STEINS PARADOX IN STATISTICS
STEINS PARADOX IN STATISTICS
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DOI：
10.1038/scientificamerican0577-119
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发表时间：
1977-01-01
影响因子：
3
通讯作者：
MORRIS, C
中科院分区：
文献类型：
--
作者：
EFRON, B;MORRIS, C
有时，即使给出了明显有效的证明，一个数学结果却与普遍持有的信念截然相反。斯坦福大学的查尔斯·斯坦（Charles Stein）在1955年发现了这样一个统计学悖论。他的结果破坏了一个半世纪以来对估计理论的研究，可以追溯到卡尔·弗里德里希·高斯（Karl Friedrich Gauss）和阿德里安·马里·勒让德（Adrien marrie Legendre）。在对斯坦因的观点进行了长时间的抵制之后，伴随着频繁的、有时甚至是愤怒的辩论，这种悖论的感觉已经减弱，斯坦因的观点正在被纳入应用和理论统计中。斯坦悖论涉及使用观测平均值来估计不可观测量。平均是统计学中第二个最基本的过程，第一个是简单的计数。在20次正式击球中打出7支安打的棒球运动员的击球率为。350. 在计算这个统计数据时，我们根据观察到的平均成功率来估计球员的真实击球能力。当被问及球员在接下来的100次击球中表现如何时，我们可能会预测他还会有35次击球。在传统的统计理论中，可以证明没有其他估计规则比观测平均值一致好。斯坦因结果的矛盾之处在于，它有时与统计理论的基本规律相矛盾。如果我们有三个或更多的棒球运动员，如果我们有兴趣预测他们每个人未来的击球率，那么有一个程序比简单地从三个单独的平均数中推断更好。这里“better”的意思很重。无论球员的真实击球能力如何，采用斯坦方法的统计学家都可以期望更准确地预测未来的平均水平。
Sometimes a mathematical result is strikingly contrary to generally held belief even though an obvi ously valid proof is given. Charles Stein of Stanford University discovered such a paradox in statistics in 1955. His result undermined a century and a half of work on estimation theory, going back to Karl Friedrich Gauss and Adrien Ma rie Legendre. After a long period of re sistance to Stein's ideas, punctuated by frequent and sometimes angry debate, the sense of paradox has diminished and Stein's ideas are being incorporated into applied and theoretical statistics. Stein's paradox concerns the use of ob served averages to estimate unobserv able quantities. Averaging is the second most basic process in statistics, the first being the simple act of counting. A base ball player who gets seven hits in 20 offi cial times at bat is said to have a batting average of. 350. In computing this sta tistic we are forming an estimate of the player's true batting ability in terms of his observed average rate of success. Asked how well the player will do in his next 100 times at bat, we would proba bly predict 35 more hits. In traditional statistical theory it can be proved that no other estimation rule is uniformly better than the observed average. The paradoxical element in Stein's re sult is that it sometimes contradicts this elementary law of statistical theory. If we have three or more baseball players, and if we are interested in predicting fu ture batting averages for each of them, then there is a procedure that is better than simply extrapolating from the three separate averages. Here" better" has a strong meaning. The statistician who employs Stein's method can expect to predict the future averages more ac curately no matter what the true bat ting abilities of the players may be.