Let X1,...,Xn1
be a
random sample from a population with mean µ
1 and variance
$\sigma_1^2$
, and X1,...,Xn1
be a random sample from
another population with mean µ
2 and variance
$\sigma_2^2$
independent of
{Xi,1 ≤ i ≤ n1
}.
Consider the two
sample t-statistic
$ T={{\bar X-\bar Y-(\mu_1-\mu_2)} \over
\sqrt{s_1^2/n_1+s_2^2/n_2}}$
.
This paper shows that
ln P(T ≥ x) ~ -x²/2 for any x := x(n1,n2)
satisfying x → ∞, x = o(n1 + n2)1/2
as n1,n2 → ∞ provided 0 < c1 ≤ n1/n2 ≤ c2
< ∞. If, in
addition, E|X1|3 < ∞, E|Y1|3 < ∞, then
$\frac{P(T \geq x)}{1-\Phi(x)} \to 1 $
holds uniformly in x ∈ (O,o((n1 + n2)1/6))