the rank of the newest entry (by timestamp) - rank of the oldest entry (by timestamp)
There are many ways to achieve this with existing functions. Like the window functions first_value() and last_value(), combined with DISTINCT or DISTINCT ON and without joins and subqueries:
SELECT DISTINCT ON (userid)
userid, last_value(rank) OVER w - first_value(rank) OVER w AS rank_delta
FROM rankings
WINDOW w AS (PARTITION BY userid ORDER BY ts
ROWS BETWEEN UNBOUNDED PRECEDING AND UNBOUNDED FOLLOWING);
Note the custom window frame!
Or you can use basic aggregate functions - with joins and a subquery:
SELECT userid, r2.rank - r1.rank AS rank_delta
FROM (
SELECT userid, min(ts) AS first_ts, max(ts) AS last_ts
FROM rankings
GROUP BY 1
) sub
JOIN rankings r1 USING (userid)
JOIN rankings r2 USING (userid)
WHERE r1.ts = first_ts
AND r2.ts = last_ts;
Assuming unique (userid, rank), or your requirements would be ambiguous.
Shichinin no samurai
... a.k.a. "7 Samurai"
The same for only the last seven rows per userid (or as many as can be found, if there are fewer) - one of the shortest ways:
SELECT DISTINCT ON (userid)
userid, first_value(rank) OVER w - last_value(rank) OVER w AS rank_delta
FROM rankings
WINDOW w AS (PARTITION BY userid ORDER BY ts DESC
ROWS BETWEEN CURRENT ROW AND 7 FOLLOWING)
ORDER BY userid, ts DESC;
Note the reversed sort order. The first row is the "newest" entry. I span a frame of (max.) 7 preceding rows and pick only the results for the newest entry per userid with DISTINCT ON.
fiddle
Old sqlfiddle 1, sqlfiddle 2