  Consider a binary mixture model of the form $F_\theta = (1-\theta)F_0 +
\theta F_1$, where $F_0$ is standard Gaussian and $F_1$ is a completely
specified heavy-tailed distribution with the same support. For a sample of $n$
independent and identically distributed values $X_i \sim F_\theta$, the maximum
likelihood estimator $\hat\theta_n$ is asymptotically normal provided that $0 <
\theta < 1$ is an interior point. This paper investigates the large-sample
behaviour for boundary points, which is entirely different and strikingly
asymmetric for $\theta=0$ and $\theta=1$. The reason for the asymmetry has to
do with typical choices such that $F_0$ is an extreme boundary point and $F_1$
is usually not extreme. On the right boundary, well known results on boundary
parameter problems are recovered, giving $\lim \mathbb{P}_1(\hat\theta_n <
1)=1/2$. On the left boundary, $\lim\mathbb{P}_0(\hat\theta_n > 0)=1-1/\alpha$,
where $1\leq \alpha \leq 2$ indexes the domain of attraction of the density
ratio $f_1(X)/f_0(X)$ when $X\sim F_0$. For $\alpha=1$, which is the most
important case in practice, we show how the tail behaviour of $F_1$ governs the
rate at which $\mathbb{P}_0(\hat\theta_n > 0)$ tends to zero. A new limit
theorem for the joint distribution of the sample maximum and sample mean
conditional on positivity establishes multiple inferential anomalies. Most
notably, given $\hat\theta_n > 0$, the likelihood ratio statistic has a
conditional null limit distribution that is not $\chi^2_1$, but is determined
by the joint limit theorem.
