An American of Asian descent is an Asian American, not an Amerasian, the latter term being largely restricted to children fathered by American servicemen stationed in Asia during the Korean and Vietnam wars. As with Latin American and Euro-American, the designation Asian American can apply to many different people with different national origins and widely varying cultural backgrounds. Wherever appropriate you should consider using a more specific term, such as Chinese American or Korean American, in place of the comprehensive one.
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As a noun, Asian American should be spelled without a hyphen. The adjective can be styled either with or without a hyphen.