exGauss

The exGauss, or the ex-Gaussian distribution, is a three-parameter probability distribution used to model positively skewed data. It arises as the convolution of a normal distribution and an exponential distribution, providing a Gaussian-like central region with an exponential tail.

Parametrization and interpretation: Let X follow a normal distribution with mean mu and standard deviation sigma, and Y follow an exponential distribution with rate lambda, where lambda = 1/tau. If Z = X + Y, then Z has the ex-Gaussian distribution with parameters mu, sigma, and tau. The mean of Z is mu + tau, and the variance is sigma^2 + tau^2. The distribution is right-skewed, with the tau parameter governing the tail heaviness.

Probability density and properties: The exGauss density combines the normal density with an exponential component and can be written in terms of the standard normal cumulative distribution function. The form reflects a Gaussian core perturbed by an exponential tail, making it well suited for data that cluster around a central value but exhibit a longer tail toward larger values.

Applications: It is widely used in modeling reaction time data in cognitive psychology, as well as other positively valued, skewed data in psychology, neuroscience, and reliability analysis. The ex-Gaussian provides a flexible framework for capturing both the central tendency and the skewness observed in such datasets.

Estimation and software: Parameter estimation is typically done by maximum likelihood, often requiring numerical optimization due to the lack of a closed-form CDF. Method-of-moments approaches are also used. Software implementations exist in various statistical environments, including R packages such as retimes and exgauss, and in Python or MATLAB with custom routines.

History: The ex-Gaussian distribution emerged in statistical modeling of reaction times and related skewed data, combining a Gaussian core with an exponential tail to reflect empirical data characteristics. See also Gaussian and Exponential distributions.