GAMMALN

Application:

Calculating the Log-Likelihood of a Gamma Distribution

A common application is calculating the log-likelihood of a set of data points assuming they follow a Gamma distribution. The probability density function (PDF) for a Gamma distribution is:

$f(x;\alpha ,\beta )=\frac{{\beta }^{\alpha }{x}^{\alpha -1}{e}^{-\beta x}}{\Gamma (\alpha )}$

where α is the shape parameter, β is the rate parameter, and $\Gamma$(α) is the Gamma function.

The log-likelihood for a single data point xi​ is ln(f(xi​)), which is:

$\ln (L(\alpha ,\beta ;x_i))=\ln ({\beta }^{\alpha })+\ln (x_i^{\alpha -1})+\ln (e^{-\beta x_i})-\ln (\Gamma (\alpha ))$

$\ln (L)=\alpha \ln (\beta )+(\alpha -1)\ln (x_i)-\beta x_i-\ln (\Gamma (\alpha ))$

The last term, −ln($\Gamma$(α)), is precisely where the GAMMALN function is used to avoid calculating the massive value of $\Gamma$(α).

Let's imagine we are a scientist studying the waiting times between major earthquakes in a particular region. Based on historical data, we hypothesize that these waiting times follow a Gamma distribution. We have a set of data points and want to calculate the log-likelihood of these parameters (α=5.5 and β=0.2) to see how well they fit the data.

Here is a table showing the calculation for three historical waiting times, demonstrating the use of the GAMMALN function.

Data and Parameters:

• Shape parameter (α): 5.5
• Rate parameter (β): 0.2
• Log-Gamma of α: GAMMALN(5.5) = 3.9578

Table of Calculations: