Modify WStat
@@ -45,27 +45,28 @@ Poisson data with background measurement
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If you not only have a measurement of counts :math:`n_{\mathrm{on}}` in the
-signal region, but also a measurement :math:`n_{\mathrm{off}}` in a background
-region you can write down the likelihood formula as
+ON region (assumed to contain signal and background), but also a measurement
+:math:`n_{\mathrm{off}}` in a OFF region (assumed to contain background only)
+you can write down the likelihood formula as
.. math::
L (n_{\mathrm{on}}, n_{\mathrm{off}}, \alpha; \mu_{\mathrm{sig}},
- \mu_{\mathrm{bkg}}) = \frac{(\mu_{\mathrm{sig}}+\alpha
+ \mu_{\mathrm{bkg}}) = \frac{(\mu_{\mathrm{sig}}+
\mu_{\mathrm{bkg}})^{n_{\mathrm{on}}}}{n_{\mathrm{on}} !}
- \exp{(-(\mu_{\mathrm{sig}}+\alpha \mu_{\mathrm{bkg}}))}\times
- \frac{(\mu_{\mathrm{bkg}})^{n_{\mathrm{off}}}}{n_{\mathrm{off}}
- !}\exp{(-\mu_{\mathrm{bkg}})},
+ \exp{(-(\mu_{\mathrm{sig}}+ \mu_{\mathrm{bkg}}))}\times
+ \frac{(\mu_{\mathrm{bkg}}/\alpha)^{n_{\mathrm{off}}}}{n_{\mathrm{off}}
+ !}\exp{(-\mu_{\mathrm{bkg}}/\alpha)},
-where :math:`\mu_{\mathrm{sig}}` is the number of expected counts in the signal
-regions, and :math:`\mu_{\mathrm{bkg}}` is the number of expected counts in the
-background region, as defined in the :ref:`stats-introduction`. By taking two
+where :math:`\mu_{\mathrm{sig}}` and :math:`\mu_{\mathrm{bkg}}` are respectively
+the number of expected signal and background counts in the ON region,
+as defined in the :ref:`stats-introduction`. By taking two
time the negative log likelihood and neglecting model independent and thus
constant terms, we define the **WStat**.
.. math::
- W = 2 \big(\mu_{\mathrm{sig}} + (1 + \alpha)\mu_{\mathrm{bkg}}
- - n_{\mathrm{on}} \log{(\mu_{\mathrm{sig}} + \alpha \mu_{\mathrm{bkg}})}
- - n_{\mathrm{off}} \log{(\mu_{\mathrm{bkg}})}\big)
+ W = 2 \big(\mu_{\mathrm{sig}} + (1 + 1/\alpha)\mu_{\mathrm{bkg}}
+ - n_{\mathrm{on}} \log{(\mu_{\mathrm{sig}} + \mu_{\mathrm{bkg}})}
+ - n_{\mathrm{off}} \log{(\mu_{\mathrm{bkg}}/alpha)}\big)
In the most general case, where :math:`\mu_{\mathrm{src}}` and
:math:`\mu_{\mathrm{bkg}}` are free the minimum of :math:`W` is at
