The edit distance between two strings $d(w_1, w_2)$ sums up the cost of edit operations that transform one string into the other. If done right, the edit distance is a metric.
In particular, the cost of an edit operation may depend on the character inserted or deleted, meaning we have costs $w_\text{insert}(a)$ and $w_\text{delete}(a)$ for all characters $a$.
Question: If the cost also depends on the position in the input string where it is applied, will the result still be a metric?
As an example, consider a spell checker that assumes mistakes in the first character of a word are rather seldom as compared to other positions, so when searching for corrected words, a change in the first position would have high cost.