def test_bias_on_array_map():
# compute mean and (biased) variance simultanously
def fn(x):
return np.mean(x), np.var(x, ddof=0)
data = (0, 1, 2)
r = bias(fn, data)
assert_almost_equal(r, (0.0, -1.0 / 3.0))
def test_bias_on_biased_order_n_minus_one():
# this "mean" has a bias of exactly O(n^{-1})
def bad_mean(x):
return (np.sum(x) + 2) / len(x)
data = (0, 1, 2)
r = bias(bad_mean, data)
mean_jk = np.mean([bad_mean([1, 2]), bad_mean([0, 2]), bad_mean([0, 1])])
# (5/2 + 4/2 + 3/2) / 3 = 12 / 6 = 2
assert mean_jk == 2.0
# f = 5/3
# (n-1) * (mean_jk - f)
# (3 - 1) * (6/3 - 5/3) = 2/3
# note: 2/3 is exactly the bias of bad_mean for n = 3
assert r == pytest.approx(2.0 / 3.0)
def test_bias_on_unbiased():
data = (0, 1, 2, 3)
# bias is exactly zero for linear functions
r = bias(np.mean, data)
assert r == 0