Estimation and Confidence Sets for Sparse Normal Mixtures

By Tommaso Cai, Jiashun Jin, Mark G. Low. 2006

110 citations recorded by the source corpus.

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.

Subfields: Artificial Intelligence, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/dc5c799d20933dc0d997801f9db104c4c3a2022d

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Citations in the corpus

Asymptotic Properties of Bayes Risk of a General Class of Shrinkage Priors in Multiple Hypothesis Testing Under Sparsity (2013, background): “See, for example, Ingster (1997), Donoho and Jin (2004), Meinshausen and Rice (2006), Cai et al. (2007) and Cai and Jin (2010), in this context.”
Optimal rates of convergence for estimating the null density and proportion of nonnull effects in large-scale multiple testing (2010, background): “See Efron (2004, 2008) and Jin and Cai (2007) for estimating fnull and ε, and see Genovese and Wasserman (2004), Meinshausen and Rice (2006), Cai, Jin and Low (2007), Jin (2008) and Celisse and Robin (2008) for estimating ε [also see Storey (2002), Efron et al. (2001), Swanepoel (1999)]. Unfortunate”
Optimal detection of heterogeneous and heteroscedastic mixtures (2011, methodology): “For example, the detection problem is the starting point for understanding estimation and large-scale multiple testing (Cai et al., 2007). The fundamental limit for detection is intimately related to the fundamental limit for classification, and the optimal procedures for detection are related to op”
PROPERTIES OF HIGHER CRITICISM UNDER STRONG DEPENDENCE (2008, background): “…and Wasserman [11], Donoho and Jin [8], Efron et al. [10] and Storey, Dai and Leek [26]; (2) Swanepoel [27], Efron et al. [10], Cai, Jin and Low [4], Genovese and Wasserman [11], Storey, Dai and Leek [26], Jin [18], Jin and Cai [19], Jin, Peng and Wang [21] and Meinshausen and Rice [24]; (3)…”
Proportion of non‐zero normal means: universal oracle equivalences and uniformly consistent estimators (2008, background): “In this paper, we concentrate on the problem of estimating the proportion of nonzero normal means (see Cai et al. (2007), Jin and Cai (2007) and Jin et al. (2007) for related studies on other settings).”
The Bayes oracle and asymptotic optimality of multiple testing procedures under sparsity (2010, methodology): “In the case of empirical Bayes methods, the asymptotic results given in [8] illustrate that consistent estimation of the mixture parameters is possible when pm ∝ m−β , with β ∈ (0,1).”
Estimating the Null and the Proportion of Nonnull Effects in Large-Scale Multiple Comparisons (2006, background): “[4] largely focuses on a very sparse setting, and so a more specific model is needed.”
Estimation of a two‐component mixture model with applications to multiple testing (2012, methodology): “In this section we draw a connection between the lower confidence bound developed in Section 4 and the Higher Criticism method of Donoho and Jin (2004) for detection of sparse heterogeneous mixtures.”
Detection boundary in sparse regression (2010, result): “This question is closely related to the previous work by several authors on detection and classification boundaries for the Gaussian sequence model [4, 8, 9, 10, 11, 12, 13, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24].”