code: def test_brockwell_davis_example_541():
      # Difference and demean the series
      endog = oshorts.copy()
  
      # Use innovations MA method to get initial coefficients for MLE
      initial, _ = innovations(endog, ma_order=1, demean=True)
  
      # Fit MLE via innovations algorithm
      p, _ = innovations_mle(
          endog,
          order=(0, 0, 1),
          demean=True,
          start_params=initial[1].params,
      )
  
      assert_allclose(p.ma_params, -0.818, atol=1e-3)
  
imports: import numpy
  import pytest
  from numpy.testing import assert_, assert_allclose, assert_warns, assert_raises
  from statsmodels.tsa.innovations.arma_innovations import arma_innovations
  from statsmodels.tsa.statespace import sarimax
  from statsmodels.tsa.arima.datasets.brockwell_davis_2002 import dowj, lake, oshorts
  from statsmodels.tsa.arima.estimators.burg import burg
  from statsmodels.tsa.arima.estimators.hannan_rissanen import hannan_rissanen
  from statsmodels.tsa.arima.estimators.innovations import innovations, innovations_mle
label_desc: Use image node_id 8 for calling a global function with example usage: test_brockwell_davis_example_541() without return types