Talk:Kurtosis

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--Cronholm¹⁴⁴ 03:57, 16 June 2007 (UTC)

1 Wikipedia inconsistancy

[edit] Wikipedia inconsistancy

Hi statistic wikipedia folks. In this page the Kurtosis definition has a "-3" in it (because the normal has a Kurtosis of 3 so this definition "normalises" things so to say). Subtracting this 3 is actually a convention, maybe this should be mentioned.

A more important point is that every single page on distributions I've encountered here does NOT include the -3 in the Kurtosis formula given on the right (correct me if I'm wrong? I didn't recalculate them all manually :)). So while this is only a matter of convention, we should at least get wikipedia consistent with its own definition conventions? The easiest way seems adapting the definition in this page.

Regards

woutersmet

The reason for this (I think!) is that people who have contributed to this page are from an econometrics background where its common to assume a conditional normal distribution. Hence the -3. —Preceding unsigned comment added by 62.30.156.106 (talk) 21:45, 14 March 2008 (UTC)

[edit] Standardized moment

If this is the "fourth standardized moment", what are the other 3 and what is a standardized moment anyway? do we need an article on it? -- Tarquin 10:39 Feb 6, 2003 (UTC)

The first three are the mean, standard deviation, and skewness, if I recall correctly.

Actually, the word "standarized" refers to the fact that the fourth moment is divided by the 4th power of the standard deviation. — Miguel 15:53, 2005 Apr 19 (UTC)

Thank you :-) It's nice when wikipedia comes up with answers so quickly! -- Tarquin 11:04 Feb 6, 2003 (UTC)

I think the term "central moments" is also used. See also http://planetmath.org/encyclopedia/Moment.htm

No, central moments are distinct from standardized moments. --MarkSweep (call me collect) 02:14, 6 December 2006 (UTC)

[edit] Peakedness

Kurtosis is a measure of the peakedness ... so what does that mean? If I have a positive kurtosis, is my distribution pointy? Is it flat? -- JohnFouhy, 01:53, 11 Nov 2004

I've tried to put the answer to this in the article: high kurtosis is 'peaked' or 'pointy', low kurtosis is 'rounded'. Kappa 05:15, 9 Nov 2004 (UTC)

[edit] Mistake

I believe the equation for the sample kurtosis is incorrect (n should be in denominator, not numerator). I fixed it. Neema Sep 7, 2005

[edit] Ratio of cumulants

The statement, "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," does not explain (to me, at least) why it is obvious that subtracting three gives the pretty sample mean result. Isn't it just a result of cranking through the algebra, and if so, should we include this explanation? More concretely, the kurtosis is a ratio of central moments, not cumulants. I don't want to change one false explanation that I don't understand to another, though. Gray 01:30, 15 January 2006 (UTC)

After thinking a little more, I'm just going to remove the sentence. Please explain why if you restore it. Gray 20:58, 15 January 2006 (UTC)

[edit] Mesokurtic

It says: "Distributions with zero kurtosis are called mesokurtic. The most prominent example of a mesokurtic distribution is the normal distribution family, regardless of the values of its parameters." Yet here: http://en.wikipedia.org/wiki/Normal_distribution, we can see that Kurtosis = 3, it's Skewness that = 0 for normal. Agree? Disagree?

Thanks, that's now fixed. There are two ways to define kurtosis (ratio of moments vs. ratio of cumulants), as explained in the article. Wikipedia uses the convention (as do most modern sources) that kurtosis is defined as a ratio of cumulants, which makes the kurtosis of the normal distribution identically zero. --MarkSweep (call me collect) 14:43, 24 July 2006 (UTC)

[edit] Unbiasedness

I have just added a discussion to the skewness page. Similar comments apply here. Unbiasedness of the given kurtosis estimator requires independence of the observations and does not therefore apply to a finite population.

The independent observations version is biased, but the bias is small. This is because, although we can make the numerator and denominator unbiased separately, the ratio will still be biased. Removing this bias can be done only for specific populations. The best we can do is either: 1 use an unbiased estimate for the fourth moment about the mean,
2 use an unbiased estimate of the fourth cumulant,
in the numerator; and either: 3 use an unbiased estimate for the variance,
4 use an unbiased estimate for the square of the variance, in the denominator.

According to the article, the given formula is 2 and 3 but I have not checked this. User:Terry Moore 11 Jun 2005

[edit] So who's Kurt?

I mean, what is the etymology of the term? -FZ 19:48, 22 Jun 2005 (UTC)

It's obviously a modern term of Greek origin (κυρτωσις, fem.). The OED gives the non-specialized meaning as "a bulging, convexity". The Liddell-Scott-Jones lexicon has "bulging, of blood-vessels", "convexity of the sea's surface" and "being humpbacked". According to the OED (corroborated by "Earliest Known Uses of Some of the Words of Mathematics" and by a search on JSTOR), the first occurrence in print of the modern technical term is in an article by Karl Pearson from June 1905. --MarkSweep 21:05, 22 Jun 2005 (UTC)

[edit] Kurtosis Excess?

I've heard of "excess kurtosis," but not vice-versa. Is "kurtosis excess" a common term? Gray 01:12, 15 January 2006 (UTC)

[edit] Diagram?

A picture would be nice ... (one is needed for skewness as well. I'd whip one up, but final projects have me beat right now. 24.7.106.155 08:27, 19 April 2006 (UTC)

[edit] Range?

Is the range -2, +infinity correct? why not -3, +infinity?

Yes, the range is correct. In general all distributions must satisfy $\gamma_2 - \gamma_1^2 + 2 \geq 0$ . The minimum value of $γ 2$ is −2. --MarkSweep (call me collect) 02:26, 6 December 2006 (UTC)

I take that back, will look into it later. --MarkSweep (call me collect) 09:32, 6 December 2006 (UTC)

[edit] Sample kurtosis

Is the given formula for the sample kurtosis really right? Isn't it supposed to have the -3 in the denominator? --60.12.8.166

In the discussion of the "D" formula, the summation seems to be over i terms, whereas the post lists: "x_i - the value of the x'th measurement" I think this should read: "x_i - the value of the i'th measurement of x" (or something close) --Twopoint718 19:25, 13 May 2007 (UTC)

[edit] Shape

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)

Is that right? How can a function have both a greater probability near the mean and a greater probability at the tails? Ditto for platykurtic distributions--DocGov 21:49, 18 November 2006 (UTC)

Yes, that's right. One typically has in mind symmetric unimodal distributions, and leptokurtic ones have a higher peak at the mode and fatter tails than the standard normal distribution. For an example have a look at the section on the Pearson type VII family I just added. --MarkSweep (call me collect) 02:29, 6 December 2006 (UTC)

On the other hand, the Cauchy distribution has a lower peak than the standard normal yet fatter tails than any density in the Pearson type VII family. However, its kurtosis and other moments are undefined. --MarkSweep (call me collect) 04:00, 6 December 2006 (UTC)

Another explanation: it's not just peaks and tails, don't forget about the shoulders. Leptokurtic density with a higher peak and fatter tails have lower shoulders than the normal distribution. Take the density of the Laplace distribution with unit variance:

$f(x) = \frac{1}{\sqrt{2}} \exp(-\sqrt{2}|x|) \!$

For reference, the standard normal density is

$g(x) = \frac{1}{\sqrt{2\pi}} \exp(-x^2/2) \!$

Now f and g intersect at four points, whose x values are $\pm\sqrt{2}\pm\sqrt{2-\ln\pi}$ . Focus on three intervals (on the positive half-line, the negative case is the same under symmetry):

Peak $(0, \sqrt2-\sqrt{2-\ln\pi}) \approx (0, 0.49)$ Here the Laplace density is greater than the normal density and so the Laplace probability of this interval (that is, the definite integral of the density) is greater (0.25 vs. 0.19 for the normal density).
Shoulder $(\sqrt2-\sqrt{2-\ln\pi}, \sqrt2+\sqrt{2-\ln\pi}) \approx (0.49,2.34)$ Here the normal density is greater than the Laplace. The normal probability of this interval is 0.30 vs. 0.23 for the Laplace.
Tail $(\sqrt2+\sqrt{2-\ln\pi},\infty) \approx (2.34,\infty)$ Here the Laplace density is again greater. Laplace probability is 0.02, normal probability is 0.01.

Because we focus on the positive half-line, the probabilities for each distribution sum to 0.5. And even though the Laplace density allocates about twice as much mass to the tail compared with the normal density, in absolute terms the difference is very small. The peak of the Laplace is acute and the region around it is narrow, hence the difference in probability between the two distributions is not very pronounced. The normal distribution compensates by having more mass in the shoulder interval (0.49,2.34). --MarkSweep (call me collect) 08:57, 6 December 2006 (UTC)

Looking at the Pearson Distribution page - isn't the example a Pearson V, not Pearson VII as stated in the title? And, if not, where is more info on Type VII - the Pearson Wikipedia page only goes up to V. 128.152.20.33 19:34, 7 December 2006 (UTC)

Obviously the article on the Pearson distributions is woefully incomplete. As the present article points out, the Pearson type VII distributions are precisely the symmetric type IV distributions. --MarkSweep (call me collect) 05:35, 8 December 2006 (UTC)

[edit] Was someone having us on ? (hoax)

"A distribution whose kurtosis is deemed unaccepatably large or small is said to be kurtoxic. Similarly, if the degree of skew is too great or little, it is said to be skewicked" – two words that had no hits in Google. I think someone was kidding us. DFH 20:33, 9 February 2007 (UTC)

Agree, zero google hits. --Salix alba (talk) 21:24, 9 February 2007 (UTC)

[edit] leptokurtic / platykurtic

I think the definitions of lepto-/platy- kurtic in the article are confusing: the prefixes are reversed. I'm not confident enough in statistics to change this. Could someone who understands the subject check that this is the correct usage?

A distribution with positive kurtosis is called leptokurtic, or leptokurtotic. 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). Examples of leptokurtic distributions include the Laplace distribution and the logistic distribution.

A distribution with negative kurtosis is called platykurtic, or platykurtotic. 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).

leptokurtic: –adjective Statistics. 1. (of a frequency distribution) being more concentrated about the mean than the corresponding normal distribution. 2. (of a frequency distribution curve) having a high, narrow concentration about the mode. [Origin: 1900–05; lepto- + irreg. transliteration of Gk kyrt(ós) swelling + -ic]

lepto- a combining form meaning "thin," "fine," "slight"

platykurtic: 1. (of a frequency distribution) less concentrated about the mean than the corresponding normal distribution. 2. (of a frequency distribution curve) having a wide, rather flat distribution about the mode. [Origin: 1900–05; platy- + kurt- (irreg. < Gk kyrtós bulging, swelling) + -ic]

platy- a combining form meaning "flat," "broad".

--Blick 19:43, 21 February 2007 (UTC)

The current usage is correct and agrees with other references, e.g. [1][2][3]. DFH 21:39, 21 February 2007 (UTC)

I'm still not sure about this. I'm suggesting that instead of describing a platykurtic distribution as one with "thin tails", we should say "broad peak". Would you agree? --Blick 07:30, 5 March 2007 (UTC)

Not really. Moments are more sensitive to the tails, because of the way powers work. The squares of 1 , 2, 3 etc. are 1, 4, 9 etc. which are successively spaced farther apart. The effect is greater for 4th powers. So, although the names playkurtic and leptokurtic are inspired by the appearance of the centre of the density function, the tails are more important. Also it is the behaviour of the tails that determine how robust statistical methods will be and the kurtosis is one diagnostic for that.203.97.74.238 00:46, 1 September 2007 (UTC)Terry Moore

[edit] L-kurtosis

I don't have the time to write about that, but I think the article should mention L-kurtosis, too. --Gaborgulya (talk) 01:13, 22 January 2008 (UTC)

[edit] why 3?

to find out if its mesokurtic, platykurtic or leptokurtic, why compare it to 3? —Preceding unsigned comment added by Reesete (talk • contribs) 10:18, 5 March 2008 (UTC)

The expected Kurtosis for sample of IID standard normal data is 3 (see the wiki article on the normal distribution for more). We tend to refer to excess kurtosis as the sample kurtosis of a series -3 for that reason.. —Preceding unsigned comment added by 62.30.156.106 (talk) 21:42, 14 March 2008 (UTC)