Merge pull request #96 from jorenham/pareto4

Pareto type IV general trimmed L-moments

docs/distributions.md

@@ -321,6 +321,8 @@ exist (in closed form).
         </a>
         <br>
         <code>pareto</code>
+        <br>
+        see eq. \( \eqref{eq:lr_pareto4} \) for \( \lmoment{r} \)
     </td>
     <td>
         \[
@@ -696,8 +698,6 @@ symmetric trimming of order 1, i.e. `trim=(1, 1)`.
         \( \approx 0.07541 \)
     </td>
 </tr>
-<!-- TODO: Pareto I -->
-<!-- TODO: Pareto III -->
 </table>
 
 
@@ -760,6 +760,43 @@ The trimmed L-moments of the GEV are
 \]
 
 
+### Pareto Type IV
+
+The [Pareto Type IV](https://wikipedia.org/wiki/Pareto_distribution) has two 
+shape parameters \( \alpha \in \mathbb{R} \) and 
+\( \gamma \in \mathbb{R}_{>0} \), and scale parameter \( \beta \).
+For \( x \ge 0 \), the CDF and its inverse (the PPF) are
+
+\[
+\begin{align*}
+    F(x) 
+        &= 1 - \left(
+            1 + \left(\frac x \beta\right)^{\frac 1 \gamma}
+        \right)^{-\alpha} \\
+    x(F) 
+        &= \beta \left(
+            (1 - F)^{-1 / \alpha} - 1
+        \right)^\gamma
+\end{align*}
+\]
+
+When \( \alpha > \gamma \), the trimmed L-moments are found to be:
+
+\[
+    \begin{equation}
+        \tlmoment{s,t}{r} 
+            = \frac{\beta \gamma}{r}
+            \sum_{k = t + 1}^{r + s + t}
+                (-1)^{k - t - 1}
+                \binom{r + k - 2}{r + t - 1}
+                \binom{r + s + t}{k}
+                \B(\gamma,\ k \alpha - \gamma)
+            \label{eq:lr_pareto4}
+    \end{equation}
+\]
+
+This distribution is currently not implemented in [`scipy.stats`][scipy.stats]
+
 
 ### Kumaraswamy