code: def test_brockwell_davis_example_525():
      # Difference and demean the series
      endog = lake.copy()
  
      # Use HR method to get initial coefficients for MLE
      initial, _ = hannan_rissanen(
          endog, ar_order=1, ma_order=1, demean=True
      )
  
      # Fit MLE via innovations algorithm
      p, _ = innovations_mle(
          endog,
          order=(1, 0, 1),
          demean=True,
          start_params=initial.params,
      )
  
      assert_allclose(p.params, [0.7446, 0.3213, 0.4750], atol=1e-4)
  
      # Fit MLE via innovations algorithm, with default starting parameters
      p, _ = innovations_mle(endog, order=(1, 0, 1), demean=True)
  
      assert_allclose(p.params, [0.7446, 0.3213, 0.4750], atol=1e-4)
  
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 7 for calling a global function with example usage: test_brockwell_davis_example_525() without return types