Backport PR #33513 on branch 1.0.x (BUG: Fix Categorical.min / max bug) (#34022)

Co-authored-by: Daniel Saxton <2658661+dsaxton@users.noreply.github.com>

Author: simonjayhawkins

doc/source/whatsnew/v1.0.4.rst

@@ -20,6 +20,7 @@ Fixed regressions
 - Bug in DataFrame reductions using ``numeric_only=True`` and ExtensionArrays (:issue:`33256`).
 - Fix performance regression in ``memory_usage(deep=True)`` for object dtype (:issue:`33012`)
 - Bug where :meth:`Categorical.replace` would replace with ``NaN`` whenever the new value and replacement value were equal (:issue:`33288`)
+- Bug where an ordered :class:`Categorical` containing only ``NaN`` values would raise rather than returning ``NaN`` when taking the minimum or maximum  (:issue:`33450`)
 - 
 
 .. _whatsnew_104.bug_fixes:

pandas/core/arrays/categorical.py

@@ -2151,7 +2151,7 @@ def min(self, skipna=True):
 
         good = self._codes != -1
         if not good.all():
-            if skipna:
+            if skipna and good.any():
                 pointer = self._codes[good].min()
             else:
                 return np.nan
@@ -2186,7 +2186,7 @@ def max(self, skipna=True):
 
         good = self._codes != -1
         if not good.all():
-            if skipna:
+            if skipna and good.any():
                 pointer = self._codes[good].max()
             else:
                 return np.nan

pandas/tests/arrays/categorical/test_analytics.py

@@ -88,6 +88,14 @@ def test_min_max_with_nan(self, skipna):
             assert _min == 2
             assert _max == 1
 
+    @pytest.mark.parametrize("function", ["min", "max"])
+    @pytest.mark.parametrize("skipna", [True, False])
+    def test_min_max_only_nan(self, function, skipna):
+        # https://github.com/pandas-dev/pandas/issues/33450
+        cat = Categorical([np.nan], categories=[1, 2], ordered=True)
+        result = getattr(cat, function)(skipna=skipna)
+        assert result is np.nan
+
     @pytest.mark.parametrize("method", ["min", "max"])
     def test_deprecate_numeric_only_min_max(self, method):
         # GH 25303