Editing Beta distribution (section)

===Kurtosis===
[[File:Excess Kurtosis for Beta Distribution as a function of variance and mean - J. Rodal.jpg|325px|thumb|Excess Kurtosis for Beta Distribution as a function of variance and mean]]

The beta distribution has been applied in acoustic analysis to assess damage to gears, as the kurtosis of the beta distribution has been reported to be a good indicator of the condition of a gear.<ref name=Oguamanam>{{cite journal |last1=Oguamanam |first1=D.C.D. |last2=Martin |first2=H. R. |last3=Huissoon |first3=J. P. |title=On the application of the beta distribution to gear damage analysis |journal=Applied Acoustics |year=1995 |volume=45 |issue=3 |pages=247–261 |doi=10.1016/0003-682X(95)00001-P}}</ref> Kurtosis has also been used to distinguish the seismic signal generated by a person's footsteps from other signals. As persons or other targets moving on the ground generate continuous signals in the form of seismic waves, one can separate different targets based on the seismic waves they generate. Kurtosis is sensitive to impulsive signals, so it's much more sensitive to the signal generated by human footsteps than other signals generated by vehicles, winds, noise, etc.<ref name=Liang>{{cite journal|author1=Zhiqiang Liang |author2=Jianming Wei |author3=Junyu Zhao |author4=Haitao Liu |author5=Baoqing Li |author6=Jie Shen |author7=Chunlei Zheng |title=The Statistical Meaning of Kurtosis and Its New Application to Identification of Persons Based on Seismic Signals |journal=Sensors |date=27 August 2008 |volume=8 |issue=8 |pages=5106–5119 |doi=10.3390/s8085106|pmid=27873804 |pmc=3705491 |bibcode=2008Senso...8.5106L |doi-access=free }}</ref>  Unfortunately, the notation for kurtosis has not been standardized. Kenney and Keeping<ref name="Kenney and Keeping">{{cite book|last=Kenney|first=J. F., and E. S. Keeping|title=Mathematics of Statistics Part Two, 2nd edition|year=1951|publisher=D. Van Nostrand Company Inc.}}</ref>  use the symbol γ<sub>2</sub> for the [[excess kurtosis]], but [[Abramowitz and Stegun]]<ref name=Abramowitz>{{cite book|last=Abramowitz|first=Milton and Irene A. Stegun|title=Handbook Of Mathematical Functions With Formulas, Graphs, And Mathematical Tables|year=1965|publisher=Dover|isbn=978-0-486-61272-0|url=https://archive.org/details/handbookofmathe000abra}}</ref>  use different terminology.  To prevent confusion<ref name=Weisstein.Kurtosi>{{cite web|last=Weisstein.|first=Eric W.|title=Kurtosis|url=http://mathworld.wolfram.com/Kurtosis.html|publisher=MathWorld--A Wolfram Web Resource|access-date=13 August 2012}}</ref>  between kurtosis (the fourth moment centered on the mean, normalized by the square of the variance) and excess kurtosis, when using symbols, they will be spelled out as follows:<ref name="Handbook of Beta Distribution">{{cite book|editor-last=Gupta|editor-first=Arjun K.|title=Handbook of Beta Distribution and Its Applications|year=2004|publisher=CRC Press|isbn=978-0824753962}}</ref><ref name=Panik>{{cite book|last=Panik|first=Michael J|title=Advanced Statistics from an Elementary Point of View|year=2005|publisher=Academic Press|isbn=978-0120884940}}</ref>

:<math>\begin{align}
\text{excess kurtosis}
     &=\text{kurtosis} - 3\\
     &=\frac{\operatorname{E}[(X - \mu)^4]}{{(\operatorname{var}(X))^{2}}}-3\\
     &=\frac{6[\alpha^3-\alpha^2(2\beta - 1) + \beta^2(\beta + 1) - 2\alpha\beta(\beta + 2)]}{\alpha \beta (\alpha + \beta + 2)(\alpha + \beta + 3)}\\
     &=\frac{6[(\alpha - \beta)^2 (\alpha +\beta + 1) - \alpha \beta (\alpha + \beta + 2)]}
{\alpha \beta (\alpha + \beta + 2) (\alpha + \beta + 3)} .
\end{align}</math>

Letting ''α'' = ''β'' in the above expression one obtains

:<math>\text{excess kurtosis} =- \frac{6}{3+2\alpha} \text{ if }\alpha=\beta </math>.

Therefore, for symmetric beta distributions, the excess kurtosis is negative, increasing from a minimum value of −2 at the limit as {''α'' = ''β''} → 0, and approaching a maximum value of zero as {''α'' = ''β''} → ∞.  The value of −2 is the minimum value of excess kurtosis that any distribution (not just beta distributions, but any distribution of any possible kind) can ever achieve.  This minimum value is reached when all the probability density is entirely concentrated at each end ''x'' = 0 and ''x'' = 1, with nothing in between: a 2-point [[Bernoulli distribution]] with equal probability 1/2 at each end (a coin toss: see section below "Kurtosis bounded by the square of the skewness" for further discussion).  The description of [[kurtosis]] as a measure of the "potential outliers" (or "potential rare, extreme values") of the probability distribution, is correct for all distributions including the beta distribution. When rare, extreme values can occur in the beta distribution, the higher its kurtosis; otherwise, the kurtosis is lower. For ''α'' ≠ ''β'', skewed beta distributions, the excess kurtosis can reach unlimited positive values (particularly for ''α'' → 0 for finite ''β'', or for ''β'' → 0 for finite ''α'') because the side away from the mode will produce occasional extreme values.  Minimum kurtosis takes place when the mass density is concentrated equally at each end (and therefore the mean is at the center), and there is no probability mass density in between the ends.

Using the [[Statistical parameter|parametrization]] in terms of mean ''μ'' and sample size ''ν'' = ''α'' + ''β'':

:<math> \begin{align}
  \alpha & {} = \mu \nu ,\text{ where }\nu =(\alpha + \beta)  >0\\
  \beta & {} = (1 - \mu) \nu , \text{ where }\nu =(\alpha + \beta)  >0.
\end{align}</math>

one can express the excess kurtosis in terms of the mean ''μ'' and the sample size ''ν'' as follows:

:<math>\text{excess kurtosis} =\frac{6}{3 + \nu}\bigg (\frac{(1 - 2 \mu)^2 (1 + \nu)}{\mu (1 - \mu) (2 + \nu)} - 1 \bigg )</math>

The excess kurtosis can also be expressed in terms of just the following two parameters: the variance var, and the sample size ''ν'' as follows:

:<math>\text{excess kurtosis} =\frac{6}{(3 + \nu)(2 + \nu)}\left(\frac{1}{\text{ var }} - 6 - 5 \nu \right)\text{ if }\text{var}< \mu(1-\mu)</math>

and, in terms of the variance ''var'' and the mean ''μ'' as follows:

:<math>\text{excess kurtosis} =\frac{6 \text{ var } (1 - \text{ var } - 5 \mu (1 - \mu) )}{(\text{var } + \mu (1 - \mu))(2\text{ var } + \mu (1 - \mu) )}\text{ if }\text{var}< \mu(1-\mu)</math>

The plot of excess kurtosis as a function of the variance and the mean shows that the minimum value of the excess kurtosis (−2, which is the minimum possible value for excess kurtosis for any distribution) is intimately coupled with the maximum value of variance (1/4) and the symmetry condition: the mean occurring at the midpoint (''μ'' = 1/2). This occurs for the symmetric case of ''α'' = ''β'' = 0, with zero skewness.  At the limit, this is the 2 point [[Bernoulli distribution]] with equal probability 1/2 at each [[Dirac delta function]] end ''x'' = 0 and ''x'' = 1 and zero probability everywhere else. (A coin toss: one face of the coin being ''x'' = 0 and the other face being ''x'' = 1.)  Variance is maximum because the distribution is bimodal with nothing in between the two modes (spikes) at each end.  Excess kurtosis is minimum: the probability density "mass" is zero at the mean and it is concentrated at the two peaks at each end.  Excess kurtosis reaches the minimum possible value (for any distribution) when the probability density function has two spikes at each end: it is bi-"peaky" with nothing in between them.

On the other hand, the plot shows that for extreme skewed cases, where the mean is located near one or the other end (''μ'' = 0 or ''μ'' = 1), the variance is close to zero, and the excess kurtosis rapidly approaches infinity when the mean of the distribution approaches either end.

Alternatively, the excess kurtosis can also be expressed in terms of just the following two parameters: the square of the skewness, and the sample size ν as follows:

:<math>\text{excess kurtosis} =\frac{6}{3 + \nu}\bigg(\frac{(2 + \nu)}{4} (\text{skewness})^2 - 1\bigg)\text{ if (skewness)}^2-2< \text{excess kurtosis}< \frac{3}{2} (\text{skewness})^2</math>

From this last expression, one can obtain the same limits published over a century ago by [[Karl Pearson]]<ref name=Pearson /> for the beta distribution (see section below titled "Kurtosis bounded by the square of the skewness"). Setting ''α''&nbsp;+&nbsp;''β''&nbsp;=&nbsp;''ν''&nbsp;=&nbsp;0 in the above expression, one obtains Pearson's lower boundary (values for the skewness and excess kurtosis below the boundary (excess&nbsp;kurtosis +&nbsp;2&nbsp;−&nbsp;skewness<sup>2</sup>&nbsp;=&nbsp;0) cannot occur for any distribution, and hence [[Karl Pearson]] appropriately called the region below this boundary the "impossible region"). The limit of ''α''&nbsp;+&nbsp;''β''&nbsp;=&nbsp;''ν''&nbsp;→&nbsp;∞ determines Pearson's upper boundary.

:<math> \begin{align}
&\lim_{\nu \to  0}\text{excess kurtosis}  = (\text{skewness})^2 - 2\\
&\lim_{\nu \to \infty}\text{excess kurtosis}  = \tfrac{3}{2} (\text{skewness})^2
\end{align}</math>

therefore:

:<math>(\text{skewness})^2-2< \text{excess kurtosis}< \tfrac{3}{2} (\text{skewness})^2</math>

Values of ''ν''&nbsp;=&nbsp;''α''&nbsp;+&nbsp;''β'' such that ''ν'' ranges from zero to infinity, 0&nbsp;<&nbsp;''ν''&nbsp;<&nbsp;∞, span the whole region of the beta distribution in the plane of excess kurtosis versus squared skewness.

For the symmetric case (''α''&nbsp;=&nbsp;''β''), the following limits apply:

:<math> \begin{align}
&\lim_{\alpha = \beta \to 0} \text{excess kurtosis} =  - 2 \\
&\lim_{\alpha = \beta \to \infty} \text{excess kurtosis} = 0 \\
&\lim_{\mu \to \frac{1}{2}} \text{excess kurtosis} = - \frac{6}{3 + \nu}
\end{align}</math>

For the unsymmetric cases (''α''&nbsp;≠&nbsp;''β'') the following limits (with only the noted variable approaching the limit) can be obtained from the above expressions:

:<math> \begin{align}
&\lim_{\alpha\to  0}\text{excess kurtosis}  =\lim_{\beta \to  0} \text{excess kurtosis}  = \lim_{\mu \to  0}\text{excess kurtosis}  = \lim_{\mu \to  1}\text{excess kurtosis}  =\infty\\
&\lim_{\alpha \to  \infty}\text{excess kurtosis}  = \frac{6}{\beta},\text{    }  \lim_{\beta \to  0}(\lim_{\alpha\to  \infty} \text{excess kurtosis})  = \infty,\text{    }  \lim_{\beta \to  \infty}(\lim_{\alpha\to  \infty} \text{excess kurtosis})  = 0\\
&\lim_{\beta \to  \infty}\text{excess kurtosis}  = \frac{6}{\alpha},\text{    }  \lim_{\alpha \to  0}(\lim_{\beta \to  \infty} \text{excess kurtosis})  = \infty,\text{    }  \lim_{\alpha \to  \infty}(\lim_{\beta \to  \infty} \text{excess kurtosis})  = 0\\
&\lim_{\nu \to  0} \text{excess kurtosis}  = - 6 + \frac{1}{\mu (1 - \mu)},\text{    }  \lim_{\mu \to  0}(\lim_{\nu \to  0} \text{excess kurtosis})  = \infty,\text{    }  \lim_{\mu \to  1}(\lim_{\nu \to  0} \text{excess kurtosis})  = \infty
\end{align}</math>

[[File:Excess Kurtosis for Beta Distribution with alpha and beta ranging from 1 to 5 - J. Rodal.jpg|325px]][[File:Excess Kurtosis for Beta Distribution with alpha and beta ranging from 0.1 to 5 - J. Rodal.jpg|325px]]