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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| Materias: | |
| Acceso en línea: | https://arxiv.org/abs/2407.20162 |
| Etiquetas: |
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- Consider a binary mixture model of the form $F_θ= (1-θ)F_0 + θF_1$, 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 $X_i \sim F_θ$, the maximum likelihood estimator $\hatθ_n$ is asymptotically normal provided that $0 < θ< 1$ is an interior point. This paper investigates the large-sample behaviour for boundary points, which is entirely different and strikingly asymmetric for $θ=0$ and $θ=1$. 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 $\lim \mathbb{P}_1(\hatθ_n < 1)=1/2$. On the left boundary, $\lim\mathbb{P}_0(\hatθ_n > 0)=1-1/α$, where $1\leq α\leq 2$ indexes the domain of attraction of the density ratio $f_1(X)/f_0(X)$ when $X\sim F_0$. For $α=1$, which is the most important case in practice, we show how the tail behaviour of $F_1$ governs the rate at which $\mathbb{P}_0(\hatθ_n > 0)$ 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 $\hatθ_n > 0$, the likelihood ratio statistic has a conditional null limit distribution $G\neqχ^2_1$ determined by the joint limit theorem. We show through this route that no advantage is gained by extending the single distribution $F_1$ to the nonparametric composite mixture generated by the same tail-equivalence class.