The reason u should generally farm in Japan's District is....it is/has never been "Over-Farmed"
The American and some of the European districts have been.
Over-Farming occurs when a area is Farmed for a specific item, like Underworld for Ectos, ANet places a 'cap' on the item and it drops less frequently. This doesnt happen in less used districts like Japan, or say Poland, since no one uses them.
Also, in addition to this the drops are of better quality. For example instead of getting 4 brown dye drops u may get 3 brown dyes and a black or white one, and the quality of the mods on a weapon or the weapons u get themselves will also generally be better or rarer.
However, if u are farming Collector Trophies for the Traveler or are feather farming Feathered Crests in Jaya Bluffs (farmed for their salvageable feathers), you should do this in the American districts since the district is over-farmed. In this instance, creatures tend to drop more Collector Trophies and gold coins since they are their default drop.
Farming with a Mini Pet out will actually increase the drop rate (though has no effect on the quality of drops). This is attributed to their being more "party members" in the zone. Though Minis are not party members there is an active bug which can at times assign drops to them instead of player characters (even if it lists it as being assigned to the miniature pet it is still retrievable by the pets owner). Generally u wont see this, it will just state that it is assigned to u, and not to your mini but it increases the drop rate none the less.