Moment-Ratio Diagrams for Univariate Distributions

Abstract

We present two moment-ratio diagrams along with guidance for their interpretation. The first moment-ratio diagram is a graph of skewness vs. kurtosis for common univariate probability distributions. The second moment-ratio diagram is a graph of coefficient of variation vs. skewness for common univariate probability distributions. Both of these diagrams, to our knowledge, are the most comprehensive to date. The diagrams serve four purposes: (1) they quantify the proximity between various univariate distributions based on their second, third, and fourth moments, (2) they illustrate the versatility of a particular distribution based on the range of values that the various moments can assume, (3) they can be used to create a short list of potential probability models based on a data set, and (4) they clarify the limiting relationships between various well-known distribution families. The use of the moment-ratio diagrams for choosing a distribution that models given data is illustrated.

Originally published in the Journal of Quality Technology, Volume 42, Number 3, in 2010, this summary of moment-ratio diagrams is yet another example of how APPL worked behind the scenes to create diagrams that can be of use in probability modeling. In this case, the primary figures were built in a two-step process. First, every distribution’s moments were calculated in APPL. In this step, the algorithm (in the appendix) was used to create the points on each moment curve for various parameter combinations of each distribution. This set of points was then imported into the statistical package R, which turned the points into curves in the figures in the paper.

Appendix

In this section, we provide exact expressions for the CV, skewness, and kurtosis, for the four distributions that occupy (two-dimensional) regions in Figures 12.2 and 12.3.

Beta

The beta family (see Johnson et al. [72, Chap. 25, page 210]) has two shape parameters p, q > 0 with

$$\displaystyle\begin{array}{rcl} \gamma _{2}& =& \frac{\sqrt{q}} {\sqrt{p^{2 } + pq + p}},\quad p,q> 0; {}\\ \gamma _{3}& =& \frac{2(q - p)\sqrt{1/p + 1/q + 1/pq}} {p + q + 2},\quad p,q> 0; {}\\ \gamma _{4}& =& 3(p + q + 1)\frac{2(p + q)^{2} + pq(p + q - 6)} {pq(p + q + 2)(p + q + 3)},\quad p,q> 0. {}\\ \end{array}$$

The regime in the (γ ₂, γ ₃) plane is bounded above by the line γ ₃ = 2γ ₂ corresponding to the gamma family, and below by the curve γ ₃ = γ ₂ − 1∕γ ₂. The regime in the (γ ₃, γ ₄) plane is bounded below by the limiting curve γ ₄ = 1 +γ ₃ ² for all distributions, and above by the curve $\gamma _{4} = 3 + \frac{3} {2}\gamma _{3}^{2}$ corresponding to the gamma family.

Inverted Beta

The beta-prime or the Pearson Type VI family (see Johnson et al. [72, Chap. 25, page 248]), also known as the inverted beta family, has two shape parameters α, β > 0 with

$$\displaystyle\begin{array}{rcl} \gamma _{2}& =& \sqrt{\frac{\alpha +\beta - 1} {\alpha (\beta -2)}},\quad \beta> 2; {}\\ \gamma _{3}& =& \sqrt{ \frac{4(\beta -2)} {(\alpha +\beta - 1)\alpha }} \cdot \frac{2\alpha +\beta -1} {\beta -3},\quad \beta> 3; {}\\ \gamma _{4}& =& \frac{3(\alpha -2 + \frac{1} {2}(\beta -3)\gamma _{2}^{2})} {\beta -4},\quad \beta> 4. {}\\ \end{array}$$

The regime in the (γ ₂, γ ₃) plane is bounded above by the curve γ ₃ = 4γ ₂∕(1 −γ ₂ ²), γ ₂ ∈ (0, 1), and below by the curve γ ₃ = 2γ ₂ corresponding to the gamma family. The regime in the (γ ₃, γ ₄) plane is bounded above by the curve

$$\displaystyle{\gamma _{3} = \frac{4\sqrt{\alpha -2}} {\alpha -3},\quad \gamma _{4} = 3 + \frac{30\alpha - 66} {(\alpha -3)(\alpha -4)},\quad \alpha> 4}$$

corresponding to the inverse gamma family, and below by the curve γ ₄ = 3 + 3γ ₃ ²∕2 corresponding to the gamma family.

Generalized Gamma

The generalized gamma family (see Johnson et al. [71, page 388]) has two shape parameters α, λ > 0 with the rth raw moment $\mu _{r}' = \Gamma (\alpha +r\lambda )/\Gamma (\alpha )$. The regime in the (γ ₂, γ ₃) plane is bounded below by the curve

$$\displaystyle{\gamma _{2} = \frac{1} {\sqrt{p(p + 2)}},\quad \gamma _{3} = \frac{1 - p} {p + 3} \cdot \frac{2} {\sqrt{1 + 2/p}},\quad p> 0}$$

corresponding to the power family, and above by the curve

$$\displaystyle{\gamma _{2} = \frac{1} {\sqrt{p(p - 2)}},\quad \gamma _{3} = \frac{1 + p} {p - 3} \cdot \frac{2} {\sqrt{1 - 2/p}},\quad p> 3}$$

corresponding to the Pareto family. The regime in the (γ ₃, γ ₄) plane is bounded above by the curve

$$\displaystyle{\gamma _{3} = \frac{1 + p} {p - 3} \frac{2} {\sqrt{1 - 2/p}},\quad \gamma _{4} = \frac{3(1 + 2/p)(3p^{2} - p + 2)} {(p + 3)(p + 4)},\quad p> 0}$$

corresponding to the power family, bounded below to the right by the curve corresponding to the generalized gamma family with λ = −0. 54, and bounded below to the left by the curve corresponding to the log gamma family. [Recall that the log gamma family with shape parameter α > 0 has the r ^th cumulant $\kappa _{r} = \Psi ^{(r)}(\alpha )$, where $\Psi ^{(r)}(z)$ is the (r + 1)^th derivative of $\ln \Gamma (z)$.]

Burr Type XII

The Burr Type XII family (see Rodriques [138]) has two shape parameters c, k > 0 with the r ^th raw moment $\mu _{r}' = \Gamma (r/c + 1)\Gamma (k - r/c)/\Gamma (k)$, c > 0, k > 0, r < ck. The regime in the (γ ₂, γ ₃) plane is bounded below by the curve corresponding to the Weibull family (r ^th raw moment $\mu _{r}' = \Gamma (r/c + 1),$ where c > 0 is the Weibull shape parameter), and above by the curve

$$\displaystyle{\gamma _{2} = \frac{1} {\sqrt{p(p - 2)}},\quad \gamma _{3} = \frac{1 + p} {p - 3} \cdot \frac{2} {\sqrt{1 - 2/p}},\quad p> 3}$$

corresponding to the Pareto family. The regime in the (γ ₃, γ ₄) plane is bounded below by the curve corresponding to the Weibull family, bounded above to the right by the curve corresponding to the Burr Type XII family with k = 1, and bounded above to the left by the curve corresponding to the Burr Type XII family with c = ∞.

APPL Code for the Diagrams

The moment-ratio diagrams in this paper were created in two steps. For each distribution, an algorithm is used to create the sets of points that will produce the curves. These sets of points are then imported into R to produce the graphs. The APPL code to generate the points for the χ ² distribution moment curves follow. Other distribution curves are produced similarly. Note the variable X establishes the χ _n ² distribution. The APPL commands in the fprintf statements find the desired moments of X for various values of n.

> file := fopen("ChiSquare.d", WRITE);

> n := 1;

> i := 1;

> while n < 30 do

>   X := ChiSquareRV(n);

>   fprintf(file, "%g %g %g %g\n", evalf(n), evalf(CoefOfVar(X)),

>           evalf(Skewness(X)),evalf(Kurtosis(X))):

>   i := i + 1:

>   n := i ^ 2:

> end do;

> for n to 100 do

>   X := ChiSquareRV(n);

>   if ‘mod‘(n, 2) <> 0 then

>     fprintf(file, "%g %g %g %g\n", evalf(n), evalf(CoefOfVar(X)),

>             evalf(Skewness(X)), evalf(Kurtosis(X))):

>   end if

> end do:

> fprintf(file, "%g %g %g %g\n", evalf(999999), evalf(0), evalf(0),

>         evalf(3)):

> fclose(file):