import numpy as np

def replace_nans_with_local_statistic(arr, window_size=3, method='mean'):
    """
    Replace NaN values in a 1D array with the local mean or median of surrounding values.

    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.

    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)  # Ensure it's a float array to support NaN
    result = arr.copy()
    
    half_window = window_size // 2
    
    for i in range(len(arr)):
        if np.isnan(arr[i]):
            start = max(0, i - half_window)
            end = min(len(arr), i + half_window + 1)
            window = arr[start:end]
            valid_window = window[~np.isnan(window)]
            
            if len(valid_window) > 0:
                if method == 'mean':
                    result[i] = np.mean(valid_window)
                else:
                    result[i] = np.median(valid_window)
            else:
                result[i] = np.nan  # No valid values in window, leave as NaN
    
    return result