Is there a closed-form solution for the tail index of a GB2 distribution? In the Generalized Beta distribution of the second kind (GB2), where a, p, and q are shape parameters and b is a scale parameter, the pdf is defined on $\mathbb{R}_+$ by:
$$
GB2(y;a,b,p,q) = \frac{|a|y^{ap-1}}{b^{ap}B(p,q)(1+(y/b)^a)^{p+q}}
$$
In the Pareto distribution, the tail index is just the single shape parameter (usually denoted by α).
In the more complicated GB2, if the four parameters are known, is there a closed-form solution for the tail index? Do all four parameters occur in it?
 A: The "tail index" of a distribution function $F$ describes the rate of decrease of the survival function $1-F(y)$.  (Thus, since the $b$ is merely a scale parameter, it cannot possibly influence the tail index.)
The question gives the derivative $f(y)=F^\prime(y)$.  We can determine the asymptotic rate of decrease of $f$ by inspection, since for very large $y$, $1$ is much smaller than $y/b$ and the rate doesn't depend on any multiplicative constants.  Thus
$$f(y) = \frac{|a|y^{ap-1}}{b^{ap}B(p,q)(1+(y/b)^a)^{p+q}} \approx C \frac{y^{ap-1}}{(y^a)^{(p+q)}} = Cy^{-aq-1}$$
where $C$ collects all the multiplicative constants.  In general, whenever a function proportional to a power $k$ of $y$ is integrated, the result is proportional to $y^{k+1}$ (provided $k\ne -1$, but that cannot be the case for otherwise $F$ would diverge instead of approaching $1$ as a limit).  Therefore $1-F$ must behave asymptotically like a multiple of $y^{-aq}$.  Such functions are said to have a "tail index" of $1/(aq)$--that is, the negative reciprocal of the power.

The foregoing can be made rigorous by expanding $f$ about $\infty$ to first order. The expansion is provided by the Binomial Theorem or Taylor's Theorem (you express $f$ in terms of $x=1/y$, expand around $x=0$, and then rewrite the result in terms of $y$) and is absolutely convergent for $|y| \gt b$, justifying integrating the expansion term-by-term when going from $f$ to $1-F$.  You need to estimate the error in this expansion and show it is a "slowly varying function," but doing so is straightforward, requiring no new ideas or techniques.
A: This is just an extended comment (and definitely not an answer) adding some details and checks to @whuber 's answer.  (Mathematica is used when some code is needed.)
From Qi (2010) the tail index of the distribution function $F$ is $1/\gamma$ defined by
$$1-F(y)=y^{-1/\gamma} L(y)$$
for $y>0$ with the function $L$ satisfying
$$\lim_{t\rightarrow \infty} {{L(t y)}\over{L(t)}} = 1$$
We start with the density function for the generalized beta distribution of the second kind:
f[y_] := (Abs[a]/(b^(a p) Beta[p, q])) y^(a p - 1)/(1 + (y/b)^a)^(p + q)

As noted by @whuber the term $\left(\left(\frac{y}{b}\right)^a+1\right)^{-p-q}$ can be replaced by $\left(\left(\frac{y}{b}\right)^a\right)^{-p-q}$ when $y$ is large:
f4LargeY[y_] := FullSimplify[f[y] //. (1 + (y/b)^a)^(-p - q) -> y^(-a (p + q)) b^(a (p + q))]

which simplifies to
$$\frac{\left| a\right|  b^{a q} y^{-a q-1}}{B(p,q)}$$
We should check on that assumption by taking the limit of the ratio of the two functions to see if that ratio approaches
1 as $y\rightarrow \infty$:
Limit[f[y]/f4LargeY[y], y -> \[Infinity], Assumptions -> {a > 0, b > 0, p > 0, q > 0}]
(* 1 *)

and the limit is 1.
We see that $1-F(y)$ is approximately $\int_y^{\infty } \text{f4LargeY}(t) \, dt$ for large enough values of $y$:
OneMinusF = Integrate[f4LargeY[t], {t, y, \[Infinity]}, 
   Assumptions -> {a > 0, b > 0, p > 0, q > 0, y > 0}] /. (b/y)^(a q) -> y^(-a q) b^(a q)

We have $1-F(y) \approx \frac{b^{a q} y^{-a q}}{q B(p,q)}$ for large $y$.  If we let $L(y)=\frac{b^{a q}}{q B(p,q)}$, then this
function satisfies the requirement in Qi (2010) as $L(y)$ is just a constant.
L[y_] := b^(a q)/(q Beta[p, q])
Limit[L[t y]/L[y], t -> \[Infinity]]
(* 1 *)

So (and, yes, this is a bit of overkill in the use of Solve) we can solve for the tail index
FullSimplify[Solve[OneMinusF == y^(-tailIndex) L[y], tailIndex], 
  Assumptions -> {y > 0, a > 0, b > 0, p > 0, q > 0}][[1, 1]]
(* tailIndex -> a q *)

