Some Nonparametric Tests of Whether the Largest Observations of a Set are too Large or too Small

Abstract

Let us consider a large number $n$ of observations which are statistically independent and drawn from continuous symmetrical populations. This paper presents some nonparametric tests of whether the $r$ largest observations of the set are too large to be consistent with the hypothesis that these populations have a common median value. Tests of whether the $r$ largest observations are too small to be consistent with this hypothesis are also considered. Here $r$ is a given integer which is independent of $n$. Subject to some weak restrictions, it is shown that the significance level of a test of the type presented tends to a value $\alpha$ as $n$ increases. For no admissible value of $n$, however, does the significance level of this test exceed $2\alpha$. If whether the largest observations are too large is considered, tests with values of $\alpha$ suitable for significance levels can be obtained for $r \geq 4$. Values of $\alpha$ suitable for significance levels can be obtained for any value of $r$ if whether the largest observations are too small is investigated ($n$ large). Properties of the power functions of these tests are considered for the special case in which the $r$ largest observations are from populations with common median $\theta$, the remaining observations are from populations with common median $\phi$, and each population has the property that the distribution of the quantity (sample value) - (population median) is independent of the value of the population median. For tests of $\theta > \phi$, the power function tends to zero as $\theta - \phi \rightarrow - \infty$ and to unity as $\theta - \phi \rightarrow \infty$. For tests of $\phi > \theta$, the power function tends to unity as $\theta - \phi \rightarrow - \infty$ and to zero as $\theta - \phi \rightarrow \infty$. Analogous tests of whether the smallest observations of a set are too small or too large can be obtained from the tests of the largest observations by symmetry considerations. If there is strong reason to believe that the set of observations is a random sample from a continuous population, the tests presented in this paper can be used to decide whether the population is symmetrical. Tests of this nature are sensitive to symmetry in the tails of the population but not to symmetry in the central part.