Speaker
Description
Consider a binary mixture model of the form $F_\theta=(1−\theta)F_0+\theta F_1$, where $F_0$ is standard normal and $F_1$ is a completely specified heavy-tailed distribution with the same support. Gaussianity of $F_0$ reflects a reduction of the raw data to a set of pivotal test statistics at each site (e.g. an energy level in a particle physics context). 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 $\theta=1$. 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 (which corresponds to no new physics) $\lim \mathbb{P}_0(\hat\theta_n>0)=1−1/\alpha$, 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, the tail behaviour of $F_1$ governs the properties of the maximum likelihood estimator and related statistics. Most notably, conditional on the event $\hat\theta_n>0$, the likelihood ratio statistic has a conditional null limit distribution that is not the usual $\chi^2_1$. In the talk I will omit technical details and focus on the conceptual points with a view to ascertaining whether the formulation is reasonable in a particle physics context.
This is joint work with Peter McCullagh and Daniel Xiang at the University of Chicago.
| Primary Field of Research | Statistics |
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