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

By Tommaso Cai, Jiashun Jin. 2010

80 citations recorded by the source corpus.

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.

Subfields: Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/a9082bbfcc2b4374f4367d47e14ed76d73cf4d70

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Citations in the corpus

Asymptotically minimax empirical Bayes estimation of a sparse normal mean vector (2013, background): “2010; Cai and Zhou 2012; Lam and Fan 2009), large-scale multiple testing (Bogdan et al. 2011; Cai and Jin 2010), and function estimation (Cai 2012; Johnstone and Silverman 2005).”
Asymptotic Properties of Bayes Risk of a General Class of Shrinkage Priors in Multiple Hypothesis Testing Under Sparsity (2013, methodology): “Note that the above estimator of τ is truncated below by 1 m and hence it is not susceptible to collapsing to zero, which is a major concern for the use of such empirical Bayes approaches as mentioned in Carvalho et al. (2009), Scott and Berger (2010), Bogdan et al.”
Distribution-free Multiple Testing (2016, background): “There is work in that direction in [7, 21].”
The Bayes oracle and asymptotic optimality of multiple testing procedures under sparsity (2010, methodology): “Similar Gaussian mixture models for multiple testing were considered, for example, in [7] and [16].”
Estimation of a two‐component mixture model with applications to multiple testing (2012, background): “…and confidence bounds for α have been proposed in the literature under certain assumptions on Fs and its density; see for example, Storey (2002), Genovese and Wasserman (2004), Meinshausen and Rice (2006), Meinshausen and Bühlmann (2005), Celisse and Robin (2010) and Langaas et al. (2005).”
Large-scale inference (2010, methodology): “Jin and Cai (2007, 2009) present a characteristic function approach to estimating the empirical null, while a more parametric normal mixture method is analyzed in Muralidharan (2009a).”