Modified Wilcoxon-Mann-Whitney tests of stochastic dominance

Abstract

Given independent samples from two univariate distributions, the one-sided Wilcoxon-Mann-Whitney statistic may be used to conduct a rank-based test of first-order stochastic dominance. We broaden the scope of applicability of such tests by showing that the bootstrap may be used to conduct valid inference in a matched pairs sampling framework permitting dependence between the two samples. Further, we show that a modified bootstrap incorporating an implicit estimate of a contact set may be used to improve power. Numerical simulations indicate that the modified bootstrap effectively controls the null rejection frequencies and delivers improved power, particularly in settings where there is strong dependence between matched pairs. We provide a brief empirical illustration involving Canadian family income data.

Figures (6)

• Figure 1: A P-P plot for two samples of equal size $n_1=n_2=10$. The one-sided WMW statistic is equal to the red shaded area multiplied by $\sqrt{n_1n_2/(n_1+n_2)}=\sqrt{5}$.
• Figure 2: Null rejection frequencies using independent samples of size $n=500$ and nominal level $\alpha=.05$. P-P curves parametrized by $\gamma$, shifting away from the 45-degree line as $\gamma$ increases, are displayed in top-left and bottom-left panels. Rejection frequencies for the modified WMW test are displayed in top-center and bottom-center panels, with $\tau_n=.5,.75,1,1.25,1.5,\infty$, and triangles superimposed on the curves for $\tau_n=\infty$. Top-right and bottom-right panels show rejection frequencies for the modified WMW test with $\tau_n=.75$ and for the DH test with tuning parameter $-.135$, with asterisks superimposed on the curve for the latter test.
• Figure 3: Null rejection frequencies using matched pair samples of size $n=500$ and nominal level $\alpha=.05$. The left, center and right columns of panels correspond to $\rho=.25,.5,.75$. Rejection frequencies are plotted for the modified WMW test with $\tau_n=.75$ (superimposed with squares) and with $\tau_n-\infty$ (superimposed with trangles), and the DH test with tuning parameter $-.139$ (superimposed with asterisks). The P-P curves, parametrized by $\gamma$, are displayed in top-left and bottom-left panels of \ref{['fig:sizeinterior']}.
• Figure 4: Alternative rejection frequencies using independent samples of size $n=500$ and nominal level $\alpha=.05$. P-P curves parametrized by $\gamma$, shifting away from the 45-degree line as $\gamma$ increases in magnitude, are displayed in top-left and bottom-left panels. Rejection frequencies for the modified WMW test are displayed in top-center and bottom-center panels, with $\tau_n=.5,.75,1,1.25,1.5,\infty$, and triangles superimposed on the curves for $\tau_n=\infty$. Top-right and bottom-right panels show rejection frequencies for the modified WMW test with $\tau_n=.75$ and for the DH test with tuning parameter $-.139$, with asterisks superimposed on the curves for the DH test.
• Figure 5: Alternative rejection frequencies using matched pair samples of size $n=500$ and nominal level $\alpha=.05$. The left, center and right panels correspond to $\rho=.25,.5,.75$. Rejection frequencies are plotted for the modified WMW test with $\tau_n=.75$ (superimposed with squares) and with $\tau_n-\infty$ (superimposed with trangles), and the DH test with tuning parameter $-.139$ (superimposed with asterisks). The P-P curves, parametrized by $\gamma$, are displayed in top-left and bottom-left panels of \ref{['fig:power']}.

Theorems & Definitions (22)

• Lemma 2.1: BK26
• Proposition 2.1
• Lemma 3.1: BK26
• Proposition 3.1
• Proposition 3.2
• Lemma 8.1
• proof
• proof : Proof of \ref{['prop:F']}
• Lemma 8.2
• Lemma 8.3