay as the square of an exponential and thus faster than an exponential toward zero, implying that the large positive or negative returns are rare event.
		The Laplace distribution is commonly used in the context of robust estimation, and so the EWMA estimator given by (13) might, therefore, be thought of as a 'robust' EWMA estimator.
		The conditional distribution of asset returns is typically found to be leptokurtic, and to have fatter tail than that of normal distribution.
	www.highbeam.com /library/docfree.asp?DOCID=1G1:126933620&ctrlInfo=Round18:Mode18c:DocG:Result&ao= (5492 words)

	ED230A Kurtosis (Site not responding. Last check: 2007-11-03)
		A leptokurtics distribution is more peaked than an normal while a platykurtic distribution is flatter.
		Converseley, a playkurtic distribution has fewer cases in the tails then would be expected in a normal distribution.
		A normal distribution is said to be mesokurtic and has a 'normal' amount of peakedness and tails that are not too heavy nor too light.
	www.gseis.ucla.edu /courses/ed230a2/kurtosis.html (401 words)

	VIAS Encyclopedia: Kurtosis (Site not responding. Last check: 2007-11-03)
		The kurtosis (or excess) measures the relative flatness of a distribution (as compared to the normal distribution, which shows a kurtosis of zero).
		A positive kurtosis indicates a tapering distribution (also called leptokurtic distribution), whereas a negative kurtosis indicates a flat distribution (platykurtic distribution).
		In this case a normal distribution would yield a kurtosis of 3.
	www.vias.org /encyclopedia/cc_kurtosis.html (124 words)

	[No title]
		Frequency distributions can be graphically presented as a frequency polygon.
		distribution takes on the form of the ideal, normal curve.
		When a distribution is not symmetrical, it is said to be skewed.
	courses.ed.asu.edu /tracey/Statall_files/sheet003.htm (106 words)

	Questions and answers about Statistics: Skewness and kurtosis
		Skewed distributions will also create problems insofar as they indicate violations of the assumption of normality that underlies many of the other statistics like correlation coefficients, t-tests, etc. used to study test validity.
		This would be especially true if the students had previously scored poorly in a positively skewed distribution (with students generally scoring very low) at the beginning of the course on the same or a similar test.
		You should also note that, when reporting central tendency for skewed distributions, it is a good idea to report the median in addition to the mean.
	www.jalt.org /test/bro_1.htm (1706 words)

	FTS Lesson: Markowitz Diversification
		If returns are described by the normal distribution then return and risk can be completely described by the first two moments, mean and variance (= volatility squared) of the return distribution.
		Negative skewness implies distribution spread is more to the left of the mean than it is to the right, and vice versa for positive skewness.
		Leptokurtic distribution --- peaked in center thin in tails (relative to a Standard Normal Distribution Kurtosis is greater than 3).
	www.ftsnet.com /Public/DiscusHTML/markowitz/ftsmkwz6.htm (715 words)

	[No title]
		A leptokurtic distribution has more observations very close to the mean and in the tails.
		In a normal distribution both g1 and g2 are equal to zero.
		A negative g2 indicates a platykurtic distribution and a positive g2 leptokurtic distribution.
	www-users.york.ac.uk /~cd9/usthand3.doc (550 words)

	[No title]
		The context intended is a unimodal distribution and peakedness is measured by the relative steepness of ascent in the neighbourhood of the mode.
		A distribution in which there are large numbers of high and low scores far from the center of the distribution is referred to as a heavy-tailed (or platykurtic) distribution.
		A distribution that is relatively thin in the tails is called a leptokurtic distribution." Well, those who associate heavy-tailedness with kurtosis usually describe the leptokurtic distribution, not the platykurtic distribution, as "heavy tailed," but heavy in the tails is defined in units that are standardized.
	core.ecu.edu /psyc/wuenschk/StatHelp/KURTOSIS.txt (4442 words)

	Stat102
		These values are often standardized for one of two purposes: (1) to allow comparison between two or more distributions or (2) to normalize the distribution.
		In a normal distribution, about 68% of all responses are found plus/minus one standard deviation from the mean, 96% are found within two standard deviations and 99.7%within three standard deviations.
		To expand vocabularies, a distribution with "large" tails is called leptokurtic, those with small tails platykurtic, and those that are normally distributed are mesokurtic.
	books.valdosta.edu /mlis/stats/stat102.htm (362 words)

	Leptokurtic
		A description of the kurtosis in a distribution in which the statistical value is positive.
		Leptokurtic distributions have higher peaks around the mean compared to normal distributions, which leads to thin tails on both sides.
		If the past return data yields a leptokurtic distribution, the stock will have a relatively low amount of variance, because return values are usually close to the mean.
	www.investopedia.com /terms/l/leptokurtic.asp (381 words)

	Kurtosis
		A normal random variable has a kurtosis of 3 irrespective of its mean or standard deviation.
		If a random variable’s kurtosis is greater than 3, it is said to be leptokurtic.
		Evans, Hastings and Peacock (2000) is a handy reference with information on numerous probability distributions, including formulas for the kurtosis of each.
	www.riskglossary.com /link/kurtosis.htm (411 words)

	Kurtosis - Biocrawler (Site not responding. Last check: 2007-11-03)
		If Y is the sum of n independent random variables, all with the same distribution as X, then Kurt[Y] = Kurt[X] / n, while the formula would be more complicated if kurtosis were defined as μ
		This is because the kurtosis as we have defined it is the ratio of the fourth cumulant and the square of the second cumulant of the probability distribution.
		Examples of platykurtic distributions include the continuous uniform distribution, and the Maxwell-Boltzmann distribution.
	www.biocrawler.com /encyclopedia/Kurtosis (612 words)

	Research Design and Analysis, Emporia State Univ.
		The normal distribution is a curve defined by equation 6.1, where Y is the height of the curve given a value of X. In the equation pi is a constant (3.14159) and e is a constant (2.71828).
		For a population to have a normal distribution, the distribution must be symmetrical.
		For example, the IQ of humans is distributed as a normal curve with a mean of 100 and a standard deviation of 15.
	academic.emporia.edu /mooredwi/rda/notes6.htm (2568 words)

	[No title] (Site not responding. Last check: 2007-11-03)
		A positive skew describes a distribution favoring the right tail, whereas a negative skew describes a distribution favoring the left tail.
		A platykurtic distribution has a smaller "peak" around the mean and "thin tails" Distributions with zero kurtosis are called mesokurtic.
		The one on the right is leptokurtic. Autocorrelation Instead of correlation between two different variables, autocorrelation is a correlation coefficient between two values of the same variable at different time periods.
	ihome.ust.hk /~imhwh/359KT3S.doc (777 words)

	79b4a (Site not responding. Last check: 2007-11-03)
		Compared to the Gaussian (bell-shaped) distribution, a leptokurtic distribution of stock market returns has an excessive number of instances near the average and at the extremes.
		Leptokurtosis is a property of a class of statistical distributions called stable Paretian.
		Whereas the Gaussian distribution is defined by two parameters, the mean and standard deviation, Paretian distributions are defined by four: a location parameter, a scale parameter, an index of skewness; and alpha, a measure of the height of extreme tail areas of the distribution.
	www.inv.com /79b4a.htm (214 words)

	Glossary of research economics
		is drawn from a distribution with mean zero and finite variance, often a normal distribution.
		Is the distribution of sums of squares of r standard normal variables.
		This is a one-parameter family of distributions, and the parameter, n, is conventionally labeled the degrees of freedom of the distribution.
	www.econterms.com /econtent.html (14590 words)

	[No title]
		Tells you how ‘spread out’ your distribution is. Question 5 on p.124-127 The Form of a Distribution Chapter 6 Number of Modes Indicates non-normality in distributions.
		Skewness Degree of symmetry Positive and negative skew A distribution is considered symmetrical if it’s skew value in SPSS is between -1 and 1.
		Kurtosis The extent to which cases are piled up around the measure of central tendency or the tails of the distribution.
	www.sfu.ca /~amalm/320lecture7.doc (165 words)

	Blackjack Encyclopedia of Casino Twenty-One: K
		A measure of how "fat" a probability distribution's tails are, measured relative to a normal distribution having the same standard deviation.
		is said to be leptokurtic if its tails are fatter than those of a corresponding normal distribution.
		It is said to be platykurtic if its tails are thinner than those of the normal distribution.
	www.bjrnet.com /member/bjapr/K.htm (317 words)

	Maximum likelihood estimation and inference in multivariate conditionally heteroscedastic dynamic regression models ...
		The second one, in contrast, specifies a parametric leptokurtic distribution for the standardized innovations, such as the Student t distribution used by Bollerslev (1987).
		Nevertheless, a non-Gaussian distribution may be indispensable when we are interested in features of the distribution of asset returns, such as its quantiles, which go beyond its conditional mean and variance.
		Similarly, in the context of multiple financial assets, one may be interested in the probability of the joint occurrence of several extreme events, which is regularly underestimated by the multivariate normal distribution, especially in larger dimensions.
	www.allbusiness.com /periodicals/article/684747-1.html (843 words)

	Article (Site not responding. Last check: 2007-11-03)
		The stable distributions generalize the Normal distribution to leptokurtic cases, with or without skewness.
		They are thus a natural extension of the Gaussian when observed errors are the cumulation of numerous unobserved contributions, eg economics and finance.
		If alpha = 1 and beta =/ 0, the distribution is what I call "afocal", and mu is just an arbitrary quantile with a simple relationship to the c.f.
	www.eco.utexas.edu /cgi-bin/http2gophermail?server=gopher.eco.utexas.edu&request=R1327080-1330619-/mailing/gaussians.archive.1999 (465 words)

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