Confidence interval for normal means in meta-analysis based on a pretest estimator

Abstract

Meta-analysis is a statistical method to summarize quantitative results from a set of published studies. Recently, a frequentist estimator was proposed for individual studies’ means in terms of the pretest (preliminary test) estimator. However, the confidence interval has not been considered yet for the pretest estimator for meta-analysis. In this paper, a novel approach to construct a CI is considered based on the pretest estimator for meta-analysis. By pivoting the cumulative distribution function of the pretest estimator, we define an explicit formula of the CI. Furthermore, we show that the coverage probability of the CI controls the nominal confidence level both theoretically and numerically. To facilitate the proposed estimator and CI, we have implemented the computational tools in the R package “meta.shrinkage”. Finally, three datasets are analyzed to illustrate the proposed CI in real meta-analyses. The R code to produce all the numerical results of the paper is given in Supplementary Materials.

Appendices

Appendices

1.1 Appendix A: The proof of Lemma 1

We let ${c}_{{\lambda }_{i}}=P\left({\mu }_{i}\in \left[{\delta }_{i}^{\mathrm{PT}}-{z}_{\gamma /2}{\sigma }_{i}, {\delta }_{i}^{\mathrm{PT}}+{z}_{\gamma /2}{\sigma }_{i}\right]\right)$.

We first consider the case of $\alpha \ge \gamma$ (Case 1). Then

$$\begin{aligned}{c}_{{\lambda_{i} }} =& P\left( { - z_{\gamma /2} \le \frac{{\delta_{i}^{{{\text{PT}}}} - \mu_{i} }}{{\sigma_{i} }} \le z_{\gamma /2} } \right)\\ =& P\left( { - z_{\gamma /2} \le \frac{{Y_{i} - \mu_{i} }}{{\sigma_{i} }} \le z_{\gamma /2} , \left| {\frac{{Y_{i} }}{{\sigma_{i} }}} \right| > z_{\alpha /2} } \right) + P\left( { - z_{\gamma /2} \le \frac{{\mu_{i} }}{{\sigma_{i} }} \le z_{\gamma /2} , \left| {\frac{{Y_{i} }}{{\sigma_{i} }}} \right| \le z_{\alpha /2} } \right)\end{aligned}$$

$$\begin{aligned} =& P\left( { - z_{\gamma /2} \le \frac{{Y_{i} - \mu_{i} }}{{\sigma_{i} }} \le z_{\gamma /2} , \left( {\frac{{Y_{i} - \mu_{i} }}{{\sigma_{i} }} + \frac{{\mu_{i} }}{{\sigma_{i} }} < - z_{\alpha /2}\quad {\text{or}}\quad z_{\alpha /2} < \frac{{Y_{i} - \mu_{i} }}{{\sigma_{i} }} + \frac{{\mu_{i} }}{{\sigma_{i} }}} \right)} \right)\\ &\quad + P\left( { - z_{\gamma /2} \le \frac{{\mu_{i} }}{{\sigma_{i} }} \le z_{\gamma /2} , - z_{\alpha /2} \le \frac{{Y_{i} - \mu_{i} }}{{\sigma_{i} }} + \frac{{\mu_{i} }}{{\sigma_{i} }} \le z_{\alpha /2} } \right).\end{aligned}$$

Putting ${Z}_{i}=\left({Y}_{i}-{\mu }_{i}\right)/{\sigma }_{i}$, we have

$$\begin{aligned}{c}_{{\lambda }_{i}}=&P\left(-{z}_{\gamma /2}\le {Z}_{i}\le {z}_{\gamma /2}, \left({Z}_{i}<-{\lambda }_{i}-{z}_{\alpha /2} \quad \mathrm{or} \quad -{\lambda }_{i}+{z}_{\alpha /2}<{Z}_{i}\right)\right)\\ &+P\left(-{z}_{\gamma /2}\le {\lambda }_{i}\le {z}_{\gamma /2}, -{\lambda }_{i}-{z}_{\alpha /2}\le {Z}_{i}\le -{\lambda }_{i}+{z}_{\alpha /2}\right). \end{aligned}$$

(A1)

Since $\alpha \ge \gamma$ is equivalent to ${z}_{\alpha /2}\le {z}_{\gamma /2}$, one has

$${c}_{{\lambda }_{i}}=\left\{\begin{array}{ll}P\left(-{z}_{\gamma /2}\le {Z}_{i}\le {z}_{\gamma /2}\right) & \quad {\rm if}\quad{\lambda }_{i}<-{z}_{\alpha /2}-{z}_{\gamma /2},\\ P\left(-{z}_{\gamma /2}\le {Z}_{i}<-{\lambda }_{i}-{z}_{\alpha /2}\right) &\quad {\rm if}\quad-{z}_{\alpha /2}-{z}_{\gamma /2}\le {\lambda }_{i}<-{z}_{\gamma /2},\\ P\left(-{z}_{\gamma /2}\le {Z}_{i}<-{\lambda }_{i}-{z}_{\alpha /2}\right)+P\left(-{\lambda }_{i}-{z}_{\alpha /2}\le {Z}_{i}\le -{\lambda }_{i}+{z}_{\alpha /2}\right) & \quad {\rm if}\quad-{z}_{\gamma /2}\le {\lambda }_{i}<{z}_{\alpha /2}-{z}_{\gamma /2},\\ P\left(-{z}_{\gamma /2}\le {Z}_{i}<-{\lambda }_{i}-{z}_{\alpha /2}\right)+P\left(-{\lambda }_{i}+{z}_{\alpha /2}<{Z}_{i}\le {z}_{\gamma /2}\right)\\ \quad +P\left(-{\lambda }_{i}-{z}_{\alpha /2}\le {Z}_{i}\le -{\lambda }_{i}+{z}_{\alpha /2}\right) & \quad {\rm if}\quad{z}_{\alpha /2}-{z}_{\gamma /2}\le {\lambda }_{i}\le {-z}_{\alpha /2}+{z}_{\gamma /2}, \\ P\left(-{\lambda }_{i}+{z}_{\alpha /2}<{Z}_{i}\le {z}_{\gamma /2}\right)+P\left(-{\lambda }_{i}-{z}_{\alpha /2}\le {Z}_{i}\le -{\lambda }_{i}+{z}_{\alpha /2}\right) & \quad {\rm if}\quad{-z}_{\alpha /2}+{z}_{\gamma /2}<{\lambda }_{i}\le {z}_{\gamma /2},\\ P\left(-{\lambda }_{i}+{z}_{\alpha /2}<{Z}_{i}\le {z}_{\gamma /2}\right) & \quad {\rm if}\quad{z}_{\gamma /2}<{\lambda }_{i}\le {z}_{\alpha /2}+{z}_{\gamma /2},\\ P\left(-{z}_{\gamma /2}\le {Z}_{i}\le {z}_{\gamma /2}\right) & \quad {\rm if}\quad{\lambda }_{i}>{z}_{\alpha /2}+{z}_{\gamma /2},\end{array}\right.$$

$$=\left\{\begin{array}{ll}1-\gamma & \quad {\rm if}\quad{\lambda }_{i}<-{z}_{\alpha /2}-{z}_{\gamma /2},\\ \Phi \left(-{\lambda }_{i}-{z}_{\alpha /2}\right)-\frac{\gamma }{2} & \quad {\rm if}\quad-{z}_{\alpha /2}-{z}_{\gamma /2}\le {\lambda }_{i}<-{z}_{\gamma /2},\\ \Phi \left(-{\lambda }_{i}-{z}_{\alpha /2}\right)-\frac{\gamma }{2}+\Phi \left(-{\lambda }_{i}+{z}_{\alpha /2}\right)-\Phi \left(-{\lambda }_{i}-{z}_{\alpha /2}\right) & \quad {\rm if}\quad-{z}_{\gamma /2}\le {\lambda }_{i}<{z}_{\alpha /2}-{z}_{\gamma /2},\\ \Phi \left(-{\lambda }_{i}-{z}_{\alpha /2}\right)-\frac{\gamma }{2}+1-\frac{\gamma }{2}-\Phi \left(-{\lambda }_{i}+{z}_{\alpha /2}\right)\\ \quad +\Phi \left(-{\lambda }_{i}+{z}_{\alpha /2}\right)-\Phi \left(-{\lambda }_{i}-{z}_{\alpha /2}\right) & \quad {\rm if}\quad{z}_{\alpha /2}-{z}_{\gamma /2}\le {\lambda }_{i}\le {-z}_{\alpha /2}+{z}_{\gamma /2},\\ 1-\frac{\gamma }{2}-\Phi \left(-{\lambda }_{i}+{z}_{\alpha /2}\right)+\Phi \left(-{\lambda }_{i}+{z}_{\alpha /2}\right)-\Phi \left(-{\lambda }_{i}-{z}_{\alpha /2}\right) & \quad {\rm if}\quad{-z}_{\alpha /2}+{z}_{\gamma /2}<{\lambda }_{i}\le {z}_{\gamma /2},\\ 1-\frac{\gamma }{2}-\Phi \left(-{\lambda }_{i}+{z}_{\alpha /2}\right) & \quad {\rm if}\quad{z}_{\gamma /2}<{\lambda }_{i}\le {z}_{\alpha /2}+{z}_{\gamma /2},\\ 1-\gamma & \quad {\rm if}\quad{\lambda }_{i}>{z}_{\alpha /2}+{z}_{\gamma /2},\end{array}\right.$$

$$=\left\{\begin{array}{ll}1-\gamma & \quad {\rm if}\quad{\lambda }_{i}<-{z}_{\alpha /2}-{z}_{\gamma /2},\\ \Phi \left(-{\lambda }_{i}-{z}_{\alpha /2}\right)-\frac{\gamma }{2} & \quad {\rm if}\quad-{z}_{\alpha /2}-{z}_{\gamma /2}\le {\lambda }_{i}<-{z}_{\gamma /2},\\ \Phi \left(-{\lambda }_{i}+{z}_{\alpha /2}\right)-\frac{\gamma }{2} & \quad {\rm if}\quad-{z}_{\gamma /2}\le {\lambda }_{i}<{z}_{\alpha /2}-{z}_{\gamma /2},\\ 1-\gamma & \quad {\rm if}\quad{z}_{\alpha /2}-{z}_{\gamma /2}\le {\lambda }_{i}\le {-z}_{\alpha /2}+{z}_{\gamma /2},\\ 1-\frac{\upgamma }{2}-\Phi \left(-{\lambda }_{i}-{z}_{\alpha /2}\right) & \quad {\rm if}\quad{-z}_{\alpha /2}+{z}_{\gamma /2}<{\lambda }_{i}\le {z}_{\gamma /2},\\ 1-\frac{\upgamma }{2}-\Phi \left(-{\lambda }_{i}+{z}_{\alpha /2}\right) & \quad {\rm if}\quad {z}_{\gamma /2}<{\lambda }_{i}\le {z}_{\alpha /2}+{z}_{\gamma /2},\\ 1-\gamma & \quad {\rm if}\quad{\lambda }_{i}>{z}_{\alpha /2}+{z}_{\gamma /2}.\end{array}\right.$$

This completes the proof for Case 1 ($\alpha \ge \gamma$).

Case 2 ($\alpha <\gamma$ and ${z}_{\alpha /2}\le 2{z}_{\gamma /2}$) and Case 3 ($\alpha <\gamma$ and ${z}_{\alpha /2}>2{z}_{\gamma /2}$) are similar to Case 1 ($\alpha \ge \gamma$). In Cases 2 and 3, the formula of ${c}_{{\lambda }_{i}}$ is computed from Eq. (A1), as well. Figure 4 shows two different cases (${z}_{\alpha /2}\le 2{z}_{\gamma /2}$ and ${z}_{\alpha /2}>2{z}_{\gamma /2}$) for the ranges of ${\lambda }_{i}$ defined by $-{z}_{\alpha /2}+{z}_{\gamma /2}\le {\lambda }_{i}\le {z}_{\alpha /2}-{z}_{\gamma /2}$. For the first case, ${c}_{{\lambda }_{i}}$ is calculated only using the first term in Eq. (A1). For the second case, ${c}_{{\lambda }_{i}}$ on $-{z}_{\alpha /2}+{z}_{\gamma /2}\le {\lambda }_{i}<{z}_{\gamma /2}$ or ${z}_{\gamma /2}<{\lambda }_{i}\le {z}_{\alpha /2}-{z}_{\gamma /2}$ is calculated using the first term, whereas ${c}_{{\lambda }_{i}}$ on $-{z}_{\gamma /2}\le {\lambda }_{i}\le {z}_{\gamma /2}$ is calculated using the first term and the second term in Eq. (A1). Thus, the proof is completed. $\hfill\square$

Fig. 4

The ranges of ${\lambda }_{i}$ for $-{z}_{\alpha /2}+{z}_{\gamma /2}\le {\lambda }_{i}\le {z}_{\alpha /2}-{z}_{\gamma /2}$ used in the first term [] and the second term [] in Eq. (A1)

1.2 Appendix B: The proof of Corollary 1

We now derive the confidence coefficient for Case 2 ($\alpha <\gamma$ and ${z}_{\alpha /2}\le 2{z}_{\gamma /2}$). Then

$$\underset{-\infty <{\lambda }_{i}<\infty }{\inf}{c}_{{\lambda }_{i}}=\mathrm{min}\left(\underset{{\lambda }_{i}<-{z}_{\alpha /2}-{z}_{\gamma /2}}{\text{inf}}{c}_{{\lambda }_{i}},\underset{-{z}_{\alpha /2}-{z}_{\gamma /2}\le {\lambda }_{i}<-{z}_{\gamma /2}}{\text{inf}}{c}_{{\lambda }_{i}},\cdots , \underset{{\lambda }_{i}>{z}_{\alpha /2}+{z}_{\gamma /2}}{\text{inf}}{c}_{{\lambda }_{i}}\right).$$

Below, we will compute all the infimums.

(i) For ${\lambda }_{i}<-{z}_{\alpha /2}-{z}_{\gamma /2}$, ${c}_{{\lambda }_{i}}=1-\gamma$. Obviously, we have

$$\underset{{\lambda }_{i}<-{z}_{\alpha /2}-{z}_{\gamma /2}}{\inf}{c}_{{\lambda }_{i}}=1-\gamma .$$

(ii) For $-{z}_{\alpha /2}-{z}_{\gamma /2}\le {\lambda }_{i}<-{z}_{\gamma /2}$, ${c}_{{\lambda }_{i}}$ is the decreasing function of ${\lambda }_{i}$. Since ${c}_{{\lambda }_{i}}$ is not continuous at ${\lambda }_{i}=-{z}_{\gamma /2}$, we take the limit ${\lambda }_{i}\uparrow -{z}_{\gamma /2}$. Therefore, we have

$$\underset{-{z}_{\alpha /2}-{z}_{\gamma /2}\le {\lambda }_{i}<-{z}_{\gamma /2}}{\inf}{c}_{{\lambda }_{i}}=\underset{{\lambda }_{i}\uparrow -{z}_{\gamma /2}}{\text{lim}}{c}_{{\lambda }_{i}}=\Phi \left({z}_{\gamma /2}-{z}_{\alpha /2}\right)-\frac{\gamma }{2}.$$

(iii) For $-{z}_{\gamma /2}\le {\lambda }_{i}<-{z}_{\alpha /2}+{z}_{\gamma /2}$, ${c}_{{\lambda }_{i}}$ is the decreasing function of ${\lambda }_{i}$. We have

$$\underset{-{z}_{\gamma /2}\!\le {\lambda }_{i}<-{z}_{\alpha /2}+{z}_{\gamma /2}}{\inf}{c}_{{\lambda }_{i}}=\underset{{\lambda }_{i}\uparrow -{z}_{\alpha /2}+{z}_{\gamma /2}}{\text{lim}}{c}_{{\lambda }_{i}}=\Phi \left({z}_{\alpha /2}-{z}_{\gamma /2}+{z}_{\alpha /2}\right)-\frac{\gamma }{2}=\Phi \left(2{z}_{\alpha /2}-{z}_{\gamma /2}\right)-\frac{\gamma }{2}.$$

(iv) For $-{z}_{\alpha /2}+{z}_{\gamma /2}\le {\lambda }_{i}\le {z}_{\alpha /2}-{z}_{\gamma /2}$, ${c}_{{\lambda }_{i}}=\Phi \left(-{\lambda }_{i}+{z}_{\alpha /2}\right)-\Phi \left(-{\lambda }_{i}-{z}_{\alpha /2}\right)$. Then, we have

$$\frac{\partial }{\partial {\lambda }_{i}}{c}_{{\lambda }_{i}}=-\phi \left(-{\lambda }_{i}+{z}_{\alpha /2}\right)+\phi \left(-{\lambda }_{i}-{z}_{\alpha /2}\right)$$

$$=\frac{1}{\sqrt{2\pi }}\mathrm{exp}\left(-\frac{{\lambda }_{i}^{2}-2{\lambda }_{i}{z}_{\alpha /2}+{z}_{\alpha /2}^{2}}{2}\right)\left\{-1+\mathrm{exp}\left(-2{\lambda }_{i}{z}_{\alpha /2}\right)\right\}.$$

From this equation, it holds that $\partial {c}_{{\lambda }_{i}}/\partial {\lambda }_{i}=0$ for ${\lambda }_{i}=0$, $\partial {c}_{{\lambda }_{i}}/\partial {\lambda }_{i}>0$ for $-{z}_{\alpha /2}+{z}_{\gamma /2}\le {\lambda }_{i}<0$ and $\partial {c}_{{\lambda }_{i}}/\partial {\lambda }_{i}<0$ for $0<{\lambda }_{i}\le {z}_{\alpha /2}-{z}_{\gamma /2}$. That is, ${c}_{{\lambda }_{i}}$ takes minimum at ${\lambda }_{i}=-{z}_{\alpha /2}+{z}_{\gamma /2}$ or ${\lambda }_{i}={z}_{\alpha /2}-{z}_{\gamma /2}$. For ${\lambda }_{i}=-{z}_{\alpha /2}+{z}_{\gamma /2}$ and ${\lambda }_{i}={z}_{\alpha /2}-{z}_{\gamma /2}$, we have ${c}_{{\lambda }_{i}}=\Phi \left(2{z}_{\alpha /2}-{z}_{\gamma /2}\right)-\gamma /2$,

$$\underset{-{z}_{\alpha /2}+{z}_{\gamma /2}\le {\lambda }_{i}\le {z}_{\alpha /2}-{z}_{\gamma /2}}{\inf}{c}_{{\lambda }_{i}}=\Phi \left(2{z}_{\alpha /2}-{z}_{\gamma /2}\right)-\frac{\gamma }{2}.$$

(v) For ${z}_{\alpha /2}-{z}_{\gamma /2}<{\lambda }_{i}\le {z}_{\gamma /2}$, ${c}_{{\lambda }_{i}}$ is the increasing function of ${\lambda }_{i}$. Therefore, we have

$$\underset{{z}_{\alpha /2}-{z}_{\gamma /2}<{\lambda }_{i}\le {z}_{\gamma /2}}{\inf}{c}_{{\lambda }_{i}}=\underset{{\lambda }_{i}\downarrow -{z}_{\alpha /2}+{z}_{\gamma /2}}{\text{lim}}{c}_{{\lambda }_{i}}=\Phi \left(2{z}_{\alpha /2}-{z}_{\gamma /2}\right)-\frac{\gamma }{2}.$$

(vi) For ${z}_{\gamma /2}<{\lambda }_{i}\le {z}_{\alpha /2}+{z}_{\gamma /2}$, ${c}_{{\lambda }_{i}}$ is the increasing function of ${\lambda }_{i}$. Since ${c}_{{\lambda }_{i}}$ is not continuous at ${\lambda }_{i}={z}_{\gamma /2}$, we take limit ${\lambda }_{i}\downarrow {z}_{\gamma /2}$.

$$\underset{{z}_{\gamma /2}\!<{\lambda }_{i}\le {z}_{\alpha /2}+{z}_{\gamma /2}}{\inf}{c}_{{\lambda }_{i}}=\underset{{\lambda }_{i}\downarrow {z}_{\gamma /2}}{\text{lim}}{c}_{{\lambda }_{i}}=1-\frac{\gamma }{2}-\Phi \left(-{z}_{\gamma /2}+{z}_{\alpha /2}\right)=\Phi \left({z}_{\gamma /2}-{z}_{\alpha /2}\right)-\frac{\gamma }{2}.$$

(vii) For ${\lambda }_{i}>{z}_{\alpha /2}+{z}_{\gamma /2}$, ${c}_{{\lambda }_{i}}=1-\gamma$. We trivially have

$$\underset{{\lambda }_{i}>{z}_{\alpha /2}+{z}_{\gamma /2}}{\inf}{c}_{{\lambda }_{i}}=1-\gamma .$$

Using (i)–(vii), and $\alpha <\gamma$ (equivalent to ${z}_{\alpha /2}>{z}_{\gamma /2}$), we have

$$\begin{aligned}\underset{-\infty <{\lambda }_{i}<\infty }{\text{inf}}{c}_{{\lambda }_{i}} =&\,\mathrm{min}\left\{1-\gamma ,\Phi \left({z}_{\gamma /2}-{z}_{\alpha /2}\right)-\frac{\gamma }{2},\Phi \left(2{z}_{\alpha /2}-{z}_{\gamma /2}\right)-\frac{\gamma }{2}\right\}\\ =&\,\Phi \left({z}_{\gamma /2}-{z}_{\alpha /2}\right)-\frac{\gamma }{2}. \end{aligned}$$

(B1)

Since ${z}_{\gamma /2}-{z}_{\alpha /2}<{z}_{\gamma /2}$ is equivalent to $\Phi \left({z}_{\gamma /2}-{z}_{\alpha /2}\right)<\Phi \left({z}_{\gamma /2}\right)$, we have

$$\begin{array}{ll}\Phi \left({z}_{\gamma /2}-{z}_{\alpha /2}\right)-\frac{\gamma }{2}<\Phi \left({z}_{\gamma /2}\right)-\frac{\gamma }{2}=1-\gamma . \end{array}$$

(B2)

This inequality holds for all $\gamma$ in Case 2 ($\alpha <\gamma$ and ${z}_{\alpha /2}\le 2{z}_{\gamma /2}$). For the proofs for Case 1 ($\alpha \ge \gamma$) and Case 3 ($\alpha <\gamma$ and ${z}_{\alpha /2}>2{z}_{\gamma /2}$), similar calculations can be performed.$\hfill\square$

1.3 Appendix C: The proof of Theorem 1

We derive the CDF of ${\delta }_{i}^{\mathrm{PT}}$ as follows:

$$\begin{aligned}F_{{\delta }_{i}^{\mathrm{PT}}}\left(d\right)&= P\left({\delta }_{i}^{\mathrm{PT}}\le d\right)&= P\left({Y}_{i}\le d, \left|\frac{{Y}_{i}}{{\sigma }_{i}}\right|>{z}_{\alpha /2}\right)+P\left(\left|\frac{{Y}_{i}}{{\sigma }_{i}}\right|\le {z}_{\alpha /2}\right)\cdot I\left(d\ge 0\right)\end{aligned}$$

$$ \begin{aligned}&= P\left(\frac{{Y}_{i}-{\mu }_{i}}{{\sigma }_{i}}\le \frac{d-{\mu }_{i}}{{\sigma }_{i}}, \left|\frac{{Y}_{i}-{\mu }_{i}}{{\sigma }_{i}}+\frac{{\mu }_{i}}{{\sigma }_{i}}\right|>{z}_{\alpha /2}\right)\\ &\quad +P\left(\left|\frac{{Y}_{i}-{\mu }_{i}}{{\sigma }_{i}}+\frac{{\mu }_{i}}{{\sigma }_{i}}\right|\le {z}_{\alpha /2}\right)\cdot I\left(d\ge 0\right).\end{aligned}$$

Putting ${Z}_{i}=\left({Y}_{i}-{\mu }_{i}\right)/{\sigma }_{i}$, we have

$$\begin{aligned}F_{{\delta }_{i}^{\mathrm{PT}}}\left(d\right)=& P\left({Z}_{i}\le \frac{d-{\mu }_{i}}{{\sigma }_{i}}, \left({Z}_{i}<-\frac{{\mu }_{i}}{{\sigma }_{i}}-{z}_{\alpha /2} \,\, {\rm or}\,\, -\frac{{\mu }_{i}}{{\sigma }_{i}}+{z}_{\alpha /2}<{Z}_{i}\right)\right)\\&+P\left(-\frac{{\mu }_{i}}{{\sigma }_{i}}-{z}_{\alpha /2}\le {Z}_{i}\le -\frac{{\mu }_{i}}{{\sigma }_{i}}+{z}_{\alpha /2}\right)\cdot I\left(d\ge 0\right)\end{aligned}$$

$$=\left\{\begin{array}{ll}\begin{array}{ll}P\left({Z}_{i}\le \frac{d-{\mu }_{i}}{{\sigma }_{i}}\right) & \quad {\rm if} \quad d \le -{z}_{\alpha /2}{\sigma }_{i},\\ P\left({Z}_{i}<-\frac{{\mu }_{i}}{{\sigma }_{i}}-{z}_{\alpha /2}\right) & \quad {\rm if}\quad -{z}_{\alpha /2}{\sigma }_{i}<d<0,\\ P\left({Z}_{i}<-\frac{{\mu }_{i}}{{\sigma }_{i}}-{z}_{\alpha /2}\right)+P\left(-\frac{{\mu }_{i}}{{\sigma }_{i}}-{z}_{\alpha /2}\le {Z}_{i}\le -\frac{{\mu }_{i}}{{\sigma }_{i}}+{z}_{\alpha /2}\right) & \quad {\rm if} \quad 0\le d<{z}_{\alpha /2}{\sigma }_{i},\\ P\left({Z}_{i}<-\frac{{\mu }_{i}}{{\sigma }_{i}}-{z}_{\alpha /2}\right)+P\left(-\frac{{\mu }_{i}}{{\sigma }_{i}}+{z}_{\alpha /2}<{Z}_{i}\le \frac{d-{\mu }_{i}}{{\sigma }_{i}}\right)\\ \quad +P\left(-\frac{{\mu }_{i}}{{\sigma }_{i}}-{z}_{\alpha /2}\le {Z}_{i}\le -\frac{{\mu }_{i}}{{\sigma }_{i}}+{z}_{\alpha /2}\right)& \quad {\rm if}\quad {z}_{\alpha /2}{\sigma }_{i}\le d.\end{array}\\ \end{array}\right.$$

Since ${Z}_{i}$ is a continuous random variable, $P\left({Z}_{i}<z\right)=P\left({Z}_{i}\le z\right)$ for $z\in (-\infty ,\infty )$. Therefore

$${F}_{{\delta }_{i}^{\mathrm{PT}}}\left(d\right)=\left\{\begin{array}{ll}P\left({Z}_{i}\le \frac{d-{\mu }_{i}}{{\sigma }_{i}}\right) & \quad {\rm if}\quad d\le -{z}_{\alpha /2}{\sigma }_{i},\\ P\left({Z}_{i}\le -\frac{{\mu }_{i}}{{\sigma }_{i}}-{z}_{\alpha /2}\right) & \quad {\rm if}\quad -{z}_{\alpha /2}{\sigma }_{i}<d<0,\\ P\left({Z}_{i}\le -\frac{{\mu }_{i}}{{\sigma }_{i}}+{z}_{\alpha /2}\right) & \quad {\rm if}\quad 0 \le d<{z}_{\alpha /2}{\sigma }_{i},\\ P\left({Z}_{i}\le \frac{d-{\mu }_{i}}{{\sigma }_{i}}\right) & \quad {\rm if}\quad {z}_{\alpha /2}{\sigma }_{i}\le d.\end{array}\right.$$

$$=\left\{\begin{array}{ll}\Phi \left(\frac{d-{\mu }_{i}}{{\sigma }_{i}}\right)& \quad {\rm if}\quad d\,\, \le -{z}_{\alpha /2}{\sigma }_{i} \,\, {\rm or} \,\, {z}_{\alpha /2}{\sigma }_{i}\le d,\\ \Phi \left(-\frac{{\mu }_{i}}{{\sigma }_{i}}-{z}_{\alpha /2}\right) & \quad {\rm if}\quad -{z}_{\alpha /2}{\sigma }_{i}<d<0,\\ \Phi \left(-\frac{{\mu }_{i}}{{\sigma }_{i}}+{z}_{\alpha /2}\right) & \quad {\rm if}\quad 0\le d<{z}_{\alpha /2}{\sigma }_{i}.\end{array}\right.$$

The proof is complete. $\hfill\square$

1.4 Appendix D: The derivation of the pivot CI

We first review the pivoting method in a general setting. The following lemma is quoted from Theorem 9.2.14 of Casella and Berger (2002):

Lemma 3 (Pivoting a CDF; Casella & Berger, 2002) Let ${\delta }_{i}$ be a discrete statistic with CDF ${F}_{{\delta }_{i}}\left(d|{\mu }_{i}\right)=P\left({\delta }_{i}\le d|{\mu }_{i}\right)$. Let $\mathcal{D}$ be the sample space of ${\delta }_{i}$. Let ${\gamma }_{1}+{\gamma }_{2}=\gamma$ with $0<\gamma <1$ be fixed values. Suppose that for each $d\in \mathcal{D}$, ${\mu }_{i\mathrm{L}}\left(d\right)$ and ${\mu }_{i\mathrm{U}}\left(d\right)$ can be defined as follows.

• i. If ${F}_{{\delta }_{i}}\left(d|{\mu }_{i}\right)$ is a decreasing function of ${\mu }_{i}$ for each $d$, define ${\mu }_{i\mathrm{L}}\left(d\right)$ and ${\mu }_{i\mathrm{U}}\left(d\right)$ by $P\left({\delta }_{i}\le d|{\mu }_{i\mathrm{U}}\left(d\right)\right)={\gamma }_{1}, P\left({\delta }_{i}\ge d|{\mu }_{i\mathrm{L}}\left(d\right)\right)={\gamma }_{2}.$

• ii. If ${F}_{{\delta }_{i}}\left(d|{\mu }_{i}\right)$ is an increasing function of ${\mu }_{i}$ for each $d$, define ${\mu }_{i\mathrm{L}}\left(d\right)$ and ${\mu }_{i\mathrm{U}}\left(d\right)$ by $P\left({\delta }_{i}\ge d|{\mu }_{i\mathrm{U}}\left(d\right)\right)={\gamma }_{1}, P\left({\delta }_{i}\le d|{\mu }_{i\mathrm{L}}\left(d\right)\right)={\gamma }_{2}.$

Then, the random interval $\left[{\mu }_{i\mathrm{L}}\left({\delta }_{i}\right), {\mu }_{i\mathrm{U}}\left({\delta }_{i}\right)\right]$ is a $100\left(1-\gamma \right)\mathrm{\%}$ CI for ${\mu }_{i}$.

Let ${\gamma }_{1}+{\gamma }_{2}=\gamma$ with $0<\gamma <1$ being a fixed value. From Eq. (1), the sample space of ${\delta }_{i}^{\mathrm{PT}}$ is ${\delta }_{i}^{\mathrm{PT}}<-{z}_{\alpha /2}{\sigma }_{i}$ or ${\delta }_{i}^{\mathrm{PT}}=0$ or ${\delta }_{i}^{\mathrm{PT}}>{z}_{\alpha /2}{\sigma }_{i}$. We derive $\left[{L}_{i}^{\mathrm{PT}}, {U}_{i}^{\mathrm{PT}}\right]=\left[{\mu }_{i\mathrm{L}}\left({\delta }_{i}^{\mathrm{PT}}\right), {\mu }_{i\mathrm{U}}\left({\delta }_{i}^{\mathrm{PT}}\right)\right]$ for each of three cases.

For each ${d}_{0}<-{z}_{\alpha /2}{\sigma }_{i}$, ${F}_{{\delta }_{i}^{\mathrm{PT}}}\left({d}_{0}|{\mu }_{i}\right)=\Phi \left\{\left({d}_{0}-{\mu }_{i}\right)/{\sigma }_{i}\right\}$ is the decreasing function of ${\mu }_{i}$. We define ${\mu }_{i\mathrm{L}}\left({d}_{0}\right)$ and ${\mu }_{i\mathrm{U}}\left({d}_{0}\right)$ by

$$P\left({\delta }_{i}^{\mathrm{PT}}\le {d}_{0}|{\mu }_{i\mathrm{U}}\left({d}_{0}\right)\right)=\Phi \left(\frac{{d}_{0}-{\mu }_{i\mathrm{U}}\left({d}_{0}\right)}{{\sigma }_{i}}\right)={\gamma }_{1},$$

$$P\left({\delta }_{i}^{\mathrm{PT}}\ge {d}_{0}|{\mu }_{i\mathrm{L}}\left({d}_{0}\right)\right)=1-P\left({\delta }_{i}^{\mathrm{PT}}<{d}_{0}|{\mu }_{i\mathrm{L}}\left({d}_{0}\right)\right)=1-\Phi \left(\frac{{d}_{0}-{\mu }_{i\mathrm{L}}\left({d}_{0}\right)}{{\sigma }_{i}}\right)={\gamma }_{2}.$$

These are equivalent to

$$\Phi \left(\frac{{d}_{0}-{\mu }_{i\mathrm{U}}\left({d}_{0}\right)}{{\sigma }_{i}}\right)={\gamma }_{1},\quad 1-\Phi \left(\frac{{d}_{0}-{\mu }_{i\mathrm{L}}\left({d}_{0}\right)}{{\sigma }_{i}}\right)={\gamma }_{2}$$

or equivalently

$${\mu }_{i\mathrm{U}}\left({d}_{0}\right)={d}_{0}+{z}_{{\gamma }_{1}}{\sigma }_{i},\quad {\mu }_{i\mathrm{L}}\left({d}_{0}\right)={d}_{0}-{z}_{{\gamma }_{2}}{\sigma }_{i}.$$

Therefore, we have $\left[{\mu }_{i\mathrm{L}}\left({d}_{0}\right), {\mu }_{i\mathrm{U}}\left({d}_{0}\right)\right]=\left[{d}_{0}-{z}_{{\gamma }_{2}}{\sigma }_{i},{d}_{0}+{z}_{{\gamma }_{1}}{\sigma }_{i}\right]$.

For ${d}_{0}=0$, ${\delta }_{i}^{\mathrm{PT}}\left({d}_{0}|{\mu }_{i}\right)=\Phi \left(-{\mu }_{i}/{\sigma }_{i}+{z}_{\alpha /2}\right)$ is the decreasing function of ${\mu }_{i}$. We define ${\mu }_{i\mathrm{L}}\left({d}_{0}\right)$ and ${\mu }_{i\mathrm{U}}\left({d}_{0}\right)$ by

$$P\left({\delta }_{i}^{\mathrm{PT}}\le 0|{\mu }_{i\mathrm{U}}\left(0\right)\right)=\Phi \left(-\frac{{\mu }_{i\mathrm{U}}\left(0\right)}{{\sigma }_{i}}+{z}_{\alpha /2}\right)={\gamma }_{1},$$

$$P\left({\delta }_{i}^{\mathrm{PT}}\ge 0|{\mu }_{i\mathrm{L}}\left(0\right)\right)=1-P\left({\delta }_{i}^{\mathrm{PT}}<0|{\mu }_{i\mathrm{L}}\left(0\right)\right)=1-\Phi \left(-\frac{{\mu }_{i\mathrm{L}}\left(0\right)}{{\sigma }_{i}}-{z}_{\alpha /2}\right)={\gamma }_{2}.$$

Expanding these equations, we have

$$\Phi \left(-\frac{{\mu }_{i\mathrm{U}}\left(0\right)}{{\sigma }_{i}}+{z}_{\alpha /2}\right)={\gamma }_{1},\quad \Phi \left(-\frac{{\mu }_{i\mathrm{L}}\left(0\right)}{{\sigma }_{i}}-{z}_{\alpha /2}\right)={\gamma }_{2.}$$

They are equivalent to

$${\mu }_{i\mathrm{U}}\left(0\right)=\left({z}_{\alpha /2}+{z}_{{\gamma }_{1}}\right){\sigma }_{i},\quad {\mu }_{i\mathrm{L}}\left(0\right)=-\left({z}_{\alpha /2}+{z}_{{\gamma }_{2}}\right){\sigma }_{i}.$$

Therefore, we have $\left[{\mu }_{i\mathrm{L}}\left(d\right), {\mu }_{i\mathrm{U}}\left(d\right)\right]=\left[-\left({z}_{\alpha /2}+{z}_{{\gamma }_{2}}\right){\sigma }_{i},\left({z}_{\alpha /2}+{z}_{{\gamma }_{1}}\right){\sigma }_{i}\right]$ for ${\delta }_{i}^{\mathrm{PT}}=0$.

For ${d}_{0}>{z}_{\alpha /2}{\sigma }_{i}$, we can derive $\left[{\mu }_{i\mathrm{L}}\left({d}_{0}\right), {\mu }_{i\mathrm{U}}\left({d}_{0}\right)\right]=\left[{d}_{0}-{z}_{{\gamma }_{2}}{\sigma }_{i},{d}_{0}+{z}_{{\gamma }_{1}}{\sigma }_{i}\right]$ as well as for ${d}_{0}<-{z}_{\alpha /2}{\sigma }_{i}$. In conclusion, we have

$$\begin{aligned}&\left[{L}_{i}^{\mathrm{PT}}, {U}_{i}^{\mathrm{PT}}\right]\\ & \quad =\left[{\mu }_{i\mathrm{L}}\left({\delta }_{i}^{\mathrm{PT}}\right), {\mu }_{i\mathrm{U}}\left({\delta }_{i}^{\mathrm{PT}}\right)\right]=\left\{\begin{array}{ll}\left[{\delta }_{i}^{\mathrm{PT}}-{z}_{{\gamma }_{2}}{\sigma }_{i}, {\delta }_{i}^{\mathrm{PT}}+{z}_{{\gamma }_{1}}{\sigma }_{i}\right] & \quad {\rm if}\quad {\delta }_{i}^{\mathrm{PT}}<-{z}_{\alpha /2}{\sigma }_{i},\\ \left[-\left({z}_{\alpha /2}+{z}_{{\gamma }_{2}}\right){\sigma }_{i}, \left({z}_{\alpha /2}+{z}_{{\gamma }_{1}}\right){\sigma }_{i}\right] & \quad {\rm if}\quad {\delta }_{i}^{\mathrm{PT}}=0,\\ \left[{\delta }_{i}^{\mathrm{PT}}-{z}_{{\gamma }_{2}}{\sigma }_{i}, {\delta }_{i}^{\mathrm{PT}}+{z}_{{\gamma }_{1}}{\sigma }_{i}\right] & \quad{\rm if}\quad {\delta }_{i}^{\mathrm{PT}}>{z}_{\alpha /2}{\sigma }_{i}.\end{array}\right.\end{aligned}$$

Thus, the derivation is complete. $\hfill\square$

1.5 Appendix E: The proof of Theorem 2

We derive the CP of the pivot CI for each of three cases. We have

$$\begin{aligned}&P\left({\mu }_{i}\in \left[{L}_{i}^{\mathrm{PT}}, {U}_{i}^{\mathrm{PT}}\right]\right)\\&=P\left({\mu }_{i}\in \left[{L}_{i}^{\mathrm{PT}}, {U}_{i}^{\mathrm{PT}}\right], {\delta }_{i}^{\mathrm{PT}}<-{z}_{\alpha /2}{\sigma }_{i}\right)+P\left({\mu }_{i}\in \left[{L}_{i}^{\mathrm{PT}}, {U}_{i}^{\mathrm{PT}}\right], {\delta }_{i}^{\mathrm{PT}}=0\right)\\&\qquad+P\left({\mu }_{i}\in \left[{L}_{i}^{\mathrm{PT}}, {U}_{i}^{\mathrm{PT}}\right], {\delta }_{i}^{\mathrm{PT}}>{z}_{\alpha /2}{\sigma }_{i}\right)\end{aligned}$$

$$\begin{aligned}&=P\left({Y}_{i}-{z}_{{\gamma }_{2}}{\sigma }_{i}\le {\mu }_{i}\le {Y}_{i}+{z}_{{\gamma }_{1}}{\sigma }_{i}, {Y}_{i}<-{z}_{\alpha /2}{\sigma }_{i}\right)\\&\qquad+P\left(-\left({z}_{\alpha /2}+{z}_{{\gamma }_{2}}\right){\sigma }_{i}\le {\mu }_{i}\le \left({z}_{\alpha /2}+{z}_{{\gamma }_{1}}\right){\sigma }_{i}, \left|\frac{{Y}_{i}}{{\sigma }_{i}}\right|\le {z}_{\alpha /2}\right)\\&\qquad+P\left({Y}_{i}-{z}_{{\gamma }_{2}}{\sigma }_{i}\le {\mu }_{i}\le {Y}_{i}+{z}_{{\gamma }_{1}}{\sigma }_{i}, {Y}_{i}>{z}_{\alpha /2}{\sigma }_{i}\right)\end{aligned}$$

$$\begin{aligned}&=P\left({\mu }_{i}-{z}_{{\gamma }_{1}}{\sigma }_{i}\le {Y}_{i}\le {\mu }_{i}+{z}_{{\gamma }_{2}}{\sigma }_{i}, {Y}_{i}<-{z}_{\alpha /2}{\sigma }_{i}\right)\\&\qquad+P\left(-\left({z}_{\alpha /2}+{z}_{{\gamma }_{2}}\right){\sigma }_{i}\le {\mu }_{i}\le \left({z}_{\alpha /2}+{z}_{{\gamma }_{1}}\right){\sigma }_{i},-{z}_{\alpha /2}{\sigma }_{i}\le {Y}_{i}\le {z}_{\alpha /2}{\sigma }_{i} \right)\\&\qquad+P\left({\mu }_{i}-{z}_{{\gamma }_{1}}{\sigma }_{i}\le {Y}_{i}\le {\mu }_{i}+{z}_{{\gamma }_{2}}{\sigma }_{i}, {Y}_{i}>{z}_{\alpha /2}{\sigma }_{i}\right)\end{aligned}$$

$$\begin{aligned}&=P\left(-{z}_{{\gamma }_{1}}\le \frac{{Y}_{i}-{\mu }_{i}}{{\sigma }_{i}}\le {z}_{{\gamma }_{2}},\frac{{Y}_{i}-{\mu }_{i}}{{\sigma }_{i}}<-\frac{{\mu }_{i}}{{\sigma }_{i}}-{z}_{\alpha /2} \right)\\&\qquad+P\left(-\left({z}_{\alpha /2}+{z}_{{\gamma }_{2}}\right)\le \frac{{\mu }_{i}}{{\sigma }_{i}}\le {z}_{\alpha /2}+{z}_{{\gamma }_{1}},-\frac{{\mu }_{i}}{{\sigma }_{i}}-{z}_{\alpha /2}\le \frac{{Y}_{i}-{\mu }_{i}}{{\sigma }_{i}}\le -\frac{{\mu }_{i}}{{\sigma }_{i}}+{z}_{\alpha /2} \right)\\&\qquad+P\left(-{z}_{{\gamma }_{1}}\le \frac{{Y}_{i}-{\mu }_{i}}{{\sigma }_{i}}\le {z}_{{\gamma }_{2}},\frac{{Y}_{i}-{\mu }_{i}}{{\sigma }_{i}}>-\frac{{\mu }_{i}}{{\sigma }_{i}}+{z}_{\alpha /2} \right)\end{aligned}$$

$$\begin{aligned}&=P\left(-{z}_{{\gamma }_{1}}\le {Z}_{i}\le {z}_{{\gamma }_{2}},{Z}_{i}<-{\lambda }_{i}-{z}_{\alpha /2} \right)\\&\qquad+P\left(-\left({z}_{\alpha /2}+{z}_{{\gamma }_{2}}\right)\le {\lambda }_{i}\le {z}_{\alpha /2}+{z}_{{\gamma }_{1}},-{\lambda }_{i}-{z}_{\alpha /2}\le {Z}_{i}\le -{\lambda }_{i}+{z}_{\alpha /2} \right)\\&\qquad+P\left(-{z}_{{\gamma }_{1}}\le {Z}_{i}\le {z}_{{\gamma }_{2}},{Z}_{i}>-{\lambda }_{i}+{z}_{\alpha /2} \right).\end{aligned}$$

Each term in the above equation can be expressed as follows. For the first term, we have

$$\begin{aligned}& P\left(-{z}_{{\gamma }_{1}}\le {Z}_{i}\le {z}_{{\gamma }_{2}},{Z}_{i}<-{\lambda }_{i}-{z}_{\alpha /2} \right)\\ & =\left\{\begin{array}{ll}1-\gamma & \quad {\rm if}\quad {\lambda }_{i}<-{z}_{\alpha /2}-{z}_{{\gamma }_{2}},\\ \Phi \left(-{\lambda }_{i}-{z}_{\alpha /2}\right)-{\gamma }_{1} &\quad {\rm if}\quad -{z}_{\alpha /2}-{z}_{{\gamma }_{2}}\le {\lambda }_{i}\le -{z}_{\alpha /2}+{z}_{{\gamma }_{1}},\\ 0 & \quad {\rm if}\quad -{z}_{\alpha /2}+{z}_{{\gamma }_{1}}\le {\lambda }_{i}\le {z}_{\alpha /2}+{z}_{{\gamma }_{1}},\\ 0 & \quad {\rm if}\quad {\lambda }_{i} >{z}_{\alpha /2}+{z}_{{\gamma }_{1}},\end{array}\right.\end{aligned}$$

and for the second term, we have

$$\begin{aligned}P\left(-\left({z}_{\alpha /2}+{z}_{{\gamma }_{2}}\right)\le {\lambda }_{i}\le {z}_{\alpha /2}+{z}_{{\gamma }_{1}},-{\lambda }_{i}-{z}_{\alpha /2}\le {Z}_{i}\le -{\lambda }_{i}+{z}_{\alpha /2} \right)=\left\{\begin{array}{ll}0 & \quad {\rm if}\quad {\lambda }_{i}<-{z}_{\alpha /2}-{z}_{{\gamma }_{2}},\\ \Phi \left(-{\lambda }_{i}+{z}_{\alpha /2}\right)-\Phi \left(-{\lambda }_{i}-{z}_{\alpha /2}\right) & \quad {\rm if}\quad -{z}_{\alpha /2}-{z}_{{\gamma }_{2}}\le {\lambda }_{i}\le {z}_{\alpha /2}+{z}_{{\gamma }_{1}},\\ 0 & \quad {\rm if}\quad {\lambda }_{i}>{z}_{\alpha /2}+{z}_{{\gamma }_{1}},\end{array}\right.\end{aligned}$$

and for the third term, we have

$$\begin{aligned}& P\left(-{z}_{{\gamma }_{1}}\le {Z}_{i}\le {z}_{{\gamma }_{2}},{Z}_{i}>-{\lambda }_{i}+{z}_{\alpha /2}\right)\\&=\left\{\begin{array}{ll}0 & \quad {\rm if}\quad {\lambda }_{i}<-{z}_{\alpha /2}-{z}_{{\gamma }_{2}},\\ 0 & \quad {\rm if}\quad -{z}_{\alpha /2}-{z}_{{\gamma }_{2}}\le {\lambda }_{i}\le {z}_{\alpha /2}-{z}_{{\gamma }_{2}},\\ 1-{\gamma }_{2}-\Phi \left(-{\lambda }_{i}+{z}_{\alpha /2}\right) & \quad {\rm if}\quad {z}_{\alpha /2}-{z}_{{\gamma }_{2}}\le {\lambda }_{i}\le {z}_{\alpha /2}+{z}_{{\gamma }_{1}},\\ 1-\gamma & \quad {\rm if}\quad {\lambda }_{i}>{z}_{\alpha /2}+{z}_{{\gamma }_{1}}.\end{array}\right.\end{aligned}$$

Summing up the three terms, we obtain the following CP. For ${z}_{\alpha /2}-{z}_{{\gamma }_{2}}<-{z}_{\alpha /2}+{z}_{{\gamma }_{1}}$

$$P\left({\mu }_{i}\in \left[{L}_{i}^{\mathrm{PT}}, {U}_{i}^{\mathrm{PT}}\right]\right)=\left\{\begin{array}{ll}1-\gamma & \quad {\rm if}\quad {\lambda }_{i}<-{z}_{\alpha /2}-{z}_{{\gamma }_{2}},\\ \Phi \left(-{\lambda }_{i}+{z}_{\alpha /2}\right)-{\gamma }_{1} & \quad {\rm if}\quad -{z}_{\alpha /2}-{z}_{{\gamma }_{2}}\le {\lambda }_{i}\le {z}_{\alpha /2}-{z}_{{\gamma }_{2}},\\ 1-\gamma & \quad {\rm if}\quad {z}_{\alpha /2}-{z}_{{\gamma }_{2}}<{\lambda }_{i}<-{z}_{\alpha /2}+{z}_{{\gamma }_{1}},\\ 1-{\gamma }_{2}-\Phi \left(-{\lambda }_{i}-{z}_{\alpha /2}\right) & \quad {\rm if}\quad -{z}_{\alpha /2}+{z}_{{\gamma }_{1}}\le {\lambda }_{i}\le {z}_{\alpha /2}+{z}_{{\gamma }_{1}},\\ 1-\gamma & \quad {\rm if}\quad {\lambda }_{i}>{z}_{\alpha /2}+{z}_{{\gamma }_{1}},\end{array}\right.$$

for ${z}_{\alpha /2}-{z}_{{\gamma }_{2}}=-{z}_{\alpha /2}+{z}_{{\gamma }_{1}}$,

$$P\left({\mu }_{i}\in \left[{L}_{i}^{\mathrm{PT}}, {U}_{i}^{\mathrm{PT}}\right]\right)=\left\{\begin{array}{ll}1-\gamma & \quad {\rm if}\quad {\lambda }_{i}<-{z}_{\alpha /2}-{z}_{{\gamma }_{2}},\\ \Phi \left(-{\lambda }_{i}+{z}_{\alpha /2}\right)-{\gamma }_{1} & \quad {\rm if}\quad -{z}_{\alpha /2}-{z}_{{\gamma }_{2}}\le {\lambda }_{i}\le {z}_{\alpha /2}-{z}_{{\gamma }_{2}},\\ 1-{\gamma }_{2}-\Phi \left(-{\lambda }_{i}-{z}_{\alpha /2}\right) & \quad {\rm if}\quad {z}_{\alpha /2}-{z}_{{\gamma }_{2}}\le {\lambda }_{i}\le {z}_{\alpha /2}+{z}_{{\gamma }_{1}},\\ 1-\gamma & \quad {\rm if}\quad {\lambda }_{i}>{z}_{\alpha /2}+{z}_{{\gamma }_{1}},\end{array}\right.$$

for ${z}_{\alpha /2}-{z}_{{\gamma }_{2}}>-{z}_{\alpha /2}+{z}_{{\gamma }_{1}}$,

$$\begin{aligned}&P\left({\mu }_{i}\in \left[{L}_{i}^{\mathrm{PT}}, {U}_{i}^{\mathrm{PT}}\right]\right)\\&=\left\{\begin{array}{ll}1-\gamma & \quad {\rm if}\quad {\lambda }_{i}<-{z}_{\alpha /2}-{z}_{{\gamma }_{2}},\\ \Phi \left(-{\lambda }_{i}+{z}_{\alpha /2}\right)-{\gamma }_{1} & \quad {\rm if}\quad -{z}_{\alpha /2}-{z}_{{\gamma }_{2}}\le {\lambda }_{i}\le -{z}_{\alpha /2}+{z}_{{\gamma }_{1}},\\ \Phi \left(-{\lambda }_{i}+{z}_{\alpha /2}\right)-\Phi \left(-{\lambda }_{i}-{z}_{\alpha /2}\right) & \quad {\rm if}\quad -{z}_{\alpha /2}+{z}_{{\gamma }_{1}}<{\lambda }_{i}<{z}_{\alpha /2}-{z}_{{\gamma }_{2}},\\ 1-{\gamma }_{2}-\Phi \left(-{\lambda }_{i}-{z}_{\alpha /2}\right) & \quad {\rm if}\quad {z}_{\alpha /2}-{z}_{{\gamma }_{2}}\le {\lambda }_{i}\le {z}_{\alpha /2}+{z}_{{\gamma }_{1}},\\ 1-\gamma & \quad {\rm if}\quad {\lambda }_{i}>{z}_{\alpha /2}+{z}_{{\gamma }_{1}}.\end{array} \right.\end{aligned}$$

$\hfill\square$

1.6 Appendix F: The proof of Theorem 3

We recall that the CP takes three cases (Cases 1–3 in Theorem 2). The first and second cases are expressed as ${z}_{\alpha /2}-{z}_{{\gamma }_{2}}<-{z}_{\alpha /2}+{z}_{{\gamma }_{1}}$ and ${z}_{\alpha /2}-{z}_{{\gamma }_{2}}=-{z}_{\alpha /2}+{z}_{{\gamma }_{1}}$, respectively. In these cases, the CP of Theorem 2 immediately yields

$$\mathop {\inf }\limits_{{ - \infty < \mu _{i} < \infty }} P\left( {\mu _{i} \in \left[ {L_{i}^{{PT}} ,~U_{i}^{{PT}} } \right]} \right) = 1 - \gamma$$

Now, we consider the last case expressed as $-{z}_{\alpha /2}+{z}_{{\gamma }_{1}}<{z}_{\alpha /2}-{z}_{{\gamma }_{2}}$. Note that $-{\lambda }_{i}+{z}_{\alpha /2}>{z}_{{\gamma }_{2}}$ is equivalent to $\Phi \left(-{\lambda }_{i}+{z}_{\alpha /2}\right)>1-{\gamma }_{2},$ and that $-{\lambda }_{i}-{z}_{\alpha /2}<-{z}_{{\gamma }_{1}}$ is equivalent to $-\Phi \left(-{\lambda }_{i}-{z}_{\alpha /2}\right)>-{\gamma }_{1}$. From these inequalities, for $-{z}_{\alpha /2}+{z}_{{\gamma }_{1}}<{\lambda }_{i}<{z}_{\alpha /2}-{z}_{{\gamma }_{2}}$, we have

$$\Phi \left(-{\lambda }_{i}+{z}_{\alpha /2}\right)-\Phi \left(-{\lambda }_{i}-{z}_{\alpha /2}\right)>1-{\gamma }_{2}-{\gamma }_{1}=1-\gamma .$$

For ${\lambda }_{i}\le -{z}_{\alpha /2}+{z}_{{\gamma }_{1}}$ and ${\lambda }_{i}\ge {z}_{\alpha /2}-{z}_{{\gamma }_{2}}$, we have

$$\underset{{\lambda }_{i}\le -{z}_{\alpha /2}+{z}_{{\gamma }_{1}},{\lambda }_{i}\ge {z}_{\alpha /2}-{z}_{{\gamma }_{2}} }{\inf}P\left({\mu }_{i}\in \left[{L}_{i}^{\mathrm{PT}}, {U}_{i}^{\mathrm{PT}}\right]\right)=1-\gamma .$$

Therefore

$$\underset{-\infty <{\mu }_{i}<\infty }{\inf}P\left({\mu }_{i}\in \left[{L}_{i}^{\mathrm{PT}}, {U}_{i}^{\mathrm{PT}}\right]\right)=1-\gamma .$$

The proof is complete. $\hfill\square$