Fix inverse_derivative2 for cloglog

CHANGELOG.rst

@@ -7,6 +7,13 @@
 Changelog
 =========
 
+3.2.2 - 2026-03-17
+------------------
+
+**Bug fix:**
+
+- Fixed incorrect formula in :meth:`~glum.CloglogLink.inverse_derivative2`. This affected observed information matrix computation and robust/clustered standard errors for models using the complementary log-log link.
+
 3.2.1 - 2026-03-16
 ------------------
 

src/glum/_link.py

@@ -253,7 +253,7 @@ def inverse_derivative2(self, lin_pred):  # noqa D
 
 
 class CloglogLink(Link):
-    """The complementary log-log link function ``log(-log(-p))``."""
+    """The complementary log-log link function ``log(-log(1-mu))``."""
 
     def __eq__(self, other):  # noqa D
         return isinstance(other, self.__class__)
@@ -279,5 +279,4 @@ def inverse_derivative(self, lin_pred):  # noqa D
         return np.exp(lin_pred - np.exp(lin_pred))
 
     def inverse_derivative2(self, lin_pred):  # noqa D
-        # TODO: check if numerical stability can be improved
-        return np.exp(np.exp(lin_pred) - lin_pred) * np.expm1(lin_pred)
+        return -np.exp(lin_pred - np.exp(lin_pred)) * np.expm1(lin_pred)

tests/glm/test_link.py

@@ -32,6 +32,15 @@ def test_link_properties(link):
 
     assert link.inverse_derivative2(x).shape == link.inverse_derivative(x).shape
 
+    # Check inverse_derivative2 against numerical second derivative
+    h = 1e-7
+    numerical_d2 = (link.inverse_derivative(x + h) - link.inverse_derivative(x - h)) / (
+        2 * h
+    )
+    np.testing.assert_allclose(
+        link.inverse_derivative2(x), numerical_d2, rtol=1e-4, atol=1e-8
+    )
+
 
 def test_equality():
     assert IdentityLink() == IdentityLink()