What often happens in an expanding population is that lower status males don’t not marry, they just marry later - so statistics about who marries who in terms of race wouldn’t necessarily capture it. So what might be happening is that the extra Asian males simply marry later, and marry younger asian females. Since the world population is still expanding that would basically take care of any reasonably sized imbalance.
Asian men are on average shorter than white women. This only is a good enough motive to refrain them to marry white women. Men mostly avoid marrying women taller than themselves, whatever the race.
That has been my observation as well. I know a couple of Asian guys that have white girlfriends; one is an average height (~5’8") Japanese fellow and the other is a very tall (>6"2") Chinese guy. Of course, it doesn’t hurt that both have good jobs and are extremely smart as well, but many of their friends are just as well off and smart and yet don’t have the “trophy” white girlfriend. 
Like I said, just an observation.
I’m an Asian man dating a white woman. With just a couple of exceptions I’ve always dated white women. My brother’s wife is a white woman. All of my cousins in the area married white people. We live in an area with a large, highly educated, affluent Asian community (DC metro area, 8.4%).
OTOH, my cousins out in L.A. all married other Asians.
Like China Guy, I’ve noticed quite a lot of Asian/white couples in the Pacific Northwest when I’ve visited that area.
It would seem that it varies greatly by region.
One of my close friends pretty much exclusively dates Asian men (she’s had three serious LTRs; one BF was from Malaysian, one from Hong Kong, and her new BF is Korean-American). She’s always been aware that this type of pairing is rare, feels like a big outlier, and people make a lot of (often rude) comments…
Asian men are not shorter than 5’4" (average for American women) on average. In most Asian countries average height for men is 5’5" - 5’8" (with urban populations being several inches taller than rural), so there are plenty of taller Asian guys. Don’t forget US average is only 5’9". And there are plenty of very short white chicks, if you’re a shorter man looking for someone small. I have three good friends under 5’1".