Ex. 18.17

Equivalence between Benjamini–Hochberg and plug-in methods.

(a) In the notation of Algorithm 18.2, show that for rejection threshold \(p_0=p_{(L)}\), a proportion of at most \(p_0\) of the permuted values \(t_j^k\) exceed \(|T|_{(L)}\) where \(|T|_{(L)}\) is the \(L\)th largest value among the \(|t_j|\). Hence show that the plug-in FDR estimate \(\widehat{\text{FDR}}\) is less than or equal to \(p_0\cdot M/L = \alpha\).

(b) Show that the cut-point \(|T|_{(L+1)}\) produces a test with estimated FDR greater than \(\alpha\).

Soln. 18.17

(a) Note that \(p_{(1)} \le p_{(2)} \le \cdots \le p_{(M)}\) and the definition of \(p_j\) in (18.41), we know \(p_{(L)}\) corresponds to \(T_{(L)}\), that is,

\[\begin{equation} p_0 = p_{(L)} = \frac{1}{MK}\sum_{j' = 1}^M \sum_{k=1}^K \bb{I}(|t_{j'}^k| > t_{(L)}).\non \end{equation}\]

Therefore, the proportion of the permuted values \(t_j^k\) exceed \(|T|_{(L)}\) is at most \(p_0\).

Recall (18.46) in Algorithm 18.3, we have

\[\begin{eqnarray} \widehat{\text{FDR}} &=& \frac{\widehat{E(V)}}{R_{\text{obs}}}\non\\ &=&\frac{\frac{1}{K}\sum_{j=1}^M\sum_{k=1}^K\bb{I}(|t_j^k| > |T|_{(L)})}{\sum_{j=1}^M\bb{I}(|t_j| > |T|_{(L)})}\non\\ &\le &\frac{p_0\cdot M}{L-1}\non\\ &=&\alpha. \label{eq:18-17a} \end{eqnarray}\]

Note that the last equality assumes \(\alpha = \frac{p_0\cdot M}{L-1}\) instead of \(\frac{p_0\cdot M}{L}\) defined in the text. Otherwise I don't see how to prove the claimed result.

(b) It follows the same arguments as \(\eqref{eq:18-17a}\) by noting the the proportion of the permuted values \(t_j^k\) exceed \(|T|_{(L+1)}\) is greater than \(p_0\).