Weibull vs. Gamma Distribution

Question

I have data comprising distances between successive points on a line (1D vector):

Traditionally in my field, such data is fitted with a gamma-distribution in an attempt to describe the distribution of the points, however, in some instances I find that a Weibull distribution fits better (higher likelihood as based on BIC/AIC) or in some instances, Weibull is the only distribution that fits with any significance. I'm not overly familiar with a Weibull distribution - what could this be revealing about my data sample? Is there a certain skew toward smaller or larger distances that is better represented with a Weibull as opposed to a Gamma? What are the central differences between a Gamma and Weibull that would be applicable here?

Answer 1

Translating to your context, "time-to-event" becomes "distance-to-point". The Weibull distribution is appropriate when the probability of observing a point increases or decreases as you move along the vector.

The probability declines over time, so most of the points occur at nearer distances:

$X \sim Wei(\lambda,k)$, where $k<1$

The probability increases over time, so most of the points are at farther distances:

$X \sim Wei(\lambda,k)$, where $k>1$

The gamma distribution would then describe the distance until $k$ points are observed - how far do you have to go to observe $k$ points. The distribution lets you determine the probability of going $X$ units before observing $k$ points. This is a Poisson process, where events (points) occur at a constant rate, $\lambda$. In the Weibull distribution, events can occur at decreasing, increasing, or constant rates (or distances).

So, in your case, if the Weibull distribution fits the data better, then you probably have a case where points are “clustering” somewhere. If your scale parameter is less than one, points are closer together at nearer distances, with distances between point increasing as you move along the vector. If it is greater than one, then points are further apart at nearer distances, with distances between points decreasing at you move along the vector.