The General Segmented Distribution

Abstract

We develop a distribution supported on a bounded interval with a probability density function that is constructed from any finite number of linear segments. With an increasing number of segments, the distribution can approach any continuous density function of arbitrary form. The flexibility of the distribution makes it a useful tool for various modeling purposes. We further demonstrate that it is capable of fitting data with considerable precision—outperforming distributions recommended by previous studies. We suggest that this distribution is particularly effective in fitting data with sufficient observations that are skewed and multimodal.

Keywords:

• Model fitting
• Probability distribution
• Trapezoidal distribution, Triangular distribution.

Mathematics Subject Classification:

• Primary 60E05
• Secondary 62E99

Notes

We provide, upon request, Mathematica code that, for any n, offers a graphic presentation of the PDF for the GSD, as well as solves for the h_i values (corresponding to any set of r_i values), the first four raw moments, and the corresponding central moments. Furthermore, the code provides for verification of the moments using numerical integration.

These data were downloaded from Simon Hix’s web site at http://personal.lse.ac.uk/hix/HixNouryRolandEPdata.HTM.

Since we use the natural logarithm of the measurement scale, we change the lower limit of the first class from zero to one, thus avoiding an infinite boundary value.

The authors use the Solver optimization tool in Excel. All analysis is available upon request.

We used the test statistics appearing in Hürlimann, however, we were unable to replicate Hürlimann’s calculations for the K-statistic, and therefore we used the calculation of K such that , where ζ_i is the upper boundary of class i for M groups.

Table 3 Comparing fit of distributions to industrial fire loss data

We note that a kernel density function (Epanechnikov, bandwidth = 0.02) was applied to the data to obtain the starting values for the algorithm.

Table 4 Starting values and parameter estimates for European parliament nominate data

Note that h₁ and h_{n + 1} are constrained to equal zero, and therefore do not constitute estimable parameters. In addition, the constraint of unit probability reduces the number of parameters by one.