```python
from typing import List, Optional

def driest_window(rain: List[int], width: int) -> int:
    if not isinstance(rain, list) or not all(isinstance(x, int) and x >= 0 for x in rain):
        raise ValueError("Rain must be a list of non-negative integers.")
    if not isinstance(width, int) or width < 1 or width > len(rain):
        raise ValueError("Width must be a positive integer less than or equal to the number of days.")
    if width == len(rain):
        return 0
    
    min_sum = float('inf')
    start_index = 0
    
    for i in range(len(rain)):
        current_sum = sum(rain[i:i+width])
        if current_sum < min_sum:
            min_sum = current_sum
            start_index = i
    
    return start_index
```