概率论与数理统计(英文)第九章

174 9. Nonparametric Statistics Statistical analyses that do not depend upon the knowledge of the distribution and parameters of the population are called nonparametric or distribution-free methods. 不依赖于总体的分布及其参数的统计方法称为非参数方法或非分布方法。 9.1 Sign Test 符号检验 1 The simplest of all nonparametric methods is the sign test, which is usually used to test the significance of the difference between two means in a paired experiment. 最简单的非参数检验是符号检验 检验两个总体均值差的显著程度 It is particularly suitable when the various pairs are observed under different conditions, a case in which the assumption of normality may not hold. However, because of its simplicity, the sign test is often used even though the populations are normally distributed. As is implied by its name in this test only the sign of the difference 175 between the paired variates is used. 若两个总体的均值相等，那么符号‘＋’、‘－’的概率一样。 D = sign of (X1－X2 ) If p denotes the probability of a difference D being positive and q the probability of its being negative, we have as hy pothesis p=1/2. appropriate test statistic is X , X~B(n, p)， X --- N(‘+”) To test the hypothesis 012:0H 112:0H we will reject 0H in favor of 1H only if the proportion of plus signs is sufficiently less than 1/2, that is , when the value x of our random variable is small. Hence, if the computed P-value 12()PP Xx when p is less than or equal to the significance level  , we reject 0H in favor of 1H . To test the hypothesis 012:0H 112:0H 176 we reject 0H in favor 1H when the proportion of plus signs is significantly less than or significantly greater than 1/2. This, of course, is equivale...