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diff --git a/‎doc/whats_new/v0.20.rst
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diff --git a/‎sklearn/preprocessing/data.py
Lines changed: 11 additions & 25 deletions b/‎sklearn/preprocessing/data.py
Lines changed: 11 additions & 25 deletions
diff --git a/‎sklearn/preprocessing/tests/test_data.py
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@@ -149,6 +149,10 @@ Changelog
   :issue:`12317` by :user:`Eric Chang <chang>`.
 
 
+- |Fix| Fixed a bug in :class:`preprocessing.PowerTransformer` where the
+  Yeo-Johnson transform was incorrect for lambda parameters outside of `[0, 2]`
+  :issue:`12522` by :user:`Nicolas Hug<NicolasHug>`.
+
 :mod:`sklearn.utils`
 ........................
 
 
@@ -2537,7 +2537,7 @@ class PowerTransformer(BaseEstimator, TransformerMixin):
     >>> print(pt.fit(data))
     PowerTransformer(copy=True, method='yeo-johnson', standardize=True)
     >>> print(pt.lambdas_)
-    [1.38668178e+00 5.93926346e-09]
+    [ 1.38668178 -3.10053309]
     >>> print(pt.transform(data))
     [[-1.31616039 -0.70710678]
      [ 0.20998268 -0.70710678]
@@ -2718,23 +2718,18 @@ def _box_cox_inverse_tranform(self, x, lmbda):
     def _yeo_johnson_inverse_transform(self, x, lmbda):
         """Return inverse-transformed input x following Yeo-Johnson inverse
         transform with parameter lambda.
-
-        Notes
-        -----
-        We're comparing lmbda to 1e-19 instead of strict equality to 0. See
-        scipy/special/_boxcox.pxd for a rationale behind this
         """
-        x_inv = np.zeros(x.shape, dtype=x.dtype)
+        x_inv = np.zeros_like(x)
         pos = x >= 0
 
         # when x >= 0
-        if lmbda < 1e-19:
+        if abs(lmbda) < np.spacing(1.):
             x_inv[pos] = np.exp(x[pos]) - 1
         else:  # lmbda != 0
             x_inv[pos] = np.power(x[pos] * lmbda + 1, 1 / lmbda) - 1
 
         # when x < 0
-        if lmbda < 2 - 1e-19:
+        if abs(lmbda - 2) > np.spacing(1.):
             x_inv[~pos] = 1 - np.power(-(2 - lmbda) * x[~pos] + 1,
                                        1 / (2 - lmbda))
         else:  # lmbda == 2
@@ -2745,27 +2740,22 @@ def _yeo_johnson_inverse_transform(self, x, lmbda):
     def _yeo_johnson_transform(self, x, lmbda):
         """Return transformed input x following Yeo-Johnson transform with
         parameter lambda.
-
-        Notes
-        -----
-        We're comparing lmbda to 1e-19 instead of strict equality to 0. See
-        scipy/special/_boxcox.pxd for a rationale behind this
         """
 
-        out = np.zeros(shape=x.shape, dtype=x.dtype)
+        out = np.zeros_like(x)
         pos = x >= 0  # binary mask
 
         # when x >= 0
-        if lmbda < 1e-19:
-            out[pos] = np.log(x[pos] + 1)
+        if abs(lmbda) < np.spacing(1.):
+            out[pos] = np.log1p(x[pos])
         else:  # lmbda != 0
             out[pos] = (np.power(x[pos] + 1, lmbda) - 1) / lmbda
 
         # when x < 0
-        if lmbda < 2 - 1e-19:
+        if abs(lmbda - 2) > np.spacing(1.):
             out[~pos] = -(np.power(-x[~pos] + 1, 2 - lmbda) - 1) / (2 - lmbda)
         else:  # lmbda == 2
-            out[~pos] = -np.log(-x[~pos] + 1)
+            out[~pos] = -np.log1p(-x[~pos])
 
         return out
 
@@ -2794,12 +2784,8 @@ def _neg_log_likelihood(lmbda):
             x_trans = self._yeo_johnson_transform(x, lmbda)
             n_samples = x.shape[0]
 
-            # Estimated mean and variance of the normal distribution
-            est_mean = x_trans.sum() / n_samples
-            est_var = np.power(x_trans - est_mean, 2).sum() / n_samples
-
-            loglike = -n_samples / 2 * np.log(est_var)
-            loglike += (lmbda - 1) * (np.sign(x) * np.log(np.abs(x) + 1)).sum()
+            loglike = -n_samples / 2 * np.log(x_trans.var())
+            loglike += (lmbda - 1) * (np.sign(x) * np.log1p(np.abs(x))).sum()
 
             return -loglike
 
 
@@ -2345,6 +2345,16 @@ def test_optimization_power_transformer(method, lmbda):
     assert_almost_equal(1, X_inv_trans.std(), decimal=1)
 
 
+def test_yeo_johnson_darwin_example():
+    # test from original paper "A new family of power transformations to
+    # improve normality or symmetry" by Yeo and Johnson.
+    X = [6.1, -8.4, 1.0, 2.0, 0.7, 2.9, 3.5, 5.1, 1.8, 3.6, 7.0, 3.0, 9.3,
+         7.5, -6.0]
+    X = np.array(X).reshape(-1, 1)
+    lmbda = PowerTransformer(method='yeo-johnson').fit(X).lambdas_
+    assert np.allclose(lmbda, 1.305, atol=1e-3)
+
+
 @pytest.mark.parametrize('method', ['box-cox', 'yeo-johnson'])
 def test_power_transformer_nans(method):
     # Make sure lambda estimation is not influenced by NaN values