Estimating the Null and the Proportion of non-Null effects in Large-scale Multiple Comparisons

An important issue raised by Efron in the context of large-scale multiple comparisons is that in many applications the usual assumption that the null distribution is known is incorrect, and seemingly negligible differences in the null may result in large differences in subsequent studies. This suggests that a careful study of estimation of the null is indispensable. In this paper, we consider the problem of estimating a null normal distribution, and a closely related problem, estimation of the proportion of non-null effects. We develop an approach based on the empirical characteristic function and Fourier analysis. The estimators are shown to be uniformly consistent over a wide class of parameters. Numerical performance of the estimators is investigated using both simulated and real data. In particular, we apply our procedure to the analysis of breast cancer and HIV microarray data sets. The estimators perform favorably in comparison to existing methods.Comment: 42 pages, 6 figure

Optimal rates of convergence for estimating the null density and proportion of nonnull effects in large-scale multiple testing

An important estimation problem that is closely related to large-scale multiple testing is that of estimating the null density and the proportion of nonnull effects. A few estimators have been introduced in the literature; however, several important problems, including the evaluation of the minimax rate of convergence and the construction of rate-optimal estimators, remain open. In this paper, we consider optimal estimation of the null density and the proportion of nonnull effects. Both minimax lower and upper bounds are derived. The lower bound is established by a two-point testing argument, where at the core is the novel construction of two least favorable marginal densities $f_1$ and $f_2$. The density $f_1$ is heavy tailed both in the spatial and frequency domains and $f_2$ is a perturbation of $f_1$ such that the characteristic functions associated with $f_1$ and $f_2$ match each other in low frequencies. The minimax upper bound is obtained by constructing estimators which rely on the empirical characteristic function and Fourier analysis. The estimator is shown to be minimax rate optimal. Compared to existing methods in the literature, the proposed procedure not only provides more precise estimates of the null density and the proportion of the nonnull effects, but also yields more accurate results when used inside some multiple testing procedures which aim at controlling the False Discovery Rate (FDR). The procedure is easy to implement and numerical results are given.Comment: Published in at http://dx.doi.org/10.1214/09-AOS696 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Nonlinear microwave response of MgB2

We calculate the intrinsic nonlinear microwave response of the two gap superconductor MgB2 in the clean and dirty limits. Due to the small value of the pi band gap, the nonlinear response at low temperatures is larger than for a single gap Bardeen-Cooper-Schrieffer (BCS) s-wave superconductor with a transition temperature of 40 K. Comparing this result with the intrinsic nonlinear d-wave response of YBa2Cu3O7 (YBCO) we find a comparable response at temperatures around 20 K. Due to its two gap nature, impurity scattering in MgB2 can be used to reduce the nonlinear response if the scattering rate in the pi band is made larger than the one in the sigma band.Comment: 4 pages, 4 figure

Effect and Compensation of Timing Jitter in Through-Wall Human Indication via Impulse Through-Wall Radar

Impulse through-wall radar (TWR) is considered as one of preferred choices for through-wall human indication due to its good penetration and high range resolution. Large bandwidth available for impulse TWR results in high range resolution, but also brings an atypical adversity issue not substantial in narrowband radars — high timing jitter effect, caused by the non-ideal sampling clock at the receiver. The fact that impulse TWR employs very narrow pulses makes little jitter inaccuracy large enough to destroy the signal correlation property and then degrade clutter suppression performance. In this paper, we focus on the timing jitter impact on clutter suppression in through-wall human indication via impulse TWR. We setup a simple timing jitter model and propose a criterion namely average range profile (ARP) contrast is to evaluate the jitter level. To combat timing jitter, we also develop an effective compensation method based on local ARP contrast maximization. The proposed method can be implemented pulse by pulse followed by exponential average background subtraction algorithm to mitigate clutters. Through-wall experiments demonstrate that the proposed method can dramatically improve through-wall human indication performance

Estimation and confidence sets for sparse normal mixtures

For high dimensional statistical models, researchers have begun to focus on situations which can be described as having relatively few moderately large coefficients. Such situations lead to some very subtle statistical problems. In particular, Ingster and Donoho and Jin have considered a sparse normal means testing problem, in which they described the precise demarcation or detection boundary. Meinshausen and Rice have shown that it is even possible to estimate consistently the fraction of nonzero coordinates on a subset of the detectable region, but leave unanswered the question of exactly in which parts of the detectable region consistent estimation is possible. In the present paper we develop a new approach for estimating the fraction of nonzero means for problems where the nonzero means are moderately large. We show that the detection region described by Ingster and Donoho and Jin turns out to be the region where it is possible to consistently estimate the expected fraction of nonzero coordinates. This theory is developed further and minimax rates of convergence are derived. A procedure is constructed which attains the optimal rate of convergence in this setting. Furthermore, the procedure also provides an honest lower bound for confidence intervals while minimizing the expected length of such an interval. Simulations are used to enable comparison with the work of Meinshausen and Rice, where a procedure is given but where rates of convergence have not been discussed. Extensions to more general Gaussian mixture models are also given.Comment: Published in at http://dx.doi.org/10.1214/009053607000000334 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Chiral rings and GSO projection in Orbifolds

The GSO projection in the twisted sector of orbifold background is sometimes subtle and incompatible descriptions are found in literatures. Here, from the equivalence of partition functions in NSR and GS formalisms, we give a simple rule of GSO projection for the chiral rings of string theory in \C^r/\Z_n, $r=1,2,3$. Necessary constructions of chiral rings are given by explicit mode analysis.Comment: 24 page