diff --git a/doc/changes/dev/13698.other.rst b/doc/changes/dev/13698.other.rst
new file mode 100644
index 00000000000..fab48e560d1
--- /dev/null
+++ b/doc/changes/dev/13698.other.rst
@@ -0,0 +1 @@
+Add optional low-variance ("hat") regularization to :func:`mne.stats.f_oneway` via new ``sigma`` and ``method`` parameters, by `Aniket Singh Yadav`_.
\ No newline at end of file
diff --git a/mne/stats/parametric.py b/mne/stats/parametric.py
index 2cc0bff2ea1..7cdbcb367ea 100644
--- a/mne/stats/parametric.py
+++ b/mne/stats/parametric.py
@@ -111,7 +111,7 @@ def ttest_ind_no_p(a, b, equal_var=True, sigma=0.0):
     return t
 
 
-def f_oneway(*args):
+def f_oneway(*args, sigma=0.0, method="relative"):
     """Perform a 1-way ANOVA.
 
     The one-way ANOVA tests the null hypothesis that 2 or more groups have
@@ -125,6 +125,16 @@ def f_oneway(*args):
     ----------
     *args : array_like
         The sample measurements should be given as arguments.
+    sigma : float
+        Regularization parameter (>= 0) added to the within-group mean
+        square to mitigate F-statistic inflation under low-variance
+        conditions. ``0`` (default) disables correction.
+    method : str
+        How *sigma* is interpreted when ``sigma > 0``. Can be
+        ``'relative'`` (default) or ``'absolute'``.
+        ``'relative'`` multiplies *sigma* by the median within-group
+        mean square (scale-invariant, recommended).
+        ``'absolute'`` uses *sigma* directly as a raw sigma squared.
 
     Returns
     -------
@@ -151,6 +161,9 @@ def f_oneway(*args):
     ----------
     .. footbibliography::
     """
+    _check_option("method", method, ["absolute", "relative"])
+    if sigma < 0:
+        raise ValueError(f"sigma must be >= 0, got {sigma}")
     n_classes = len(args)
     n_samples_per_class = np.array([len(a) for a in args])
     n_samples = np.sum(n_samples_per_class)
@@ -168,6 +181,12 @@ def f_oneway(*args):
     dfwn = n_samples - n_classes
     msb = ssbn / float(dfbn)
     msw = sswn / float(dfwn)
+    if sigma > 0.0:
+        if method == "relative":
+            sigma_sq = sigma * np.median(msw)
+        else:
+            sigma_sq = float(sigma)
+        msw = (sswn + sigma_sq) / float(dfwn)
     f = msb / msw
     return f
 
diff --git a/mne/stats/tests/test_parametric.py b/mne/stats/tests/test_parametric.py
index 61ecbc43af3..79d274ca4ba 100644
--- a/mne/stats/tests/test_parametric.py
+++ b/mne/stats/tests/test_parametric.py
@@ -11,7 +11,7 @@
 from numpy.testing import assert_allclose, assert_array_almost_equal, assert_array_less
 
 import mne
-from mne.stats.parametric import _map_effects, f_mway_rm, f_threshold_mway_rm
+from mne.stats.parametric import _map_effects, f_mway_rm, f_oneway, f_threshold_mway_rm
 
 # hardcoded external test results, manually transferred
 test_external = {
@@ -175,3 +175,57 @@ def theirs(*a, **kw):
         # something to the divisor (var)
         assert_allclose(got, want, rtol=2e-1, atol=1e-2)
         assert_array_less(np.abs(got), np.abs(want))
+
+
+@pytest.mark.parametrize("sigma", [0.0, 1e-3])
+@pytest.mark.parametrize("method", ["absolute", "relative"])
+@pytest.mark.parametrize("seed", [0, 42, 1337])
+def test_f_oneway_hat(sigma, method, seed):
+    """Test f_oneway hat (low-variance) regularization."""
+    rng = np.random.RandomState(seed)
+    X1 = rng.randn(10, 50)
+    X2 = rng.randn(10, 50)
+
+    f_ours = f_oneway(X1, X2, sigma=0.0, method=method)
+    f_scipy = scipy.stats.f_oneway(X1, X2)[0]
+    assert_allclose(f_ours, f_scipy, rtol=1e-7, atol=1e-6)
+
+    if sigma > 0:
+        f_reg = f_oneway(X1, X2, sigma=sigma, method=method)
+        f_unreg = f_oneway(X1, X2, sigma=0.0)
+        pos = f_unreg > 0
+        assert_array_less(f_reg[pos], f_unreg[pos])
+
+
+def test_f_oneway_hat_small_variance():
+    """Test that f_oneway hat stabilizes F-values for near-zero variance."""
+    rng = np.random.RandomState(0)
+    X1 = rng.normal(0, 1e-6, (10, 100))
+    X2 = rng.normal(1, 1e-6, (10, 100))
+
+    f_unreg = f_oneway(X1, X2, sigma=0.0)
+    f_abs = f_oneway(X1, X2, sigma=1e-3, method="absolute")
+    f_rel = f_oneway(X1, X2, sigma=1e-3, method="relative")
+
+    assert np.median(f_unreg) > 1e6
+    assert np.median(f_abs) < np.median(f_unreg)
+    assert np.median(f_rel) < np.median(f_unreg)
+    assert np.all(np.isfinite(f_abs))
+    assert np.all(np.isfinite(f_rel))
+
+
+def test_f_oneway_hat_input_validation():
+    """Test f_oneway input validation for sigma and method."""
+    rng = np.random.RandomState(0)
+    X1 = rng.randn(5, 10)
+    X2 = rng.randn(5, 10)
+
+    f_plain = f_oneway(X1, X2, sigma=0.0)
+    f_scipy = scipy.stats.f_oneway(X1, X2)[0]
+    assert_allclose(f_plain, f_scipy, rtol=1e-7)
+
+    with pytest.raises(ValueError, match="sigma must be >= 0"):
+        f_oneway(X1, X2, sigma=-0.1)
+
+    with pytest.raises(ValueError, match="method"):
+        f_oneway(X1, X2, sigma=1e-3, method="invalid")