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5This article concerns tests for sphericity when data dimension is larger than the sample size. The existing multivariate-sign-based procedure (Hallin & Paindaveine, 2006) for sphericity is not robust against high dimensionality, producing tests with type I error rates much larger than nominal levels. This is mainly due to bias from estimating the location parameter. We develop a correction that makes the existing test statistic robust against high dimensionality. We show that the proposed test statistic is asymp-10 totically normal under elliptical distributions. The proposed method allows dimensionality to increase as the square of sample size. Simulations show that it has good size and power for a wide range of settings.

[1] Characteristics of convective and dynamical instabilities in the mesopause region (between 85 and 100 km) over Maui, Hawaii (20.7°N, 156.3°W) are investigated using 19 nights, $133 hours of high-resolution wind and temperature data obtained by the University of Illinois Na wind/temperature lidar during the Maui Mesosphere and Lower Thermosphere (Maui MALT) campaigns. The mean probabilities of convective and dynamical instabilities are observed to be $3 and 10%, respectively, but there is considerable night-to-night variation. At any given time the probability that an unstable condition is found at some altitudes in the 85-100 km range is $90%. The Maui MALT data exhibit a distinct trend for N 2 to increase with wind shear and vice versa. This correlation has important implications in the understanding of the development of instabilities. The night of 11 April 2002 is studied in detail in order to investigate the spatial and temporal structures of N 2 , wind shear, and convective and dynamical instabilities. A close linkage between instability and the mesosphere inversion layers (MILs) is identified. Most of the convectively and dynamically unstable regions are located above the MILs, with a tendency for dynamical instability to develop below convective instability. It is found that the vertical variations of N 2 are often correlated with those of wind shear, but with a phase shift such that the maxima and minima of N 2 are located $0.5-1 km below those of wind shear. Because of this shift, dynamical instability tends to develop in the region above the maximum wind shear, where relatively small N 2 is observed to be associated with large wind shear. We also found that the wind shear is dominated by the contribution of the meridional wind, especially when the wind shear is strong. Possible mechanisms for the observed features are discussed.

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This article concerns tests for the two-sample location problem when data dimension is larger than the sample size. Existing multivariate-sign-based procedures are not robust against high dimensionality, producing tests with type I error rates far away from nominal levels. This is mainly due to the biases from estimating location parameters. We propose a novel test to overcome this issue by using the "leave-one-out" idea. The proposed test statistic is scalarinvariant and thus is particularly useful when different components have different scales in high-dimensional data. Asymptotic properties of the test statistic are studied. Compared with other existing approaches, simulation studies show that the proposed method behaves well in terms of sizes and power.