城市的Gibrat定律：Pareto对对数正态分布的一致最有效无偏检验
Gibrat's law for cities: uniformly most powerful unbiased test of the Pareto against the lognormal
摘要：
我们解决的一般问题，测试幂律分布与对数正态分布的统计数据。2000年美国城市规模普查的分布说明了这个普遍问题。Eeckhout（2004，2009）和Levy（2009）关于Zipf定律（尾部指数为1的特殊帕累托定律）是否适用于描述美国城市规模分布尾部的争论，我们提供了明确的结果。由于Eeckhout和Levy之间的分歧源于他们检验的有限功效，因此我们对对数正态分布的Pareto分布的零假设执行{ \em一致最大功效无偏检验}。$p$值和希尔估计值作为城市规模下限的函数，无疑证实了1000个最大城市的规模分布是帕累托分布，其中包括美国总人口的一半以上，但我们排除了尾部指数，估计为$1.4 \pm 0.1$，等于1。对于较大的排名，$p$值变得非常小，希尔的估计系统地衰减与降低排名，资格的对数正态分布作为更好的模型为一组较小的城市。这两个结果调和了Eeckhout（2004，2009）和Levy（2009）的相反观点。我们解释了如何Gibrat的法律比例增长的帕累托分布和对数正态分布的基础上，并强调其差异的城市\cite{Gabaix99,Eeckhout2004}的标准随机增长模型的起源的关键成分。
英文摘要：
We address the general problem of testing a power law distribution versus a log-normal distribution in statistical data. This general problem is illustrated on the distribution of the 2000 US census of city sizes. We provide definitive results to close the debate between Eeckhout (2004, 2009) and Levy (2009) on the validity of Zipf's law, which is the special Pareto law with tail exponent 1, to describe the tail of the distribution of U.S. city sizes. Because the origin of the disagreement between Eeckhout and Levy stems from the limited power of their tests, we perform the {\em uniformly most powerful unbiased test} for the null hypothesis of the Pareto distribution against the lognormal. The $p$-value and Hill's estimator as a function of city size lower threshold confirm indubitably that the size distribution of the 1000 largest cities or so, which include more than half of the total U.S. population, is Pareto, but we rule out that the tail exponent, estimated to be $1.4 \pm 0.1$, is equal to 1. For larger ranks, the $p$-value becomes very small and Hill's estimator decays systematically with decreasing ranks, qualifying the lognormal distribution as the better model for the set of smaller cities. These two results reconcile the opposite views of Eeckhout (2004, 2009) and Levy (2009). We explain how Gibrat's law of proportional growth underpins both the Pareto and lognormal distributions and stress the key ingredient at the origin of their difference in standard stochastic growth models of cities \cite{Gabaix99,Eeckhout2004}.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
