Power Bounds and Asymptotic Minimax Results for One-sample Rank Tests

Abstract

Let $X_1,\cdots, X_N$ be identically independently distributed with the common continuous distribution function $F$, and let $r_1 < r_2 < \cdots < r_s$ denote the ordered ranks of the positive $X$'s among $|X_1|,\cdots, |X_N|$. Two problems are considered. The first is the location problem where the null hypothesis $H_0$ that $F$ is symmetric about zero (or more generally $F(x) \geqq 1 - F(-x)$ for all $x$) is tested against the alternative that $X$ is stochastically larger than $-X$ (i.e. $F(x) \leqq 1 - F(-x)$ for all $x$ with strict inequality for some $x$). Departure from the null hypothesis will be measured in terms of the distribution difference $D(x) = \lbrack 1 - F(-x) \rbrack - F(x)$, i.e. the difference of the distributions of $-X$ and $X$. We let $\delta(F) = \sup_x |D(x)|$ denote the Kolmogorov distance between these distributions and consider the class $\Omega(\Delta) = \{F:D(x) \geqq 0$ and $\delta (F) \geqq \Delta\}$ of one-sided alternatives with this distance at least $\Delta$. Lower bounds on the power of monotone rank tests are given for $F$ in $\Omega(\Delta)$ (Theorem 2.1), and similarly, upper bounds on the power of monotone rank tests are found for $F$ in $\bar{\Omega}(\Delta) = \{F:\delta(F) \leqq \Delta\}$ (see Theorem 2.2, Corollary 2.2 and Corollary 2.3). $\Omega(\Delta)$ and $\bar{\Omega}(\Delta)$ are of interest for paired comparison and one-sample experiments in which the location of $F$ is the main concern. Note that the above location alternatives are not necessarily shift alternatives. The second problem considered is the symmetry problem where the hypothesis that $F$ is symmetric about zero is to be tested against the alternative that it is skewed to the right. Lower (Theorem 3.1) and upper (Corollary 3.2) bounds on the power of monotone rank tests are found for $F$ in $\Omega_s(\Delta) = \{F:F \in\Omega (\Delta)$ and $F$ has median $0\}$ and $\bar{\Omega}_s(\Delta) = \{ F:\delta(F) \leqq \Delta$ and $F$ has median $0\}$, respectively. Hoeffding (1951) and Ruist (1954) considered alternative classes of the type $\Gamma(q) = \{F:F$ is symmetric and $(F(0) - \frac{1}{2}) \leqq q\}$ and found that the Sign test is minimax (maximizes the minimum power, i.e., minimizes the maximum risk = (1-power)) for $\Gamma(q)$. For the location alternative, we consider the problem of maximizing the minimum power over $\Omega(\Delta)$ and find (Theorem 4.1) that for a class of statistics of the form $\sum^S_{i=1} J_N(r_i/(N + 1))$, a solution is asymptotically given by the statistics $T(\Delta) = W + \lbrack (\frac{1}{2} \Delta (N - 1)-1)/(N + 1) \rbrack S$, where $S$ is the number of positive $X$'s and where $W = \sum^s_{i=1} r_i/(N + 1)$ is the one-sample Wilcoxon statistic. This result contrasts with the two-sample result of [6] in which the Wilcoxon statistic is asymptotically the uniformly (in $\Delta$) unique minimax solution. For the symmetry problem, a class of statistics of the form $\sum^s_{i=1} J(r_i/(N + 1))$ is considered, and it is shown (Theorem 4.2) that $V = W - \frac{1}{2} S$ asymptotically maximizes the minimum power over $\Omega_s(\Delta)$. $T(\Delta)$ and $V$ are functions of the one-sample Wilcoxon statistic $W$ and the Sign statistic $S$. Such statistics have also been considered by Ruist (1954). See also Hodges and Lehmann (1962, page 495). $V$ has been considered by Gross (1966, page 76) and is asymptotically equivalent to the statistic considered by Gupta (1967). The power bounds are tabulated exactly or estimated using Monte Carlo methods for sample sizes 10 and 20 in Section 2B. In particular, the minimum power over $\Omega(\Delta)$ of the statistics $W, T^{(1)} = T(\Delta_1)$ and $T^{(2)} = T(\Delta_2)$, where $\Delta_1 = (1.645/(3N)^{\frac{1}{2}})^{\frac{1}{2}}$ and $\Delta_2 = (2.326/(3N)^{\frac{1}{2}})^{\frac{1}{2}}$, is given or estimated using Monte Carlo power methods. These choices of $\Delta$ are discussed in Section 4, Remark 4.3. The results show that for $N = 10, T^{(1)}$ and $T^{(2)}$ do not improve on the minimum power of the Wilcoxon statistic $W$, while for $\alpha = .05$ and $N = 20$, the asymptotic results are in effect in the sense that $T^{(1)}$ and $T^{(2)}$ are improvements on $W$. The statistic $V$ is similarly compared with the statistic [Gross (1966)] $T_G = \sum^S_{i=1} r^2_i/(N + 1)^2 - (\frac{1}{3})S$ and it is found (Section 3) that in terms of minimum power over $\Omega_s(\Delta), V$ is much better than $T_G$ already for sample sizes $N = 10$ and 20. Power bounds similar to the ones obtained in this paper have been obtained by Birnbaum (1953) and Chapman (1958) for the goodness-of-fit problem, by Bell, Moser and Thompson (1966) for the two-sample problem, and by Bell and Doksum (1967) for the independence problem. In Section 5, tables of the null distributions of $T^{(1)}, V$ and $T_G$ and given for $N \leqq 10$, and the Monte Carlo powers of $T^{(1)}, T^{(2)}$ and $W$ are compared for normal, double exponential and logistic shift alternatives for $N = 10$ and 20. For the double exponential distribution, $T^{(1)}$ and $T^{(2)}$ are slightly better than $W$ when $N = 20$. For the normal distribution, $W$ is slightly better than $T^{(1)}$ and $T^{(2)}$. Thus $T^{(1)}$ and $T^{(2)}$ appear to be better than $W$ for "heavy" tail shift alternatives, while the opposite holds for "light" tail shift alternatives. Although there is essentially no difference in the power of $T^{(1)}$ and $T^{(2)}$ for the various models considered, $T^{(1)}$ is recommended because the normal approximation to the rejection limits of $(N + 1)T^{(1)} = (N + 1)W + \lbrack .487(N - 1)N^{-\frac{1}{4}}- 1\rbrack S$ are closer to the true limits.