Optimal Rates of Convergence for Estimating the Null Density and
Proportion of non-Null Effects in Large-Scale Multiple Testing
The Annals of Statistics 38, 100-145, (2010).Tony Cai and Jiashun Jin
Optimal Rates of Convergence for Estimating the Null Density and
Proportion of non-Null Effects in Large-Scale Multiple Testing
The Annals of Statistics 38, 100-145, (2010).Tony Cai and Jiashun Jin
- Abstract: An important estimation problem that is closely related to large-scale multiple testing is that of estimating the null density and the proportion of non-null 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 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.
- Paper: pdf file.
- Other related paper:
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).
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The Annals of Statistics 35, 2421-2449.