def _confidence_interval_bca(theta: float, thetas: np.ndarray,
j_thetas: np.ndarray,
alpha_half: float) -> Tuple[float, float]:
norm = stats.norm
# bias correction
prop_less = np.mean(
thetas < theta) # proportion of replicates less than obs
z_naught = norm.ppf(prop_less)
# acceleration
j_thetas -= np.mean(j_thetas)
num = np.sum((-j_thetas)**3)
den = np.sum(j_thetas**2)
acc = num / (6 * den**1.5)
z_low = z_naught + norm.ppf(alpha_half)
z_high = z_naught + norm.ppf(1 - alpha_half)
p_low = norm.cdf(z_naught + z_low / (1 - acc * z_low))
p_high = norm.cdf(z_naught + z_high / (1 - acc * z_high))
quant = quantile_function_gen(thetas)
return quant(p_low), quant(p_high)
def test_quantile_out_of_bounds_is_nan(arg):
q = quantile_function_gen(np.array([0, 1, 2, 3]))
assert np.isnan(q(arg))
def test_quantile_is_inverse_of_cdf(rng):
x = rng.normal(size=30)
y = cdf_gen(x)(x)
assert_equal(quantile_function_gen(x)(y), x)
def test_quantile_on_array():
x = np.arange(4)
q = quantile_function_gen(x)
prob = (x + 1) / len(x)
assert_equal(q(prob), x)
def test_quantile_simple_cases():
q = quantile_function_gen([0, 1, 2, 3])
assert q(0.25) == 0
assert q(0.5) == 1
assert q(0.75) == 2
assert q(1.0) == 3
def _confidence_interval_percentile(
thetas: np.ndarray,
alpha_half: float,
) -> Tuple[float, float]:
quant = quantile_function_gen(thetas)
return quant(alpha_half), quant(1 - alpha_half)