From 31b679eaee2a8cf5ec5307c0e33e00ac3e2aee7c Mon Sep 17 00:00:00 2001
From: "Jack Y. Araz" <jackaraz@gmail.com>
Date: Tue, 12 Mar 2024 10:19:32 -0400
Subject: [PATCH] update docstring

---
 .../hypothesis_testing/test_statistics.py     | 21 +++++++++++++++++++
 1 file changed, 21 insertions(+)

diff --git a/src/spey/hypothesis_testing/test_statistics.py b/src/spey/hypothesis_testing/test_statistics.py
index 70dbd91..a728a24 100644
--- a/src/spey/hypothesis_testing/test_statistics.py
+++ b/src/spey/hypothesis_testing/test_statistics.py
@@ -16,6 +16,13 @@ def qmu_tilde(
     r"""
     Alternative test statistics, :math:`\tilde{q}_{\mu}`, see eq. (62) of :xref:`1007.1727`.
 
+    .. math::
+
+        \tilde{q}_{\mu} = \begin{cases}
+            0 & \text{if}\ \hat{\mu} > \mu\ , \\
+            -2\log\left( \frac{\mathcal{L}(\mu, \theta_\mu)}{\mathcal{L}(\max(\hat{\mu}, 0), \hat{\theta})} \right) & \text{otherwise}
+        \end{cases}
+
     .. warning::
 
         Note that this assumes that :math:`\hat\mu\geq0`, hence :obj:`allow_negative_signal`
@@ -48,6 +55,13 @@ def qmu(
 ) -> float:
     r"""
     Test statistic :math:`q_{\mu}`, see eq. (54) of :xref:`1007.1727`
+    
+    .. math::
+
+        q_{\mu} = \begin{cases}
+            0 & \text{if}\ \hat{\mu} > \mu\ ,\\
+            -2\log\left( \frac{\mathcal{L}(\mu, \theta_\mu)}{\mathcal{L}(\hat{\mu}, \hat{\theta})} \right) & \text{otherwise}
+        \end{cases}
 
     Args:
         mu (``float``): parameter of interest, :math:`\mu`.
@@ -68,6 +82,13 @@ def q0(
     r"""
     Discovery test statistics, :math:`q_{0}` see eq. (47) of :xref:`1007.1727`.
 
+    .. math::
+
+        q_0 = \begin{cases}
+            0 & \text{if}\ \hat{\mu} < 0\ ,\\
+            -2\log\left( \frac{\mathcal{L}(0, \theta_0)}{\mathcal{L}(\hat{\mu}, \hat{\theta})} \right) & \text{otherwise}
+        \end{cases}
+
     Args:
         mu (``float``): parameter of interest, :math:`\mu`.