Unlimited simultaneous discrimination intervals in regression.
Unlimited simultaneous discrimination intervals in regression.
复制标题
回归中无限的同时判别间隔。
作者：
Gerald J. Lieberman;Rupert G. Miller;M. A. Hamilton
判别问题可以描述为：统计学家有n对值(x1，yi)，(x2，y2)，…，(xn，yj)，根据这些值估计回归直线Ac+，Fx。他现在观察到K个附加观测值Y*，4Y*，…，Yk，其中对应的自变量值x1‘，4，…，xk未知。统计学家希望估计这些x的值，并用同时的可信区间将它们括起来。Mandel(1958)首先讨论了这个问题，Miller(1966)给出了另一种解决方案。当K未知并且可能任意大时，这些结果不适用。对于任意K的这一问题，给出了无限同时判别区间的解决方案。提出了无限同时判别区间[D(P)，D+*(P)]，其基于相同的估计线性回归，并且具有至少循环百分比的判别区间将包含置信度为1-Ca的真x的性质。本文给出了获得无限同时分辨间隔的两种技术。第一种方法是通过Bonferroni不等式得到的过程，而第二种方法是基于Lieberman&Miller(1963)的思想。最后给出了一个数值算例。对这两种求无限同时判别区间的方法进行了一般性的讨论和比较。
The discrimination problem can be described as follows: The statistician has n pairs of values (xl, YI), (x2, Y2), ... , (xn, YJ) from which he estimates the regression line ac +,fx. He now observes K additional observations Y*, 4Y*, ... , YK for which the corresponding independent variable values x1', 4,..., XK are unknown. The statistician wishes to estimate these values of x and bracket them by means of simultaneous confidence intervals. This problem was first treated by Mandel (1958) and another solution was given by Miller (1966). When K is unknown and possibly arbitrarily large, these results do not apply. A solution to this problem of arbitrary K is given in terms of unlimited simultaneous discrimination intervals. Unlimited simultaneous discrimination intervals [D(P), D+*(P)] are presented which are based upon the same estimated linear regression and which have the property that at least lOOP per cent of the discrimination intervals will contain the true x's with confidence 1-ca. In this paper two techniques for obtaining unlimited simultaneous discrimination intervals are given. The first method is a procedure obtained through the Bonferroni inequality, while the second technique is based upon an idea of Lieberman & Miller (1963). A numerical example is analyzed. A general discussion and comparison of the two methods for finding unlimited simultaneous discrimination intervals is given.