Are the Feller-Pareto and the generalized beta distributions really the same?

The Feller-Pareto distribution was originally is defined in terms of a transformed beta distribution. If $Y\sim \beta(\gamma_1, \gamma_2)$ then $W=\mu + \sigma\left(\left(1/Y\right) - 1\right)^\gamma=W(\mu, \sigma, \gamma_1, \gamma_2, \gamma)$ is the five-parameter Feller-Pareto. In a number of places it is asserted that this is the same distribution as the one referred to as the generalized beta distribution (GB), with the iconic tree of distributions diagram in McDonald & Xu, "A generalization of the beta distribution with applications," Journal of Econometrics 66 (1995) pp. 133-152, showing the GB as the common parent of the GB1 and the GB2.

This identity is not obvious to me, and it is consequential, as there are a number of distributions claimed as special cases for the Feller-Pareto that McDonald has not claimed for the GB, such as the Pareto type IV.

Are these two five-parameter distributions really the same? And if so, how do the parameters of the transformation-based definition of the Feller-Pareto relate to to the parameters of McDonald & Xu’s top-of-the-distributional-hierarchy definition?

Summary

Comparing parameterized families of distributions is usually a matter of algebraic manipulation of the most convenient form of expression (usually a density (PDF) or the distribution function (CDF) but sometimes a characteristic function or cumulant generating function).

The procedure can be simplified to the point where the comparison can almost always be performed by inspecting the formulas, using these basic techniques:

• Ignore all normalizing constants.

• Recognize common location and scale parameters and set them to values that simplify the formulas (almost always $0$ for the location and $1$ for the scale).

• Compare their supports, if these might vary within either family.

• Isolate algebraic factors common to both expressions and equate corresponding elements (often powers).

These techniques are illustrated through their application to the present question. The conclusions section at the end summarizes the result of this comparison.

The Feller-Pareto Distribution

For analytical purposes, the alternative construction

$$W = \mu + \sigma\left(\frac{X}{Y}\right)^\gamma$$

may be more congenial, where independently

$$X\sim\Gamma(\alpha);\quad Y \sim\Gamma(\beta).$$

Since the support of both $X$ and $Y$ is the set of positive real numbers, it is immediate that the support of $W$ is $(\mu, \infty)$ for $\sigma\gt 0$ and $(-\infty, \mu)$ otherwise. In what follows, all expressions for density functions are understood as applying only for values within the support: the densities otherwise are zero.

Clearly $\mu$ is a location parameter and $\sigma$ is a scale parameter (with negative values of $\sigma$ reversing the distribution). To understand $W$ we may therefore begin with $\mu=0$ and $\sigma=1$. As such, it is recognizable as being a multiple of the $\gamma$ power of an F-Ratio distribution with parameters $(2\alpha,2\beta).$

To understand probability distributions we may ignore constant factors, because in the end we know the distribution will integrate to unity. Consider, then, the density of the F-ratio distribution (without its scaling constants),

$$f(z;2\alpha, 2\beta)\frac{dz}{z} \ \propto\ z^{\alpha}(1+z)^{-(\alpha+\beta)}\frac{dz}{z}.$$

Writing $z=w^{1/\gamma}$ produces

$$\frac{dz}{z} = d\log(z) = \frac{1}{\gamma} d\log(w) = \frac{1}{\gamma} \frac{dw}{w}\ \propto\ \frac{dw}{dw},$$

whence the density of $w$ is found simply by substituting $z=w^{1/\gamma}$ into $f$:

$$f(w^{1/\gamma};\alpha,\beta)\frac{dw}{w} \ \propto\ w^{\alpha/\gamma}(1+w^{1/\gamma})^{-(\alpha+\beta)}\frac{dw}{w}.$$

The Generalized Beta (GB) Distribution

Again ignoring constants, the density of a distribution in this family is

$$g(w; a,b,c,p,q)\frac{dw}{w}\ \propto\ w^{ap}\left(1 - (1-c)(w/b)^a\right)^{q-1}\left(1 + c(w/b)^a\right)^{-(p+q)}\frac{dw}{w}$$

where $0 \lt w^a \lt b^a/(1-c),$ $0\le c\le 1,$ and $b, p, q$ are positive. Obviously $b$ is a (positive) scale parameter and plays the role of $\sigma$ (up to a multiple). Let us therefore consider $b=1.$

Comparing the Distributions

A distribution will occur in both families provided its density has an expression of the form $f(w;2\alpha,2\beta)dw/w$ (for some $\mu$) and $g(w;a,1,c,p,q)dw/w.$

By comparing the supports of these two families, we must have $c=1$ (so that $W$ is unbounded) and $\sigma\gt 0$ (so that $W$ is not negative).

Equate these two forms of density function (neglecting the common differential element $dw/w$) and solve for the parameters:

$$w^{\alpha/\gamma}(1+w^{1/\gamma})^{-(\alpha+\beta)} = w^{ap}\left(1 - (1-c)w^a\right)^{q-1}\left(1 + cw^a\right)^{-(p+q)}.$$

Comparing powers of $w$ gives

$$\alpha/\gamma = ap.$$

Since the left hand side has no power of $(1-\lambda w)$ (for any nonzero constant $\lambda$), neither can the right hand side, whence

$$1-c=0,$$

a conclusion we arrived at previously by considering the support of $W.$

Comparing the remaining terms with $c=1$ shows

$$(1+w^{1/\gamma})^{-(\alpha+\beta)} = \left(1 + w^a\right)^{-(p+q)}$$

implying

$$a = \frac{1}{\gamma};\ \alpha+\beta=p+q.$$

Collecting these results we find

$$\alpha=p;\ \beta = q;\ \gamma=1/a;\ c=1.$$

There is no analog of the location parameter $\mu$ in the GB distribution.

Conclusions

For $\sigma\gt 0,$ members of the Feller-Pareto distribution with parameters $(\mu=0,\sigma=1,\gamma,\alpha,\beta)$ are also members of the GB distribution with parameters

$$(a,b,c,p,q) = (1/\gamma, 1, 1, \alpha,\beta).$$

Moreover, $\sigma$ and $b$ are both scale parameters.

The two five-parameter families are not the same, but they share a four-parameter family of distributions. The Feller-Pareto includes a location parameter $\mu$ not present in the GB family and the GB includes another shape parameter $c$ not present in the Feller-Pareto.

• Would I be correct in concluding from your answer that if I were to add μ to the 5-parameter Generalized Beta, the resulting 6-parameter distribution would encompass all the distributions subsidiary to either the Feller-Pareto or the Generalized Beta? Would I need to adjust the constant as well? I find the treatment of the support for these two distributions confusing (the original treatment, not your treatment). The GB & FP are both defined only over the positive half-plane (shifted for the FP), but both claim that the normal is a special case for particular parameter restrictions. – andrewH Mar 12 '18 at 23:32
• Yes, I believe you're correct. I don't see how any Normal distribution could possibly be expressed in either the GB or FP form, but--as is the case with many families of distributions--the Normal will show up as a suitable limit (in many ways) of the GB or FP. As a more intuitive example, consider extending the Beta family by including scale and location parameters. That enables you to standardize the distributions. In the limit as the Beta parameters are equal and grow large, a suitably standardized Beta distribution becomes Normal. – whuber Mar 12 '18 at 23:38
• When the GB is fit to US income data, it accepts the restriction that c=0, where c is the parameter that flips between the GB1 and the GB2 (or similarly, if you estimate c it is trivially different from zero). That suggests that the five-parameter distribution you get by adding a location parameter to the GB and zeroing out c might be an interesting one for income data. I wonder if that distribution would include the Pareto IV? I’ll play with the math. Thanks! – andrewH Mar 12 '18 at 23:41