In this paper, we consider optimal estimation of the null density and the proportion of non-null 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 f₁ and f₂ . The density f₁ is heavy-tailed both in the spatial and frequency domains and f₂ is a perturbation of f₁ such that the characteristic functions associated with f₁ and f₂ 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 non-null effects, but also yields more accurate results in subsequent studies including the control of the False Discovery Rate (FDR). The procedure is easy to implement. Numeric results are reported both with simulated data and SNP data on the Parkinson's Disease.

Sun, W. & Cai, T. (2007).
Oracle and adaptive compound decision rules for false discovery rate control.
J. American Statistical Association 102, 901-912.

Jin, J. & Cai, T. (2007).
Estimating the null and the proportion of non-null effects in large-scale multiple comparisons.
J. American Statistical Association 102, 495-506.

Cai, T., Jin, J. & Low, M. (2007).
Estimation and confidence sets for sparse normal mixtures.
The Annals of Statistics 35, 2421-2449.