For each element of x, compute the probability density function (PDF) of the generalized Pareto distribution with shape parameter k, scale parameter sigma, and location parameter theta. The size of y is the common size of p, k, sigma, and theta. A scalar input functions as a constant matrix of the same size as the other inputs.

When k = 0 and theta = 0, the Generalized Pareto is equivalent to the exponential distribution. When k > 0 and theta = k / k the Generalized Pareto is equivalent to the Pareto distribution. The mean of the Generalized Pareto is not finite when k >= 1 and the variance is not finite when k >= 1/2. When k >= 0, the Generalized Pareto has positive density for x > theta, or, when theta < 0, for 0 <= (x - theta) / sigma <= -1 / k.

Further information about the generalized Pareto distribution can be found at https://en.wikipedia.org/wiki/Generalized_Pareto_distribution

Input arguments must be double or single; integer, logical, and character arrays are rejected. MATLAB accepts a character array and evaluates it at the character codes, which Octave deliberately does not, since a character array is an integer type and integers are refused too.

With a negative shape parameter the support is the closed interval [theta, theta - sigma/k], and the density at its upper endpoint follows the limit of the density there: 0 for -1 < k < 0, 1/sigma at k = -1, and unbounded for k < -1. MATLAB returns 0 at that endpoint whatever the shape, which contradicts its own unifpdf: the generalized Pareto with k = -1 is the uniform distribution on [theta, theta + sigma], for which MATLAB’s unifpdf returns 1/sigma at the same point. This implementation returns the limit, and so agrees with unifpdf.

See also: gpcdf, gpinv, gprnd, gpfit, gplike, gpstat

Source Code: gppdf