diff --git a/sklearn/utils/estimator_checks.py b/sklearn/utils/estimator_checks.py
index 3633479672cde..1db6031e8d702 100644
--- a/sklearn/utils/estimator_checks.py
+++ b/sklearn/utils/estimator_checks.py
@@ -88,6 +88,7 @@ def _yield_non_meta_checks(name, estimator):
     yield check_dtype_object
     yield check_sample_weights_pandas_series
     yield check_sample_weights_list
+    yield check_sample_weights_invariance
     yield check_estimators_fit_returns_self
     yield partial(check_estimators_fit_returns_self, readonly_memmap=True)
     yield check_complex_data
@@ -554,6 +555,40 @@ def check_sample_weights_list(name, estimator_orig):
         estimator.fit(X, y, sample_weight=sample_weight)
 
 
+@ignore_warnings(category=(DeprecationWarning, FutureWarning))
+def check_sample_weights_invariance(name, estimator_orig):
+    # check that the estimators yield same results for
+    # unit weights and no weights
+    if (has_fit_parameter(estimator_orig, "sample_weight") and
+            not (hasattr(estimator_orig, "_pairwise")
+                 and estimator_orig._pairwise)):
+        # We skip pairwise because the data is not pairwise
+
+        estimator1 = clone(estimator_orig)
+        estimator2 = clone(estimator_orig)
+        set_random_state(estimator1, random_state=0)
+        set_random_state(estimator2, random_state=0)
+
+        X = np.array([[1, 3], [1, 3], [1, 3], [1, 3],
+                      [2, 1], [2, 1], [2, 1], [2, 1],
+                      [3, 3], [3, 3], [3, 3], [3, 3],
+                      [4, 1], [4, 1], [4, 1], [4, 1]], dtype=np.dtype('float'))
+        y = np.array([1, 1, 1, 1, 2, 2, 2, 2,
+                      1, 1, 1, 1, 2, 2, 2, 2], dtype=np.dtype('int'))
+
+        estimator1.fit(X, y=y, sample_weight=np.ones(shape=len(y)))
+        estimator2.fit(X, y=y, sample_weight=None)
+
+        for method in ["predict", "transform"]:
+            if hasattr(estimator_orig, method):
+                X_pred1 = getattr(estimator1, method)(X)
+                X_pred2 = getattr(estimator2, method)(X)
+                assert_allclose(X_pred1, X_pred2, rtol=0.5,
+                                err_msg="For %s sample_weight=None is not"
+                                        " equivalent to sample_weight=ones"
+                                        % name)
+
+
 @ignore_warnings(category=(DeprecationWarning, FutureWarning, UserWarning))
 def check_dtype_object(name, estimator_orig):
     # check that estimators treat dtype object as numeric if possible