Editing Beta distribution (section)

====Geometric variance and covariance====
[[File:Beta distribution log geometric variances front view - J. Rodal.png|thumb|log geometric variances vs. ''α'' and ''β'']]
[[File:Beta distribution log geometric variances back view - J. Rodal.png|thumb|log geometric variances vs. ''α'' and ''β'']]

The logarithm of the geometric variance, ln(var<sub>''GX''</sub>), of a distribution with [[random variable]] ''X'' is the second moment of the logarithm of ''X'' centered on the geometric mean of ''X'', ln(''G<sub>X</sub>''):

:<math>\begin{align}
\ln \operatorname{var}_{GX} &= \operatorname{E} \left [(\ln X - \ln G_X)^2 \right ] \\
&= \operatorname{E}[(\ln X - \operatorname{E}\left [\ln X])^2 \right] \\
&= \operatorname{E}\left[(\ln X)^2 \right] - (\operatorname{E}[\ln X])^2\\
&= \operatorname{var}[\ln X]
\end{align}</math>

and therefore, the geometric variance is:

:<math>\operatorname{var}_{GX} = e^{\operatorname{var}[\ln X]}</math>

In the [[Fisher information]] matrix, and the curvature of the log [[likelihood function]], the logarithm of the geometric variance of the [[reflection formula|reflected]] variable 1&nbsp;−&nbsp;''X'' and the logarithm of the geometric covariance between ''X'' and 1&nbsp;−&nbsp;''X'' appear:

:<math>\begin{align}
\ln \operatorname{var_{G(1-X)}} &= \operatorname{E}[(\ln (1-X) - \ln G_{1-X})^2] \\
&= \operatorname{E}[(\ln (1-X) - \operatorname{E}[\ln (1-X)])^2] \\
&= \operatorname{E}[(\ln (1-X))^2] - (\operatorname{E}[\ln (1-X)])^2\\
&= \operatorname{var}[\ln (1-X)] \\
& \\
\operatorname{var_{G(1-X)}} &= e^{\operatorname{var}[\ln (1-X)]} \\
& \\
\ln \operatorname{cov_{G{X,1-X}}} &= \operatorname{E}[(\ln X - \ln G_X)(\ln (1-X) - \ln G_{1-X})] \\
&= \operatorname{E}[(\ln X - \operatorname{E}[\ln X])(\ln (1-X) - \operatorname{E}[\ln (1-X)])] \\
&= \operatorname{E}\left[\ln X \ln(1-X)\right] - \operatorname{E}[\ln X]\operatorname{E}[\ln(1-X)]\\
&= \operatorname{cov}[\ln X, \ln(1-X)] \\
& \\
\operatorname{cov}_{G{X,(1-X)}} &= e^{\operatorname{cov}[\ln X, \ln(1-X)]}
\end{align}</math>

For a beta distribution, higher order logarithmic moments can be derived by using the representation of a beta distribution as a proportion of two gamma distributions and differentiating through the integral. They can be expressed in terms of higher order poly-gamma functions. See the section {{section link||Moments of logarithmically transformed random variables}}.  The [[variance]] of the logarithmic variables and [[covariance]] of ln&nbsp;''X'' and ln(1−''X'') are:

: <math>\operatorname{var}[\ln X]= \psi_1(\alpha) - \psi_1(\alpha + \beta)</math>
: <math>\operatorname{var}[\ln (1-X)] = \psi_1(\beta) - \psi_1(\alpha + \beta)</math>
: <math>\operatorname{cov}[\ln X, \ln(1-X)] = -\psi_1(\alpha+\beta)</math>

where the '''[[trigamma function]]''', denoted ''ψ''<sub>1</sub>(''α''), is the second of the [[polygamma function]]s, and is defined as the derivative of the [[digamma function]]:

:<math>\psi_1(\alpha) = \frac{d^2\ln\Gamma(\alpha)}{d\alpha^2}= \frac{d \, \psi(\alpha)}{d\alpha}.</math>

Therefore,

:<math> \ln \operatorname{var}_{GX}=\operatorname{var}[\ln X]= \psi_1(\alpha) - \psi_1(\alpha + \beta) </math>
:<math> \ln \operatorname{var}_{G(1-X)} =\operatorname{var}[\ln (1-X)] = \psi_1(\beta) - \psi_1(\alpha + \beta)</math>
:<math> \ln \operatorname{cov}_{GX,1-X} =\operatorname{cov}[\ln X, \ln(1-X)] = -\psi_1(\alpha+\beta)</math>

The accompanying plots show the log geometric variances and log geometric covariance versus the shape parameters ''α'' and ''β''.  The plots show that the log geometric variances and log geometric covariance are close to zero for shape parameters ''α'' and ''β'' greater than 2, and that the log geometric variances rapidly rise in value for shape parameter values ''α'' and ''β'' less than unity. The log geometric variances are positive for all values of the shape parameters. The log geometric covariance is negative for all values of the shape parameters, and it reaches large negative values for ''α'' and ''β'' less than unity.

Following are the limits with one parameter finite (non-zero) and the other approaching these limits:

:<math> \begin{align}
&\lim_{\alpha\to  0} \ln \operatorname{var}_{GX} =  \lim_{\beta\to  0} \ln \operatorname{var}_{G(1-X)}  =\infty \\
&\lim_{\beta \to  0} \ln \operatorname{var}_{GX} = \lim_{\alpha \to  \infty} \ln \operatorname{var}_{GX} = \lim_{\alpha \to  0} \ln \operatorname{var}_{G(1-X)} = \lim_{\beta\to  \infty} \ln \operatorname{var}_{G(1-X)} = \lim_{\alpha\to  \infty} \ln \operatorname{cov}_{GX,(1-X)} =  \lim_{\beta\to  \infty} \ln \operatorname{cov}_{GX,(1-X)} = 0\\
&\lim_{\beta \to  \infty} \ln \operatorname{var}_{GX} =  \psi_1(\alpha)\\
&\lim_{\alpha\to  \infty}  \ln \operatorname{var}_{G(1-X)} =  \psi_1(\beta)\\
&\lim_{\alpha\to  0} \ln \operatorname{cov}_{GX,(1-X)} = - \psi_1(\beta)\\
&\lim_{\beta\to  0}  \ln \operatorname{cov}_{GX,(1-X)} = - \psi_1(\alpha)
\end{align}</math>

Limits with two parameters varying:

:<math> \begin{align}
&\lim_{\alpha\to  \infty}( \lim_{\beta \to \infty} \ln \operatorname{var}_{GX}) = \lim_{\beta \to  \infty}( \lim_{\alpha\to  \infty} \ln \operatorname{var}_{G(1-X)}) = \lim_{\alpha\to  \infty} (\lim_{\beta \to  0} \ln \operatorname{cov}_{GX,(1-X)}) = \lim_{\beta\to  \infty}( \lim_{\alpha\to  0} \ln \operatorname{cov}_{GX,(1-X)}) =0\\
&\lim_{\alpha\to  \infty} (\lim_{\beta \to  0} \ln \operatorname{var}_{GX}) = \lim_{\beta\to \infty} (\lim_{\alpha\to  0} \ln \operatorname{var}_{G(1-X)}) = \infty\\
&\lim_{\alpha\to  0} (\lim_{\beta \to  0} \ln \operatorname{cov}_{GX,(1-X)}) = \lim_{\beta\to 0} (\lim_{\alpha\to  0} \ln \operatorname{cov}_{GX,(1-X)}) = - \infty
\end{align}</math>

Although both ln(var<sub>''GX''</sub>) and ln(var<sub>''G''(1&nbsp;−&nbsp;''X'')</sub>) are asymmetric, when the shape parameters are equal, α = β, one has: ln(var<sub>''GX''</sub>) = ln(var<sub>''G(1−X)''</sub>). This equality follows from the following symmetry displayed between both log geometric variances:

:<math>\ln \operatorname{var}_{GX}(\Beta(\alpha, \beta))=\ln \operatorname{var}_{G(1-X)}(\Beta(\beta, \alpha)).</math>

The log geometric covariance is symmetric:

:<math>\ln \operatorname{cov}_{GX,(1-X)}(\Beta(\alpha, \beta) )=\ln \operatorname{cov}_{GX,(1-X)}(\Beta(\beta, \alpha))</math>