ENH: stats.vonmises.fit: treat case in which location likelihood equa…

…tion has no solution(#18190)

* ENH: stats.vonmises.fit: treat case in which location likelihood equation has no solution

scipy/stats/_continuous_distns.py

@@ -10029,12 +10029,24 @@ def find_kappa(data, loc):
             r = np.sum(np.cos(loc - data))/len(data)
             # See gh-18128 for more information.
 
-            def solve_for_kappa(kappa):
-                return sc.i1e(kappa)/sc.i0e(kappa) - r
+            if r > 0:
+                def solve_for_kappa(kappa):
+                    return sc.i1e(kappa)/sc.i0e(kappa) - r
 
-            root_res = root_scalar(solve_for_kappa, method="brentq",
-                                   bracket=(1e-8, 1e12))
-            return root_res.root
+                root_res = root_scalar(solve_for_kappa, method="brentq",
+                                    bracket=(1e-8, 1e12))
+                return root_res.root
+            else:
+                # if the provided floc is very far from the circular mean,
+                # the mean resultant length r can become negative.
+                # In that case, the equation
+                # I[1](kappa)/I[0](kappa) = r does not have a solution.
+                # The maximum likelihood kappa is then 0 which practically
+                # results in the uniform distribution on the circle. As
+                # vonmises is defined for kappa > 0, return instead the
+                # smallest floating point value.
+                # See gh-18190 for more information
+                return np.finfo(float).tiny
 
         # location likelihood equation has a solution independent of kappa
         loc = floc if floc is not None else find_mu(data)

scipy/stats/tests/test_distributions.py

@@ -244,6 +244,19 @@ def test_vonmises_fit_shape(self):
         assert scale_fit == 1
         assert_allclose(kappa, kappa_fit, rtol=1e-2)
 
+    @pytest.mark.xslow
+    @pytest.mark.parametrize('loc', [-0.5 * np.pi, -0.9 * np.pi])
+    def test_vonmises_fit_bad_floc(self, loc):
+        data = [-0.92923506, -0.32498224, 0.13054989, -0.97252014, 2.79658071,
+                -0.89110948, 1.22520295, 1.44398065, 2.49163859, 1.50315096,
+                3.05437696, -2.73126329, -3.06272048, 1.64647173, 1.94509247,
+                -1.14328023, 0.8499056, 2.36714682, -1.6823179, -0.88359996]
+        data = np.asarray(data)
+        kappa_fit, loc_fit, scale_fit = stats.vonmises.fit(data, floc=loc)
+        assert kappa_fit == np.finfo(float).tiny
+        _assert_less_or_close_loglike(stats.vonmises, data,
+                                      stats.vonmises.nnlf, fscale=1, floc=loc)
+
     @pytest.mark.parametrize('sign', [-1, 1])
     def test_vonmises_fit_unwrapped_data(self, sign):
         rng = np.random.default_rng(6762668991392531563)