I am working on a project where I need to chart statistical data and related, skewed distributions a la http://en.wikipedia.org/wiki/Skew_normal_distribution.
Unlike with normal distributions, in these charts, when there is skew, neither the mean nor the median appear to be at the maximal point of the probability distribution curve drawn along the graph's domain, which is fine, but does this point have a name? And more importantly, is there a more direct way to discover the value along the domain of values such that the PDF(value) has the maximal value?
I can currently find this point with a
binary search hill-climbing algorithim, but, this seems a rather round-about way to go about it.
While this point may or may not have any intrinsic value in statistics, from the standpoint of scaling the resulting graphs, I'd like to calculate the highest point of the distribution.
I am not a statistician, so it could very well be that the answer is on the very Wikipedia page I posted above, but I am not familiar with the parlance of this level of Statistics.
The consensus is that there is no closed form of computing the mode from an arbitrary Skew Normal Distribution, so, a Hill Climbing algorithm is probably a sensible approach. A good starting point for the algorithm, seems to be the distribution's median, which does have a closed form, and except in cases of extreme skew, is fairly close to the mode. At extreme skews, the location is probably the best starting point. I have created a nifty animation in R to visualise this:
PS: R is fun! =)