import numpy as np

def replace_nans_with_local_statistic(arr, window_size=3, method='mean', max_iterations=10):
    """
    Replace NaN values in a 1D array with the local mean or median of surrounding values,
    including handling of consecutive NaN segments via iterative refinement.

    Parameters:
    - arr: Input 1D array with NaN values (list or numpy array).
    - window_size: Size of the window to compute local statistic (must be odd).
    - method: 'mean' or 'median' to specify the replacement method.
    - max_iterations: Number of iterations to try filling NaNs (for consecutive NaNs).

    Returns:
    - Modified array with NaNs replaced by local statistics.
    """
    if method not in ['mean', 'median']:
        raise ValueError("Method must be 'mean' or 'median'")
    
    if window_size % 2 == 0:
        raise ValueError("Window size must be an odd number")

    arr = np.array(arr, dtype=float)
    result = arr.copy()
    n = len(arr)
    half_window = window_size // 2

    for iteration in range(max_iterations):
        modified = False  # Track whether any NaNs were filled in this pass

        for i in range(n):
            if np.isnan(result[i]):
                # Create a local window around the NaN
                start = max(0, i - half_window)
                end = min(n, i + half_window + 1)
                window = result[start:end]

                valid_values = window[~np.isnan(window)]
                if len(valid_values) > 0:
                    if method == 'mean':
                        result[i] = np.mean(valid_values)
                    else:
                        result[i] = np.median(valid_values)
                    modified = True

        if not modified:
            # No changes in this iteration → stop
            break

    return result