SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Table A5: The FDR and power performance of the four methods with effect size  $\alpha_{j} = 0.2$ ,  $\beta_{j} = 0.3$  under dense alternatives scenario. The nominal FDR level is 0.05, and the number of mediators is 15000.
Method	FDR	Power
n=200	AMDP	0.0471	0.0331
JS-mixture	0.0348	0.0206
DACT (Efron)	0.0449	0.0379
DACT (JC)	0.1156	0.1673
n=500	AMDP	0.0460	0.4768
JS-mixture	0.0504	0.4168
DACT (Efron)	0.0333	0.3488
DACT (JC)	0.1640	0.6839
n=1000	AMDP	0.0487	0.8734
JS-mixture	0.0541	0.8208
DACT (Efron)	0.0315	0.7636
DACT (JC)	0.0818	0.8674
n=5000	AMDP	0.0501	1
JS-mixture	0.0544	1
DACT (Efron)	0.0116	1
DACT (JC)	0.1085	1
sampling distribution of  $(p_{1j}, p_{2j})$  is symmetrical around  $p^{(1)} = 0.5$  and  $p^{(2)} = 0.5$ . Under  $H_{01}$ ,  $p_{1j}$  still obeys the uniform distribution, but  $p_{2j}$  does not, leading to  $(p_{1j}, p_{2j})$  being only symmetrical about  $p^{(1)} = 0.5$  on  $[0,1] \times [0,1]$ . Similarly, under  $H_{10}$ ,  $p_{2j}$  obeys the uniform distribution, but  $p_{1j}$  does not, resulting in  $(p_{1j}, p_{2j})$  being only symmetrical about  $p^{(2)} = 0.5$  on  $[0,1] \times [0,1]$ . It is essential to emphasize that Assumption 1 specifically applies to the null mediators.
For Assumption 2, a non-null p-value theoretically lies within  $[0,0.5)$ . Therefore, as the sample size  $n$  tends to infinity, the probability of p-values under alternative hypotheses falling within  $[0.5,1]$  approaches zero. For example, as  $n$  goes to infinity, p-values under  $H_{11}$  and  $H_{10}$  are not expected to fall within the region  $\tilde{D}_{01} = [0.5,1] \times [0,0.5)$  because non-null  $p_{1j}$  not lies within  $[0.5,1]$  theoretically, therefore the region  $\tilde{D}_{01}$  only contains p-values under  $H_{00}$  and  $H_{01}$ . Similarly, the region  $\tilde{D}_{10}$  theoretically only includes p-values under  $H_{00}$  and  $H_{10}$ .
E Proofs
E.1 Proof of Theorem 1
For any rejection region  $S \in [0,1]^2$ , the global FDR in mediation analysis is defined as follows  $\mathrm{gFDR}(S) = \mathbb{P}(H_{00} \cup H_{01} \cup H_{10} = 1 \mid p_j \in S)$
$$
\begin{array}{l} = \frac {\pi_ {0 0} \mathbb {P} (p _ {j} \in S \mid H _ {0 0} = 1) + \pi_ {0 1} \mathbb {P} (p _ {j} \in S \mid H _ {0 1} = 1) + \pi_ {1 0} \mathbb {P} (p _ {j} \in S \mid H _ {1 0} = 1)}{\pi_ {0 0} \mathbb {P} (p _ {j} \in S \mid H _ {0 0} = 1) + \pi_ {0 1} \mathbb {P} (p _ {j} \in S \mid H _ {0 1} = 1) + \pi_ {1 0} \mathbb {P} (p _ {j} \in S \mid H ^ {\prime} _ {1 0} = 1) + \pi_ {1 1} \mathbb {P} (p _ {j} \in S \mid H _ {1 1} = 1)} \\ = \frac {\pi_ {0 0} \int_ {S} f _ {0 0} (p) d p + \pi_ {0 1} \int_ {S} f _ {0 1} (p) d p + \pi_ {1 0} \int_ {S} f _ {1 0} (p) d p}{\pi_ {0 0} \int_ {S} f _ {0 0} (p) d p + \pi_ {0 1} \int_ {S} f _ {0 1} (p) d p + \pi_ {1 0} \int_ {S} f _ {1 0} (p) d p + \pi_ {1 1} \int_ {S} f _ {1 1} (p) d p}. \tag {A.1} \\ \end{array}
$$
We introduce some notations
$$
D _ {0 0} (S) = \int_ {S} f _ {0 0} (p) d p, D _ {0 1} (S) = \int_ {S} f _ {0 1} (p) d p, D _ {1 0} (S) = \int_ {S} f _ {1 0} (p) d p, D _ {1 1} (S) = \int_ {S} f _ {1 1} (p) d p.
$$
Thus,  $\mathrm{gFDR}(S)$  is transformed into
$$
\begin{array}{l} \operatorname {g F D R} (S) = \frac {\pi_ {0 0} D _ {0 0} (S) + \pi_ {0 1} D _ {0 1} (S) + \pi_ {1 0} D _ {1 0} (S)}{\pi_ {0 0} D _ {0 0} (S) + \pi_ {0 1} D _ {0 1} (S) + \pi_ {1 0} D _ {1 0} (S) + \pi_ {1 1} D _ {1 1} (S)} \tag {A.2} \\ = \frac {1}{1 + \left\{D _ {1 1} (S) / \left(\gamma_ {0 0} D _ {0 0} (S) + \gamma_ {0 1} D _ {0 1} (S) + \gamma_ {1 0} D _ {1 0} (S)\right) \right\}}, \\ \end{array}
$$
where  $\gamma_{00} = \frac{\pi_{00}}{\pi_{11}},\gamma_{01} = \frac{\pi_{01}}{\pi_{11}},\gamma_{10} = \frac{\pi_{10}}{\pi_{11}}$  For any threshold  $\zeta \in (0,1]$  , define the rejection region  $S(\zeta)$  as
$$
\begin{array}{l} S (\zeta) = \left\{p: \frac {\pi_ {0 0} f _ {0 0} (p) + \pi_ {0 1} f _ {0 1} (p) + \pi_ {1 0} f _ {1 0} (p)}{\pi_ {0 0} f _ {0 0} (p) + \pi_ {0 1} f _ {0 1} (p) + \pi_ {1 0} f _ {1 0} (p) + \pi_ {1 1} f _ {1 1} (p)} \leq \zeta \right\} \tag {A.3} \\ = \left\{p: \frac {1}{1 + \left\{f _ {1 1} (p) / \left(\gamma_ {0 0} f _ {0 0} (p) + \gamma_ {0 1} f _ {0 1} (p) + \gamma_ {1 0} f _ {1 0} (p)\right) \right\}} \leq \zeta \right\}. \\ \end{array}
$$
Here we prove that  $\mathrm{gFDR}(S(\zeta))$  is a non-decreasing function of  $\zeta$ . Suppose  $\zeta_2 > \zeta_1$ , considering two cases:
Case 1  $\nu (S(\zeta_2) - S(\zeta_1)) = 0$ . We derive that  $\mathrm{gFDR}(S(\zeta_1)) = \mathrm{gFDR}(S(\zeta_2))$ .
Case 2  $\nu (S(\zeta_2) - S(\zeta_1)) > 0$ . We can prove that  $\mathrm{gFDR}(S(\zeta))$  is a non-decreasing function of  $\zeta$  if
$$
\begin{array}{l} \frac {D _ {1 1} \left(S \left(\zeta_ {2}\right) - S \left(\zeta_ {1}\right)\right)}{\gamma_ {0 0} D _ {0 0} \left(S \left(\zeta_ {2}\right) - S \left(\zeta_ {1}\right)\right) + \gamma_ {0 1} D _ {0 1} \left(S \left(\zeta_ {2}\right) - S \left(\zeta_ {1}\right)\right) + \gamma_ {1 0} D _ {1 0} \left(S \left(\zeta_ {2}\right) - S \left(\zeta_ {1}\right)\right)} \tag {A.4} \\ <   \frac {D _ {1 1} (S (\zeta_ {1}))}{\gamma_ {0 0} D _ {0 0} (S (\zeta_ {1})) + \gamma_ {0 1} D _ {0 1} (S (\zeta_ {1})) + \gamma_ {1 0} D _ {1 0} (S (\zeta_ {1}))} \\ \end{array}
$$
holds, the reason is as follows. Let
$$
m _ {1} = \sup  \left\{\frac {f _ {1 1} (p)}{\gamma_ {0 0} f _ {0 0} (p) + \gamma_ {0 1} f _ {0 1} (p) + \gamma_ {1 0} f _ {1 0} (p)}: p \in S (\zeta_ {2}) - S (\zeta_ {1}) \right\},
$$
$$
m _ {2} = \inf  \left\{\frac {f _ {1 1} (p)}{\gamma_ {0 0} f _ {0 0} (p) + \gamma_ {0 1} f _ {0 1} (p) + \gamma_ {1 0} f _ {1 0} (p)}: p \in S \left(\zeta_ {1}\right) \right\}.
$$
By the definition of region  $S(\zeta)$ , we have  $m_2 > m_1$  obviously. Therefore, we have
$$
\begin{array}{l} \frac {D _ {1 1} (S (\zeta_ {1}))}{\gamma_ {0 0} D _ {0 0} (S (\zeta_ {1})) + \gamma_ {0 1} D _ {0 1} (S (\zeta_ {1})) + \gamma_ {1 0} D _ {1 0} (S (\zeta_ {1}))} \\ \geq m _ {2} \frac {\gamma_ {0 0} D _ {0 0} (S (\zeta_ {1})) + \gamma_ {0 1} D _ {0 1} (S (\zeta_ {1})) + \gamma_ {1 0} D _ {1 0} (S (\zeta_ {1}))}{\gamma_ {0 0} D _ {0 0} (S (\zeta_ {1})) + \gamma_ {0 1} D _ {0 1} (S (\zeta_ {1})) + \gamma_ {1 0} D _ {1 0} (S (\zeta_ {1}))} \tag {A.5} \\ > m _ {1} \frac {\gamma_ {0 0} D _ {0 0} (S (\zeta_ {2}) - S (\zeta_ {1})) + \gamma_ {0 1} D _ {0 1} (S (\zeta_ {2}) - S (\zeta_ {1})) + \gamma_ {1 0} D _ {1 0} (S (\zeta_ {2}) - S (\zeta_ {1}))}{\gamma_ {0 0} D _ {0 0} (S (\zeta_ {2}) - S (\zeta_ {1})) + \gamma_ {0 1} D _ {0 1} (S (\zeta_ {2}) - S (\zeta_ {1})) + \gamma_ {1 0} D _ {1 0} (S (\zeta_ {2}) - S (\zeta_ {1})))} \\ \geq \frac {D _ {1 1} (S (\zeta_ {2}) - S (\zeta_ {1}))}{\gamma_ {0 0} D _ {0 0} (S (\zeta_ {2}) - S (\zeta_ {1})) + \gamma_ {0 1} D _ {0 1} (S (\zeta_ {2}) - S (\zeta_ {1})) + \gamma_ {1 0} D _ {1 0} (S (\zeta_ {2}) - S (\zeta_ {1}))}. \\ \end{array}
$$
Furthermore, we decompose the region  $S(\zeta_2)$  as follows
$$
\begin{array}{l} \frac {D _ {1 1} (S (\zeta_ {2}))}{\gamma_ {0 0} D _ {0 0} (S (\zeta_ {2})) + \gamma_ {0 1} D _ {0 1} (S (\zeta_ {2})) + \gamma_ {1 0} D _ {1 0} (S (\zeta_ {2}))} \\ = \left\{D _ {1 1} \left(S \left(\zeta_ {2}\right) - S \left(\zeta_ {1}\right)\right) + D _ {1 1} \left(S \left(\zeta_ {1}\right)\right) \right\} / \left\{ \begin{array}{l} \gamma_ {0 0} D _ {0 0} \left(S \left(\zeta_ {2}\right) - S \left(\zeta_ {1}\right)\right) + \gamma_ {0 1} D _ {0 1} \left(S \left(\zeta_ {2}\right) - S \left(\zeta_ {1}\right)\right) \\ + \gamma_ {1 0} D _ {1 0} \left(S \left(\zeta_ {2}\right) - S \left(\zeta_ {1}\right)\right) + \gamma_ {0 0} D _ {0 0} \left(S \left(\zeta_ {1}\right)\right) \\ + \gamma_ {0 1} D _ {0 1} \left(S \left(\zeta_ {1}\right)\right) + \gamma_ {1 0} D _ {1 0} \left(S \left(\zeta_ {1}\right)\right) \end{array} \right\}. \tag {A.6} \\ \end{array}
$$
Combined with (A.4), we obtain
$$
\frac {D _ {1 1} \left(S \left(\zeta_ {1}\right)\right)}{\gamma_ {0 0} D _ {0 0} \left(S \left(\zeta_ {1}\right)\right) + \gamma_ {0 1} D _ {0 1} \left(S \left(\zeta_ {1}\right)\right) + \gamma_ {1 0} D _ {1 0} \left(S \left(\zeta_ {1}\right)\right)} > \frac {D _ {1 1} \left(S \left(\zeta_ {2}\right)\right)}{\gamma_ {0 0} D _ {0 0} \left(S \left(\zeta_ {2}\right)\right) + \gamma_ {0 1} D _ {0 1} \left(S \left(\zeta_ {2}\right)\right) + \gamma_ {1 0} D _ {1 0} \left(S \left(\zeta_ {2}\right)\right)}. \tag {A.7}
$$
Moreover, by the definition of  $\mathrm{gFDR}(S)$ , it holds that
$$
\operatorname {g F D R} \left(S \left(\zeta_ {1}\right)\right) <   \operatorname {g F D R} \left(S \left(\zeta_ {2}\right)\right).
$$
Under the Assumption (ii) in Theorem 1, for a given  $\alpha \in (0,1)$ , there exists a threshold  $\zeta^{\star} > 0$ , s.t.  $\mathrm{gFDR}(S(\zeta^{\star})) = \alpha$ . For the ease of presentation, we denote  $S(\zeta^{\star})$  as  $S^{\star}$ . In the following, we will prove that  $S^{\star}$  is the optimal rejection region.
Considering any set  $T$  that satisfies  $D_{11}(T) > D_{11}(S^{\star})$ . Let  $R_{T} = T - S^{\star}$  and  $R_{S} = S^{\star} - T$ . We can derive that
$$
D _ {1 1} (T) = D _ {1 1} \left(T \cap S ^ {\star}\right) + D _ {1 1} \left(R _ {T}\right), \tag {A.8}
$$
$$
D _ {1 1} (S) = D _ {1 1} \left(T \cap S ^ {\star}\right) + D _ {1 1} \left(R _ {S}\right).
$$
Then, we have  $D_{11}(R_T) > D_{11}(R_S)$ . By the definition of  $S^{\star}$ , we have
$$
\left. \inf  \left\{\frac {\gamma_ {0 0} f _ {0 0} (p) + \gamma_ {0 1} f _ {0 1} (p) + \gamma_ {1 0} f _ {1 0} (p)}{f _ {1 1} (p)}: p \in R _ {T} \right\} > \sup  \left\{\frac {\gamma_ {0 0} f _ {0 0} (p) + \gamma_ {0 1} f _ {0 1} (p) + \gamma_ {1 0} f _ {1 0} (p)}{f _ {1 1} (p)}: p \in R _ {S} \right\}. \right. \tag {A.9}
$$
$$
\frac {\gamma_ {0 0} D _ {0 0} \left(R _ {T}\right) + \gamma_ {0 1} D _ {0 1} \left(R _ {T}\right) + \gamma_ {1 0} D _ {1 0} \left(R _ {T}\right)}{D _ {1 1} \left(R _ {T}\right)} > \frac {\gamma_ {0 0} D _ {0 0} \left(R _ {S}\right)) + \gamma_ {0 1} D _ {0 1} \left(R _ {S}\right) + \gamma_ {1 0} D _ {1 0} \left(R _ {S}\right)}{D _ {1 1} \left(R _ {S}\right)}. \tag {A.10}
$$
In a similar way, we can derive that
$$
\frac {\gamma_ {0 0} D _ {0 0} \left(R _ {T}\right) + \gamma_ {0 1} D _ {0 1} \left(R _ {T}\right) + \gamma_ {1 0} D _ {1 0} \left(R _ {T}\right)}{D _ {1 1} \left(R _ {T}\right)} > \frac {\gamma_ {0 0} D _ {0 0} \left(T \cap S ^ {\star}\right)) + \gamma_ {0 1} D _ {0 1} \left(\left(T \cap S ^ {\star}\right) + \gamma_ {1 0} D _ {1 0} \left(\left(T \cap S ^ {\star}\right) \right. \right.}{D _ {1 1} \left(\left(T \cap S ^ {\star}\right)\right)}. \tag {A.11}
$$
Finally, we have
$$
\begin{array}{l} \frac {\gamma_ {0 0} D _ {0 0} (T) + \gamma_ {0 1} D _ {0 1} (T) + \gamma_ {1 0} D _ {1 0} (T)}{D _ {1 1} (T)} \\ = \frac {\gamma_ {0 0} D _ {0 0} (T \cap S ^ {\star}) + \gamma_ {0 1} D _ {0 1} (T \cap S ^ {\star}) + \gamma_ {1 0} D _ {1 0} (T \cap S ^ {\star}) + \gamma_ {0 0} D _ {0 0} (R _ {T}) + \gamma_ {0 1} D _ {0 1} (R _ {T}) + \gamma_ {1 0} D _ {1 0} (R _ {T})}{D _ {1 1} (T \cap S ^ {\star}) + D _ {1 1} (R _ {T})} \\ > \frac {\gamma_ {0 0} D _ {0 0} (T \cap S ^ {\star}) + \gamma_ {0 1} D _ {0 1} (T \cap S ^ {\star}) + \gamma_ {1 0} D _ {1 0} (T \cap S ^ {\star}) + \gamma_ {0 0} D _ {0 0} (R _ {S}) + \gamma_ {0 1} D _ {0 1} (R _ {S}) + \gamma_ {1 0} D _ {1 0} (R _ {S})}{D _ {1 1} (T \cap S ^ {\star}) + D _ {1 1} (R _ {S})} \\ = \frac {\gamma_ {0 0} D _ {0 0} \left(S ^ {\star}\right) + \gamma_ {0 1} D _ {0 1} \left(S ^ {\star}\right) + \gamma_ {1 0} D _ {1 0} \left(S ^ {\star}\right)}{D _ {1 1} \left(S ^ {\star}\right)}. \tag {A.12} \\ \end{array}
$$
The second inequality holds because  $D_{11}(R_T) > D_{11}(R_S)$ , implying  $\mathrm{gFDR}(T) > \mathrm{gFDR}(S(\zeta^{\star})) = \alpha$ . Therefore, we can conclude that the rejection region  $S(\zeta^{\star})$  is optimal.
E.2 Proof of Theorem 2
E.2.1 The consistent estimator of local FDR
To justify Assumption (i) for the corresponding local FDR estimator in Theorem 2, we first prove the consistency of local FDR estimator under  $L_{\infty}$  norm in Proposition 1.
Proposition 1. Assume that the smoothing parameter  $b$  satisfies
$$
\lim  _ {J \to \infty} b = 0 a n d \lim  _ {J \to \infty} J b ^ {2} = + \infty .
$$
$$
\sup  _ {p _ {j} \in [ 0, 1 ] ^ {2}} \left| \widehat {\operatorname {f d r}} (p _ {j}) - \operatorname {f d r} (p _ {j}) \right| \xrightarrow {P} 0 a s n, J \to \infty .
$$
Let  $g$  be a probability density on  $[0,1]$ , and  $\hat{g}$  be the beta kernel estimator:
$$
\hat {g} \left(p ^ {(i)}\right) = J ^ {- 1} \sum_ {j = 1} ^ {J} K _ {p ^ {(i)}, b} ^ {\star} \left(p _ {i j}\right), \quad i = 1, 2.
$$
To prove the consistency of the beta kernel estimator  $\hat{g}$ , i.e
$$
\sup  _ {p ^ {(i)} \in [ 0, 1 ]} \left| \hat {g} \left(p ^ {(i)}\right) - g \left(p ^ {(i)}\right)\right| \xrightarrow {P} 0 \text {a s} J \rightarrow \infty , \tag {A.13}
$$
we first need to establish the uniform convergence of its bias on the interval  $[0, 1]$ .
Lemma 1. Let  $g$  be the probability density on  $[0,1]$ , and  $\hat{g}$  be the beta kernel estimator. We have
$$
\sup  _ {p ^ {(i)} \in [ 0, 1 ]} \left| \operatorname {E} \left\{\hat {g} (p ^ {(i)}) \right\} - g (p ^ {(i)}) \right| \to 0 a s b \to 0, \quad i = 1, 2.
$$
Proof of Lemma 1. Without loss of generality, we replace  $p^{(i)}, i = 1,2$  with  $p$  for simplifying the proof steps, and discuss three cases in the following.
Case 1  $p \in (2b, 1 - 2b)$  Denote  $\mu_1$  and  $\sigma_1^2$  are mean and variance of  $P$ , a variable following  $\mathrm{Beta}(p / b, (1 - p) / b)$ . According to Johnson et al. [26], there exists a constant  $C$  such that
$$
\mu_ {1} = p, \tag {A.14}
$$
$$
\sigma_ {1} ^ {2} = b p (1 - p) + R _ {2} (p), \tag {A.15}
$$
where  $R_{2}(p) \leq C b^{2}$ . Because  $f$  is a probability density on  $[0,1]$ , for  $\varepsilon > 0$ , there exists a  $\delta > 0$  such that
$$
\left| g (t) - g (p) \right| <   \varepsilon \quad \text {f o r} | p - t | <   \delta \tag {A.16}
$$
for all  $p\in (2b,1 - 2b)$  ; According to (A.14), we have
$$
\left| \mu_ {1} - p \right| <   \delta / 2 \text {f o r a l l} p \in (2 b, 1 - 2 b). \tag {A.17}
$$
Therefore, we can derive that
$$
\begin{array}{l} | E \left\{\hat {g} (p) \right\} - g (p) | = \left| \int_ {2 b} ^ {1 - 2 b} \left\{g (t) - g (p) \right\} K \left(t, \frac {p}{b}, \frac {1 - p}{b}\right) d t \right| \\ \leq \int_ {| t - \mu_ {1} | <   \delta / 2} | g (t) - g (p) | K \left(t, \frac {p}{b}, \frac {1 - p}{b}\right) d t \\ + \int_ {| t - \mu_ {1} | > \delta / 2} | g (t) - g (p) | K \left(t, \frac {p}{b}, \frac {1 - p}{b}\right) d t \\ \leq \int_ {| t - \mu_ {1} | <   \delta / 2} | g (t) - g (p) | K \left(t, \frac {p}{b}, \frac {1 - p}{b}\right) d t \\ + 2 \sup  _ {t \in (2 b, 1 - 2 b)} | g (t) | \int_ {| t - \mu_ {1} | > \delta / 2} K \left(t, \frac {p}{b}, \frac {1 - p}{b}\right) d t \\ \equiv \mathcal {M} _ {1} + \mathcal {M} _ {2}. \\ \end{array}
$$
According (A.16) and (A.17), we obtain
$$
\mathcal {M} _ {1} \leq \varepsilon . \tag {A.18}
$$
Combining the Chebyshev's inequality and (A.15), and there also exists  $b_{\varepsilon}$  such that
$$
\mathcal {M} _ {2} \leq \left\{8 \sup  _ {t \in (2 b, 1 - 2 b)} | g (t) | \sigma_ {1} ^ {2} \right\} / \delta^ {2} \leq \left\{2 \sup  _ {t \in (2 b, 1 - 2 b)} | g (t) | (b + 4 C b ^ {2}) \right\} / \delta^ {2} \leq \varepsilon \text {f o r a l l} b \leq b _ {\varepsilon}. \tag {A.19}
$$
Thus, from (A.18) and (A.19), we conclude that
$$
\sup  _ {p \in (2 b, 1 - 2 b)} | \operatorname {E} \left\{\hat {g} (p) \right\} - g (p) | <   2 \varepsilon \quad \text {f o r a l l} b \leq b _ {\varepsilon}.
$$
Case 2  $p \in [0, 2b]$  Based on the notations of Case 1, we have
$$
\mu_ {2} = p + \xi (p, b), \tag {A.20}
$$
$$
\sigma_ {2} ^ {2} = R _ {2} (p), \tag {A.21}
$$
where  $\mu_{2}$  and  $\sigma_2^2$  are mean and variance of P, a variable following Beta  $(\rho (p,b),(1 - p) / b)$ ,  $\xi (p,b) = (1 - p)\{\rho (p,b) - p / b\} /\{1 + b\rho (p,b) - p\}$ , and  $R_{2}(p)\leq Cb^{2}$ . For  $\varepsilon >0$ , there exists a  $\delta >0$  such that
$$
\left| g (t) - g (p) \right| <   \varepsilon \quad \text {f o r} | p - t | <   \delta \tag {A.22}
$$
for all  $p \in [0, 2b]$ ; According to (A.20), since  $\xi(p, b)$  is a bounded function for  $p \in [0, 2b]$ , there also exists  $b_{\delta}$  such that
$$
\left| \mu_ {2} - p \right| <   \delta / 2 \text {f o r} b \leq b _ {\delta} \quad \text {f o r a l l} p \in [ 0, 2 b ]. \tag {A.23}
$$
$$
\begin{array}{l} \left| E \left\{\hat {g} (p) \right\} - g (p) \right| = \left| \int_ {0} ^ {2 b} \left\{g (t) - g (p) \right\} K \left(t, \rho (p, b), \frac {1 - p}{b}\right) d t \right| \\ \leq \int_ {| t - \mu_ {2} | <   \delta / 2} | g (t) - g (p) | K \left(t, \rho (p, b), \frac {1 - p}{b}\right) d t \\ + \int_ {| t - \mu_ {2} | > \delta / 2} | g (t) - g (p) | K \left(t, \rho (p, b), \frac {1 - p}{b}\right) d t \\ \leq \int_ {| t - \mu_ {2} | <   \delta / 2} | g (t) - g (p) | K \left(t, \rho (p, b), \frac {1 - p}{b}\right) d t \\ + 2 \sup  _ {t \in [ 0, 2 b ]} | g (t) | \int_ {| t - \mu_ {2} | > \delta / 2} K \left(t, \rho (p, b), \frac {1 - p}{b}\right) d t \\ \equiv \mathcal {M} _ {1} + \mathcal {M} _ {2}. \\ \end{array}
$$
According to (A.22) and (A.23), there exists  $b_{\varepsilon}^{(1)}$  such that
$$
\mathcal {M} _ {1} \leq \varepsilon \text {f o r a l l} b \leq b _ {\varepsilon} ^ {(1)}. \tag {A.24}
$$
Combining the Chebyshev's inequality and (A.21), and there also exists  $b_{\varepsilon}^{(2)}$  such that
$$
\mathcal {M} _ {2} \leq \left\{8 \sup  _ {t \in [ 0, 2 b ]} | g (t) | \sigma_ {2} ^ {2} \right\} / \delta^ {2} \leq \left\{8 \sup  _ {t \in [ 0, 2 b ]} | g (t) | C b ^ {2} \right\} / \delta^ {2} \leq \varepsilon \text {f o r a l l} b \leq b _ {\varepsilon} ^ {(2)}. \tag {A.25}
$$
Thus, from (A.24) and (A.25), we conclude that
$$
\sup_{p\in [0,2b]}|\operatorname {E}\left\{\hat{g} (p)\right\} -g(p)| <   2\varepsilon \quad \text{for all} b\leq \min \left(b_{\varepsilon}^{(1)},b_{\varepsilon}^{(2)}\right).
$$
Case 3  $p \in [1 - 2b, 1]$  Case 3 can be proven a similar procedure. We note that
$$
\mu_ {3} = p - b \cdot \xi (1 - p, b), \tag {A.26}
$$
$$
\sigma_ {3} ^ {2} = R _ {2} (p), \tag {A.27}
$$
where  $\mu_3$  and  $\sigma_3^2$  are mean and variance of  $\mathrm{P}$ , a variable following  $\mathrm{Beta}(p / b, \rho(1 - p, b))$ , and  $R_2(p) \leq C b^2$ .
This completes the proof of Lemma 1.
Proof of Proposition 1. To prove the consistency of the beta kernel estimator, we use the inequality:
$$
\sup  _ {p \in [ 0, 1 ]} | \hat {g} (p) - g (p) | \leq \sup  _ {p \in [ 0, 1 ]} | \hat {g} (p) - \mathrm {E} \left\{\hat {g} (p) \right\} | + \sup  _ {p \in [ 0, 1 ]} | \mathrm {E} \left\{\hat {g} (p) \right\} - g (p) |. \tag {A.28}
$$
From Lemma 1, the second term converges to zero. In the following, we prove that
$$
\sup  _ {p \in [ 0, 1 ]} | \hat {g} (p) - \mathrm {E} \left\{\hat {g} (p) \right\} | \xrightarrow {P} 0 \text {a s} J \rightarrow \infty . \tag {A.29}
$$
We also consider three cases:
Case 1  $p \in (2b, 1 - 2b)$  The beta kernel estimator  $\hat{g}(p)$  is expressed as
$$
\hat {g} (p) = \int_ {2 b} ^ {1 - 2 b} K \left(t, \frac {p}{b}, \frac {1 - p}{b}\right) d F _ {n} (t), \tag {A.30}
$$
where  $F_{n}$  is the empirical distribution. The expectation of the beta kernel estimator is
$$
E \left\{\hat {g} (p) \right\} = \int_ {2 b} ^ {1 - 2 b} K \left(t, \frac {p}{b}, \frac {1 - p}{b}\right) d F (t). \tag {A.31}
$$
Thus, for  $p \in (2b, 1 - 2b)$ , we can derive that
$$
\begin{array}{l} \left| \hat {g} (p) - \mathrm {E} \left\{\hat {g} (p) \right\} \right| = \left| \int_ {2 b} ^ {1 - 2 b} K \left(t, \frac {p}{b}, \frac {1 - p}{b}\right) d \left\{F _ {n} (t) - F (t) \right\} \right| \tag {A.32} \\ \leq \sup  _ {t \in (b, 1 - 2 b)} | F _ {n} (t) - F (t) | \int_ {2 b} ^ {1 - 2 b} \left| d K \left(t, \frac {p}{b}, \frac {1 - p}{b}\right) \right|. \\ \end{array}
$$
Note that the integral in (A.32) is bounded above by
$$
\frac {1 - b}{b} \int_ {2 b} ^ {1 - 2 b} \left| K \left(t, \frac {p}{b} - 1, \frac {1 - p}{b}\right) - K \left(t, \frac {p}{b}, \frac {1 - p}{b} - 1\right) \right| d t \leq 2 \frac {1 - b}{b}. \tag {A.33}
$$
$$
| \hat {g} (p) - E \left\{\hat {g} (p) \right\} | \leq 2 \frac {1 - b}{b} \sup  _ {t \in (2 b, 1 - 2 b)} \left| F _ {n} (t) - F (t) \right|. \tag {A.34}
$$
From Dvoretzky et al. [16], we can obtain
$$
\begin{array}{l} \left. \right. \operatorname {P} \left[ \sup  _ {p \in (2 b, 1 - 2 b)} | \hat {g} (p) - \mathrm {E} \left\{\hat {g} (p) \right\} | \geq \varepsilon \right] \leq \operatorname {P} \left\{\sup  _ {t \in (2 b, 1 - 2 b)} \left| F _ {n} (t) - F (t) \right| \geq \frac {\varepsilon}{2} \cdot \frac {b}{1 - b} \right\} \tag {A.35} \\ \leq 2 \exp \left\{- J \frac {\varepsilon^ {2}}{2} \frac {b ^ {2}}{(1 - b) ^ {2}} \right\}. \\ \end{array}
$$
By utilizing the Borel-Cantelli Lemma, it is shown that under the beta kernel estimator is consistent.
Case 2  $p \in [0, 2b]$  Case 2 can be proven a similar procedure of Case 1. Note that, for all  $p \in [0, 2b]$ ,
$$
\begin{array}{l} \left| \hat {g} (p) - \mathrm {E} \left\{\hat {g} (p) \right\} \right| = \left| \int_ {0} ^ {2 b} K (t, \rho (p, b), \frac {1 - p}{b}) d \left\{F _ {n} (t) - F (t) \right\} \right| \tag {A.36} \\ \leq \sup  _ {t \in [ 0, 2 b ]} | F _ {n} (t) - F (t) | \int_ {0} ^ {2 b} \left| d K \left(t, \rho (p, b), \frac {1 - p}{b}\right) \right|. \\ \end{array}
$$
Since  $\rho(p, b)$  is monotonic increasing in  $[0, 2b]$ ,  $\rho(0, b) = 1$ ,  $\rho(2b, b) = 2$ . For  $p \in (0, 2b]$ , the integral in (A.36) is bounded above by  $2\frac{1 + b}{b}$ . For  $p = 0$ , it is bounded above by
$$
\left. \left(\rho (p, b) + \frac {1 - p}{b} - 1\right) \int_ {0} ^ {2 b} \left| K \left(t, \rho (p, b), \frac {1 - p}{b} - 1\right) \right| d t = \frac {1 + b}{b}. \right. \tag {A.37}
$$
Thus, we can obtain
$$
\begin{array}{l} \left. \right. \operatorname {P} \left[ \sup  _ {p \in [ 0, 2 b ]} | \hat {g} (p) - \mathrm {E} \left\{\hat {g} (p) \right\} | \geq \varepsilon \right] \leq \operatorname {P} \left\{\sup  _ {t \in [ 0, 2 b ]} | F _ {n} (t) - F (t) | \geq \frac {\varepsilon}{2} \cdot \frac {b}{1 + b} \right\} \tag {A.38} \\ \leq 2 \exp \left\{- J \frac {\varepsilon^ {2}}{2} \frac {b ^ {2}}{(1 + b) ^ {2}} \right\}, \\ \end{array}
$$
which concludes the proof of the consistency of beta kernel estimator in Case 2.
Case 3  $p \in [1 - 2b, 1]$  Case 3 can be proven a similar procedure of Case 1. Note that for all  $p \in [1 - 2b, 1]$ ,
$$
\begin{array}{l} | \hat {g} (p) - \mathrm {E} \left\{\hat {g} (p) \right\} | = \left| \int_ {1 - 2 b} ^ {1} K \left(t, \frac {p}{b}, \rho (1 - p, b)\right) d \left\{F _ {n} (t) - F (t) \right\} \right| \tag {A.39} \\ \leq \sup  _ {t \in [ 1 - 2 b, 1 ]} | F _ {n} (t) - F (t) | \int_ {1 - 2 b} ^ {1} \left| d K \left(t, \frac {p}{b}, \rho (1 - p, b)\right) \right|. \\ \end{array}
$$
For  $p \in [1 - 2b, 1)$ , the integral in (A.39) is bounded above by  $2\frac{1 + b}{b}$ . And for  $p = 1$ , it is bounded above by
$$
\left. \left(\rho (1 - p, b) + \frac {p}{b} - 1\right) \int_ {1 - 2 b} ^ {1} \left| K \left(t, \frac {p}{b} - 1, \rho (1 - p, b)\right) \right| d t = \frac {1 + b}{b}. \right. \tag {A.40}
$$
Thus for all  $p\in [1 - 2b,1]$
$$
| \hat {g} (p) - E \left\{\hat {g} (p) \right\} | \leq 2 \frac {1 + b}{b} \sup  _ {t \in [ 1 - 2 b, 1 ]} | F _ {n} (t) - F (t) |. \tag {A.41}
$$
Similarly, we obtain
$$
\begin{array}{l} \left. \right. \operatorname {P} \left[ \sup  _ {p \in [ 1 - 2 b, 1 ]} | \hat {g} (p) - \mathrm {E} \left\{\hat {g} (p) \right\} | \geq \varepsilon \right] \leq \operatorname {P} \left\{\sup  _ {t \in [ 1 - 2 b, 1 ]} | F _ {n} (t) - F (t) | \geq \frac {\varepsilon}{2} \cdot \frac {b}{1 + b} \right\} \tag {A.42} \\ \leq 2 \exp \left\{- J \frac {\varepsilon^ {2}}{2} \frac {b ^ {2}}{(1 + b) ^ {2}} \right\}, \\ \end{array}
$$
which concludes the proof of the consistency of beta kernel estimator in Case 3.
From Dai et al. [12], for a fixed  $J$  and  $\lambda$ , the biases of  $\hat{\pi}_{00}$ ,  $\hat{\pi}_{10}$ , and  $\hat{\pi}_{01}$  go to zero as  $n \to \infty$ :
$$
\lim  _ {n \rightarrow \infty} \hat {\pi} _ {0 0} = \pi_ {0 0}, \quad \lim  _ {n \rightarrow \infty} \hat {\pi} _ {0 1} = \pi_ {0 1}, \quad \lim  _ {n \rightarrow \infty} \hat {\pi} _ {1 0} = \pi_ {1 0}. \tag {A.43}
$$
And we can derive