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 article we consider the problem of estimating a null normal distribution, and a closely related problem, estimation of the proportion of nonnull 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. We investigate the numerical performance of the estimators using both simulated and real data. In particular, we apply our procedure to the analysis of breast cancer and human immunodeficiency virus microarray datasets. The estimators perform favorably compared with existing methods.