```python
from typing import List

def driest_window(rain: List[int], width: int) -> int:
    if not isinstance(rain, list):
        raise ValueError("rain must be a list")
    if not all(isinstance(x, int) for x in rain):
        raise ValueError("all elements of rain must be integers")
    if any(x < 0 for x in rain):
        raise ValueError("all elements of rain must be non-negative")
    if not isinstance(width, int) or width <= 0:
        raise ValueError("width must be a positive integer")
    if width > len(rain):
        raise ValueError("width cannot exceed the number of days")

    min_sum = float('inf')
    start_index = -1

    for i in range(len(rain) - width + 1):
        current_sum = sum(rain[i:i+width])
        if current_sum < min_sum:
            min_sum = current_sum
            start_index = i

    return start_index
```