Give the example some love

doc/cookbook/source/examples/statistical_testing/quadratic_time_mmd.rst

@@ -2,11 +2,17 @@
 Quadratic Time MMD
 ==================
 
-The unbiased statistic is given by
+The Quadratic time MMD implements a nonparametric statistical hypothesis test to reject the null hypothesis that to distributions :math:`p` and :math:`q`, only observed via :math:`n` and :math:`m` samples respectively, are the same:
 
 .. math::
 
-  \frac{1}{n(n-1)}\sum_{i=1}^n\sum_{j=1}^n k(x_i,x_i) + k(x_j, x_j) - 2k(x_i,x_j).
+  H_0:p=q.
+
+The (biased) test statistic is given by
+
+.. math::
+
+  \frac{1}{nm}\sum_{i=1}^n\sum_{j=1}^m k(x_i,x_i) + k(x_j, x_j) - 2k(x_i,x_j).
   
 
 See :cite:`gretton2012kernel` for a detailed introduction.
@@ -15,24 +21,38 @@ See :cite:`gretton2012kernel` for a detailed introduction.
 Example
 -------
 
-Imagine we have samples from :math:`p` and :math:`p`. We create CDenseFeatures (here 64 bit floats aka RealFeatures)as
+Imagine we have samples from :math:`p` and :math:`q`. We create CDenseFeatures (here 64 bit floats aka RealFeatures)as
 
 .. sgexample:: quadratic_time_mmd.sg:create_features
 
-We create an instance of :sgclass:`CQuadraticTimeMMD`, passing it data and the kernel to use, a :sgclass:`CGaussianKernel` here.
+We create an instance of :sgclass:`CQuadraticTimeMMD`, passing it data the kernel, and the test significance level :math:`\alpha`
 
 .. sgexample:: quadratic_time_mmd.sg:create_instance
 
-Computing the statistic is done as
+We can select multiple ways to compute the test statistic, see :sgclass:`CQuadraticTimeMMD` for details. Unbiased statistic
+
+.. sgexample:: quadratic_time_mmd.sg:estimate_mmd_unbiased
+
+Biased statistic
 
-.. sgexample:: quadratic_time_mmd.sg:estimate_mmd
+.. sgexample:: quadratic_time_mmd.sg:estimate_mmd_biased
 
-We can perform the hypothesis test as
+There are multiple ways to perform the actual hypothesis test. The permutation version simulates from :math:`H_0` via repeatedly permuting the samples from :math:`p` and :math:`q`:
 
-.. sgexample:: quadratic_time_mmd.sg:perform_test
+.. sgexample:: quadratic_time_mmd.sg:perform_test_permutation
+
+The spectrum version simulates from :math:`H_0` via approximating the Eigenspectrum of the underlying kernel:
+
+.. sgexample:: quadratic_time_mmd.sg:perform_test_spectrum
+
+The Gamma version fit a Gamma cumulative distribution function to :math:`H_0` via moment matching:
+
+.. sgexample:: quadratic_time_mmd.sg:perform_test_gamma
 
 ----------
 References
 ----------
 .. bibliography:: ../../references.bib
     :filter: docname in docnames
+
+:wiki:`Statistical_hypothesis_testing`