\begin{algorithm}[]
	\caption{Signal Proportion Estimator}\label{alg}	
	\begin{algorithmic}[1]
		\Statex {\bf Input:} $p$-values of the observed test statistics and bounding sequences $c_{m, 0.5}$ and $c_{m, 1}$
		\Statex {\bf Output:} a proportion estimate $\hat \pi$
		\State Rank the variables by their $p$-values so that $p_{(1)} < p_{(2)} < \ldots < p_{(m)}$	
		\State Compute 
		\[
		\hat{\pi}_{0.5}= \max_{1 < j < m}\frac{ j/m-p_{(j)}-c_{m,0.5} \cdot \sqrt{p_{(j)}}} {1-p_{(j)}} \qquad \mbox{and} \qquad \hat{\pi}_{1}= \max_{1 < j < m}\frac{ j/m-p_{(j)} -c_{m,1} \cdot p_{(j)}} {1-p_{(j)}}
		\]
		\State Obtain $\hat \pi = \max\{\hat{\pi}_{0.5}, \hat{\pi}_{1}\}$
	\end{algorithmic}
\end{algorithm}	

\begin{algorithm}
[]
\caption{Signal Proportion Estimator} \begin{algorithmic}
[1]
\Statex {\bf Input:} $p$-values of the observed test statistics and bounding sequences $c_{m, 0.5}$ and $c_{m, 1}$
\Statex {\bf Output:} a proportion estimate $\hat \pi$
\State Rank the variables by their $p$-values so that $p_{(1)} < p_{(2)} < \ldots < p_{(m)}$
\State Compute
\[
\hat{\pi}_{0.5}= \max_{1 < j < m}\frac{ j/m-p_{(j)}-c_{m,0.5} \cdot \sqrt{p_{(j)}}} {1-p_{(j)}} \qquad \mbox{and} \qquad \hat{\pi}_{1}= \max_{1 < j < m}\frac{ j/m-p_{(j)} -c_{m,1} \cdot p_{(j)}} {1-p_{(j)}}
\]
\State Obtain $\hat \pi = \max\{\hat{\pi}_{0.5}, \hat{\pi}_{1}\}$
\end{algorithmic}
\end{algorithm}
{
    "title": "weak signal inclusion under dependence and applications in genome-wide   association study",
    "id": "2212.13574",
    "abstract": "motivated by the inquiries of weak signals in underpowered genome-wide association studies (gwass), we consider the problem of retaining true signals that are not strong enough to be individually separable from a large amount of noise. we address the challenge from the perspective of false negative control and present false negative control (fnc) screening, a data-driven method to efficiently regulate false negative proportion at a user-specified level. fnc screening is developed in a realistic setting with arbitrary covariance dependence between variables. we calibrate the overall dependence through a parameter whose scale is compatible with the existing phase diagram in high-dimensional sparse inference. utilizing the new calibration, we asymptotically explicate the joint effect of covariance dependence, signal sparsity, and signal intensity on the proposed method. we interpret the results using a new phase diagram, which shows that fnc screening can efficiently select a set of candidate variables to retain a high proportion of signals even when the signals are not individually separable from noise. finite sample performance of fnc screening is compared to those of several existing methods in simulation studies. the proposed method outperforms the others in adapting to a user-specified false negative control level. we implement fnc screening to empower a two-stage gwas procedure, which demonstrates substantial power gain when working with limited sample sizes in real applications.",
    "categories": "stat.me",
    "doi": "",
    "created": "2022-12-27",
    "updated": "2024-02-02",
    "authors": [
        "x. jessie jeng",
        "yifei hu",
        "quan sun",
        "yun li"
    ],
    "affiliation": [],
    "url": "https://arxiv.org/abs/2212.13574"
}
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