Some Significance Tests Based on Order Statistics

Abstract

In this paper significance tests are developed whose application requires only the determination of one order statistic and the computation of sums of sample values. The simplest case considered is that of testing a new sample value $x$ on the basis of $m$ previous sample values $y_1, \cdots, y_m,$ all sample values being assumed from normal populations with the same variance. Two separate tests of whether the mean of the new population from which $x$ was taken exceeds the mean of the population from which $y_1, \cdots, y_m$ were drawn consist in accepting the alternative that the new population mean exceeds the old population mean if \begin{equation*}\tag{1} x > \big(\frac{\sqrt{m + 1} + 1}{m}\big)\sum^m_1 y_i - \sqrt{m + 1} y_{(u)}\end{equation*} \begin{equation*}\tag{2} x > \big(\frac{\sqrt{m + 1} - 1}{m}\big)\sum^m_1 y_i + \sqrt {m + 1} y_{(m+1-u)},\end{equation*} where $y_{(u)}$ is the $u$th largest of $y_1, \cdots, y_m$. It can be shown that both of these tests have the same power so that either one might be equally well selected for use. In practical application, however, there may exist reasons for preferring one test to the other. Similarly, the alternative that the new population mean is less than the old population mean will be accepted if \begin{equation*}\tag{3} x < \big( \frac{\sqrt{m+1} + 1}{m}\big) \sum^m_1 y_i - \sqrt{m + 1} y_{(m+1-u)}\end{equation*} \begin{equation*}\tag{4} x < \big(\frac{\sqrt {m + 1} - 1}{m}\big)\sum^m_1 y_i + \sqrt{m + 1} y_{(u)}.\end{equation*} All four of these significance tests have the same power, also the same significance level $\alpha (u, m)$. By appropriate choice of $u$ and $m$ the significance level can be made to assume values suitable for significance tests. For example, $\alpha (1, 6) = .0156,\quad\alpha (2, 10) = .0107$ $\alpha (3, 13) = .0110,\quad \alpha (4, 16) = .0107.$ The above tests are still valid if each of $x, y_1, \cdots, y_m$ equals a sum of $r$ sample values. These order statistic tests are generalized to the case where $x$ is a sum of $r$ new sample values; $y_1, \cdots, y_m$ each equals a sum of $s$ past sample values and another sum of relatively weighted past sample values is utilized but not as an order statistic. The introduction of this relatively weighted sum allows less reliable past information to be lumped together and weighted according to its relative importance. In comparing the order statistic tests with the most powerful tests which could be used for these alternatives it is found that the size of the samples used must be increased in order to bring the efficiency of the order statistic test up to that of the corresponding most powerful test. Thus the advisability of using the order statistic test will depend upon whether it is more desirable to take larger samples but have less computation.