关于线性回归的Hannan-Quinn命题
The Hannan-Quinn Proposition for Linear Regression
摘要：
研究了线性回归中的变量选择问题。假设我们有一组随机变量$X_1,...,X_m,Y,ε$使得$Y=\sum_{k\in π}α_kX_k+ε$，其中$π\subseteq \{1,...,m\}$和$α_k\in {\mathbb R}$未知，并且$ε$独立于$X_1,...,X_m$的任何线性组合。给定从$(X_1,...,X_m, Y)$发射的实际发射的$n$示例$\{(x_{i,1}...,x_{i,m},y_i)\}_{i=1}^n$，我们希望使用$H+(k/2)d_n$形式的信息准则来估计真实的$π$，其中$H$是关于$π$的似然性乘以-1，并且$\{d_n\}$是正实数序列。如果$d_n$太小，我们不能获得一致性，因为高估。对于自回归，Hannan-Quinn证明了，在他们的$H$和$k$设置中，速率$d_n=2\log\log n$是满足强一致性的最小值。本文也解决了线性回归的肯定命题，但设置完全不同。
英文摘要：
We consider the variable selection problem in linear regression. Suppose that we have a set of random variables $X_1,...,X_m,Y,ε$ such that $Y=\sum_{k\in π}α_kX_k+ε$ with $π\subseteq \{1,...,m\}$ and $α_k\in {\mathbb R}$ unknown, and $ε$ is independent of any linear combination of $X_1,...,X_m$. Given actually emitted $n$ examples $\{(x_{i,1}...,x_{i,m},y_i)\}_{i=1}^n$ emitted from $(X_1,...,X_m, Y)$, we wish to estimate the true $π$ using information criteria in the form of $H+(k/2)d_n$, where $H$ is the likelihood with respect to $π$ multiplied by -1, and $\{d_n\}$ is a positive real sequence. If $d_n$ is too small, we cannot obtain consistency because of overestimation. For autoregression, Hannan-Quinn proved that, in their setting of $H$ and $k$, the rate $d_n=2\log\log n$ is the minimum satisfying strong consistency. This paper solves the statement affirmative for linear regression as well which has a completely different setting.
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