标题： 高维逻辑回归中的似然比检验渐近于一个重新标度的卡方分布 显示英文标题

标题： The Likelihood Ratio Test in High-Dimensional Logistic Regression Is Asymptotically a Rescaled Chi-Square

摘要： Logistic regression is used thousands of times a day to fit data, predict future outcomes, and assess the statistical significance of explanatory variables. When used for the purpose of statistical inference, logistic models produce p-values for the regression coefficients by using an approximation to the distribution of the likelihood-ratio test. Indeed, Wilks' theorem asserts that whenever we have a fixed number $p$ of variables, twice the log-likelihood ratio (LLR) $2\Lambda$ is distributed as a $\chi^2_k$ variable in the limit of large sample sizes $n$; here, $k$ is the number of variables being tested. In this paper, we prove that when $p$ is not negligible compared to $n$, Wilks' theorem does not hold and that the chi-square approximation is grossly incorrect; in fact, this approximation produces p-values that are far too small (under the null hypothesis). 假设$n$和$p$增大，并且对于某个常数$\kappa < 1/2$满足$p/n\rightarrow\kappa$。 我们证明了对于一类逻辑模型，LLR 收敛于一个尺度变换的卡方分布，即 $2\Lambda~\stackrel{\mathrm{d}}{\rightarrow}~\alpha(\kappa)\chi_k^2$，其中尺度因子$\alpha(\kappa)$大于 1 只要维度比$\kappa$为正。 因此，LLR 比通常假设的更大。 例如，当$\kappa=0.3$时， $\alpha(\kappa)\approx1.5$。 一般而言，我们展示了如何通过求解一个含有两个未知数的非线性方程组来计算尺度因子。 我们的数学论证涉及多个领域，包括近似消息传递理论、非渐近随机矩阵理论和凸几何技术。 此外，我们还通过展示新极限分布对于有限样本大小的准确性来补充我们的数学研究。 最后，本文中的所有结果都可推广到其他一些回归模型，例如 probit 回归模型。 显示英文摘要

摘要： Logistic regression is used thousands of times a day to fit data, predict future outcomes, and assess the statistical significance of explanatory variables. When used for the purpose of statistical inference, logistic models produce p-values for the regression coefficients by using an approximation to the distribution of the likelihood-ratio test. Indeed, Wilks' theorem asserts that whenever we have a fixed number $p$ of variables, twice the log-likelihood ratio (LLR) $2\Lambda$ is distributed as a $\chi^2_k$ variable in the limit of large sample sizes $n$; here, $k$ is the number of variables being tested. In this paper, we prove that when $p$ is not negligible compared to $n$, Wilks' theorem does not hold and that the chi-square approximation is grossly incorrect; in fact, this approximation produces p-values that are far too small (under the null hypothesis). Assume that $n$ and $p$ grow large in such a way that $p/n\rightarrow\kappa$ for some constant $\kappa < 1/2$. We prove that for a class of logistic models, the LLR converges to a rescaled chi-square, namely, $2\Lambda~\stackrel{\mathrm{d}}{\rightarrow}~\alpha(\kappa)\chi_k^2$, where the scaling factor $\alpha(\kappa)$ is greater than one as soon as the dimensionality ratio $\kappa$ is positive. Hence, the LLR is larger than classically assumed. For instance, when $\kappa=0.3$, $\alpha(\kappa)\approx1.5$. In general, we show how to compute the scaling factor by solving a nonlinear system of two equations with two unknowns. Our mathematical arguments are involved and use techniques from approximate message passing theory, non-asymptotic random matrix theory and convex geometry. We also complement our mathematical study by showing that the new limiting distribution is accurate for finite sample sizes. Finally, all the results from this paper extend to some other regression models such as the probit regression model.