code: def distance_covariance_test(x, y, B=None, method="auto"):
      r"""The Distance Covariance (dCov) test
  
      Apply the Distance Covariance (dCov) test of independence to `x` and `y`.
      This test was introduced in [1]_, and is based on the distance covariance
      statistic. The test is applicable to random vectors of arbitrary length
      (see the notes section for more details).
  
      Parameters
      ----------
      x : array_like, 1-D or 2-D
          If `x` is 1-D than it is assumed to be a vector of observations of a
          single random variable. If `x` is 2-D than the rows should be
          observations and the columns are treated as the components of a
          random vector, i.e., each column represents a different component of
          the random vector `x`.
      y : array_like, 1-D or 2-D
          Same as `x`, but only the number of observation has to match that of
          `x`. If `y` is 2-D note that the number of columns of `y` (i.e., the
          number of components in the random vector) does not need to match
          the number of columns in `x`.
      B : int, optional, default=`None`
          The number of iterations to perform when evaluating the null
          distribution of the test statistic when the `emp` method is
          applied (see below). if `B` is `None` than as in [1]_ we set
          `B` to be ``B = 200 + 5000/n``, where `n` is the number of
          observations.
      method : {'auto', 'emp', 'asym'}, optional, default=auto
          The method by which to obtain the p-value for the test.
  
          - `auto` : Default method. The number of observations will be used to
            determine the method.
          - `emp` : Empirical evaluation of the p-value using permutations of
            the rows of `y` to obtain the null distribution.
          - `asym` : An asymptotic approximation of the distribution of the test
            statistic is used to find the p-value.
  
      Returns
      -------
      test_statistic : float
          The value of the test statistic used in the test.
      pval : float
          The p-value.
      chosen_method : str
          The method that was used to obtain the p-value. Mostly relevant when
          the function is called with `method='auto'`.
  
      Notes
      -----
      The test applies to random vectors of arbitrary dimensions, i.e., `x`
      can be a 1-D vector of observations for a single random variable while
      `y` can be a `k` by `n` 2-D array (where `k > 1`). In other words, it
      is also possible for `x` and `y` to both be 2-D arrays and have the
      same number of rows (observations) while differing in the number of
      columns.
  
      As noted in [1]_ the statistics are sensitive to all types of departures
      from independence, including nonlinear or nonmonotone dependence
      structure.
  
      References
      ----------
      .. [1] Szekely, G.J., Rizzo, M.L., and Bakirov, N.K. (2007)
         "Measuring and testing by correlation of distances".
         Annals of Statistics, Vol. 35 No. 6, pp. 2769-2794.
  
      Examples
      --------
      >>> from statsmodels.stats.dist_dependence_measures import
      ... distance_covariance_test
      >>> data = np.random.rand(1000, 10)
      >>> x, y = data[:, :3], data[:, 3:]
      >>> x.shape
      (1000, 3)
      >>> y.shape
      (1000, 7)
      >>> distance_covariance_test(x, y)
      (1.0426404792714983, 0.2971148340813543, 'asym')
      # (test_statistic, pval, chosen_method)
  
      """
      x, y = _validate_and_tranform_x_and_y(x, y)
  
      n = x.shape[0]
      stats = distance_statistics(x, y)
  
      if method == "auto" and n <= 500 or method == "emp":
          chosen_method = "emp"
          test_statistic, pval = _empirical_pvalue(x, y, B, n, stats)
  
      elif method == "auto" and n > 500 or method == "asym":
          chosen_method = "asym"
          test_statistic, pval = _asymptotic_pvalue(stats)
  
      else:
          raise ValueError(f"Unknown 'method' parameter: {method}")
  
      # In case we got an extreme p-value (0 or 1) when using the empirical
      # distribution of the test statistic under the null, we fall back
      # to the asymptotic approximation.
      if chosen_method == "emp" and pval in [0, 1]:
          msg = (
              f"p-value was {pval} when using the empirical method. "
              "The asymptotic approximation will be used instead"
          )
          warnings.warn(msg, HypothesisTestWarning)
          _, pval = _asymptotic_pvalue(stats)
  
      return test_statistic, pval, chosen_method
  
file_path: statsmodels/statsmodels/stats/dist_dependence_measures.py
imports: from collections import namedtuple
  import warnings
  import numpy
  from scipy.spatial.distance import pdist, squareform
  from scipy.stats import norm
  from statsmodels.tools.sm_exceptions import HypothesisTestWarning
label_desc: Use image node_id 1 for calling a global function with example usage: distance_covariance_test(x, y, B, method) and returns: test_statistic,  pval,  chosen_method