NSDate is a bit of a misnomer. It is a not a simple date, it is a date a time, with subsecond level precision. The earlierDate: method is a comparison function that returns which ever of the two dates is earlier, it does not create an earlier date. If all you care about is a date that is before your current one you can do:
NSDate *earlierDate = [date addTimeInterval:-1.0];
Which will return a date 1 second before date. Likewise you can do
NSDate *laterDate = [date addTimeInterval:1.0];
for a date 1 second in the future.
If you want a day earlier you can add a days worth of seconds. Of course that is approximate unless you deal with all the gregorian calendar issues, but if you just want a quick approximation:
NSDate *earlierDate = [date addTimeInterval:(-1.0*24*60*60)];
That doesn't work with leap seconds, etc, but for most uses it is probably fine.