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用稳定指数 α 衡量尾部厚度：一项批判AI 解读

Stable laws are often fit to outlier-prone data and, if the index $\alpha$ is estimated to be much less than two, then the normal law is rejected in favor of an infinite-variance stable law. This paper derives the theoretical properties of such a procedure. When the true distribution is stable, the distribution of the m.l.e. of $\alpha$ is non-regular if $\alpha = 2$. When the true distribution is not stable, the estimate of $\alpha$ is not a robust measure of the rate of…

分层样本多元回归中调查权重的使用AI 解读

The rationale for the use of sample survey weights in a least squares regression analysis is examined with respect to four increasingly general specifications of the population regression model. The appropriateness of the weighted regression estimate depends on which model is chosen. A proposal is made to use the difference between the weighted and unweighted estimates as an aid in choosing the appropriate model and hence the appropriate estimator. When applied to an…

结合人类与其他物种癌症研究结果的贝叶斯方法AI 解读

We propose a class of Bayesian statistical methods for interspecies extrapolation of dose-response functions. The methods distinguish formally between the conventional sampling error within each dose-response experiment and a novel error of uncertain relevance between experiments. Through a system of hierarchical prior distributions similar to that of Lindley and Smith (1972), the dose-response data from many substances and species are used to estimate the interexperimental…

稳定分布抽样下最大似然估计的渐近正态性AI 解读

The large-sample distributions of the maximum-likelihood estimates for the index, skewness, scale, and location parameters (respectively $\alpha, \beta, c$, and $\delta$) of a stable distribution are studied. It is shown that if both $\alpha$ and $\delta$ are unknown, then the likelihood function $L$ will have no maximum within $0 < \alpha \leqq 2, -\infty < \delta < \infty$, but that $L(\alpha, \delta) \rightarrow \infty$ as $(\alpha, \delta) \rightarrow (0, x_k)$ where $x_k$…

统计推断中的稳定分布（一）：对称稳定分布与其他对称长尾分布的比较AI 解读

The hypothesis that very long-tailed data come from a symmetric stable distribution can be generalized, namely, that the observations are generated by a mixture of a normal distribution and another symmetric stable distribution. The difficulty of statistical distinction of the two cases is computed as a function of the various parameters involved. Also, computations show that convolutions of these mixtures can be very slow to converge to the limiting stable distribution or…