Appendix 1: Proofs of Local Powers of Six Exact Tests
We begin by stating a result related to the distribution of a P-value under the alternative hypothesis \(H_0: \mu =\mu _1\), which will be crucial for providing the main results on local power of all tests based on the P-values. We denote \(F_\nu (\cdot )\) to represent the cdf of a central t-distribution with \(\nu \) degrees of freedom.
Lemma 1
$$\begin{aligned} Pr\{P > c |H_1 \} \approx (1 - c) + \frac{n \Delta ^2}{2 \sigma ^2} \xi _{\nu }(c). \end{aligned}$$
(7)
Proof
$$\begin{aligned} Pr\{P> c |H_1 \}= & {} Pr \bigg \{ Pr \bigg [|t_{\nu }|> |\frac{\sqrt{n}(\bar{X} - \mu _0\big )}{S}| \bigg ]> c |H_1 \bigg \} \nonumber \\= & {} Pr \bigg \{1 - \bigg [ F_{\nu } \bigg (|\frac{\sqrt{n}(\bar{X} - \mu _0\big )}{S}| \bigg ) - F_{\nu } \bigg (- |\frac{\sqrt{n}(\bar{X} - \mu _0}{S}| \bigg ) \bigg ] > c |H_1 \bigg \} \nonumber \\= & {} Pr \bigg \{\bigg [ F_{\nu } \bigg (|\frac{\sqrt{n}(\bar{X} - \mu _0\big )}{S}| \bigg ) - F_{\nu } \bigg (- |\frac{\sqrt{n}(\bar{X} - \mu _0}{S}| \bigg ) \bigg ]< 1- c |H_1 \bigg \} \nonumber \\= & {} Pr \bigg \{ |\frac{\sqrt{n}(\bar{X} - \mu _0\big )}{S}|< t_{\nu } \big ( \frac{c}{2} \big ) |H_1 \bigg \} \nonumber \\= & {} Pr \bigg \{-t_{\nu } \big ( \frac{c}{2} \big )< \frac{\sqrt{n}(\bar{X} - \mu _0 \big )}{S}< t_{\nu } \big ( \frac{c}{2} \big ) |H_1 \bigg \} \nonumber \\= & {} Pr \bigg \{-t_{\nu } \big ( \frac{c}{2} \big )< t_{\nu }(\delta ) < t_{\nu } \big ( \frac{c}{2} \big ) |H_1 \nonumber \bigg \} \nonumber \\= & {} \int _{-t_{\nu } (\frac{c}{2})}^{t_{\nu } (\frac{c}{2})} f(x |\nu , \delta )\, dx \quad \quad \quad \bigg [f(x |\nu , \delta ) \sim \text {non-central} \quad t_\nu \bigg (\delta =\frac{\sqrt{n}}{\sigma }\Delta \bigg ) \bigg ] \nonumber \\\approx & {} \int _{-t_{\nu } (\frac{c}{2})}^{t_{\nu } (\frac{c}{2})} \bigg \{ f (x |\nu , 0) + \delta \bigg (\frac{\partial f}{\partial \delta }\bigg )\Bigr |_{\delta = 0} + \frac{\delta ^2}{2} \bigg (\frac{\partial ^2 f}{\partial \delta ^2}\bigg )\Bigr |_{\delta = 0} \bigg \}dx \nonumber \\\approx & {} (1-c) + \frac{n}{2 \sigma ^2} \Delta ^2 \int _{-t_{\nu } (\frac{c}{2})}^{t_{\nu } (\frac{c}{2})} \bigg \{ \frac{\partial ^2 f(x |\nu , \delta )}{\partial \delta ^2} \Bigr |_{\delta = 0} \bigg \} dx \nonumber \\\approx & {} (1 - c) + \frac{n \Delta ^2}{2 \sigma ^2} \xi _{\nu }(c) \quad \nonumber \end{aligned}$$
where \(\xi _{\nu }(c)=\int _{-t_{\nu } (\frac{c}{2})}^{t_{\nu } (\frac{c}{2})} \bigg \{ \frac{\partial ^2 f(x |\nu , \delta )}{\partial \delta ^2} \Bigr |_{\delta = 0} \bigg \} dx= \frac{\Gamma {(\frac{\nu +1}{2})}}{\Gamma {(\frac{\nu }{2})}\sqrt{\nu \pi }} \int _{-t_{\nu } (\frac{c}{2})}^{t_{\nu } (\frac{c}{2})} \bigg (\frac{x^2 - 1}{[\frac{x^2}{\nu }+1]^{\frac{\nu +3}{2}}}\bigg ) dx\). It turns out that \(\xi _{\nu }(c)<0\). Â Â Â \(\square \)
1.1 I. Local Power of Tippett’s Test [LP(T)]
Recall that Tippett’s test rejects the null hypothesis if \(P_{(1)} < \big [1-(1-\alpha )^{\frac{1}{k}}\big ] = a_{\alpha } \). This leads to
$$\begin{aligned} \text {Power} = 1- \prod _{i=1}^k Pr\big \{P_i > a_{\alpha } |H_1 \big \} . \nonumber \end{aligned}$$
Applying Lemma 1, the local power of Tippett’s test is calculated as follows:
$$\begin{aligned} \text {Local power}\approx & {} 1- \prod _{i=1}^k \bigg [(1- a_{\alpha }) + \frac{\Delta ^2}{2} \bigg (\frac{n_i}{\sigma _i^2} \xi _{\nu _i T}(a_{\alpha }) \bigg ) \bigg ] \nonumber \\\approx & {} 1- \prod _{i=1}^k \bigg [(1- \alpha )^{\frac{1}{k}} + \frac{\Delta ^2}{2} \bigg (\frac{n_i}{\sigma _i^2} \xi _{\nu _i T}(a_{\alpha })\bigg ) \bigg ] \nonumber \\\approx & {} 1- \bigg [(1-\alpha )+(1- \alpha )^{\frac{k-1}{k}} \frac{\Delta ^2}{2} \bigg (\sum _{i=1}^k \frac{n_i}{\sigma _i^2} \xi _{\nu _i T}(a_{\alpha }) \bigg ) \bigg ] \nonumber \\\approx & {} \alpha + (1- \alpha )^{\frac{k-1}{k}} \frac{\Delta ^2}{2} \bigg (\sum _{i=1}^k \frac{n_i}{\sigma _i^2} |\xi _{\nu _i T}(a_{\alpha })| \bigg ) . \nonumber \end{aligned}$$
For the special case \(n_1=\cdots =n_k=n\); \(\nu _1=\cdots =\nu _k=\nu =n-1\) and \( \xi _{\nu _1 T}(a_{\alpha })=\cdots = \xi _{\nu _k T}(a_{\alpha }) = \xi _{\nu T}(a_{\alpha })\), the local power of Tippett’s test reduces to
$$\begin{aligned} \text {LP(T)}\approx & {} \alpha + (1- \alpha )^{\frac{k-1}{k}} \frac{n \Delta ^2}{2} |\xi _{\nu T}(a_{\alpha })| \bigg (\sum _{i=1}^k \frac{1}{\sigma _i^2} \bigg ) \nonumber \\= & {} \alpha + \bigg [ \frac{n \Delta ^2}{2} \Psi \bigg ] \bigg [(1-\alpha )^{\frac{k-1}{k}}\bigg ] |\xi _{\nu T}(a_{\alpha })| \quad \text {where} \quad \Psi =\sum _{i=1}^k \frac{1}{\sigma _i^2}. \nonumber \end{aligned}$$
1.2 II. Local Power of Wilkinson’s Test \([LP(W_r)]\)
Using \(r^{th}\) smallest p-value \(P_{(r)}\) as a test statistic, the null hypothesis will be rejected if \( P_{(r)} < d_{r, \alpha }\), where \(P_{(r)}\) \(\sim \) Beta\([r, k-r+1]\) under \(H_0\) and \(d_{r, \alpha }\) satisfies \(\alpha =Pr\{ P_{(r)} < d_{r, \alpha } | H_0 \}=\int _0^{d_{r, \alpha }} \frac{u^{r-1} (1-u)^{k-r}}{B[r, k-r+1]}du\). This leads to
$$\begin{aligned} \text {Power}= & {} Pr[P_{(r)}< d_{r, \alpha } |H_1 ] \nonumber \\= & {} \sum _{l=r}^k Pr \{ P_{i_{1}} , \ldots , P_{i_{l}}< d_{r, \alpha } < P_{i_{l+1}} , \ldots , P_{i_{k}} |H_1 \} \nonumber \end{aligned}$$
where \((i_1, \cdots , i_l, i_{l+1}, \cdots , i_k)\) is a permutation of \((1, \cdots , k)\). Applying Lemma 1, we get
$$\begin{aligned} Pr \{ P_{i_{1}} ,&\ldots&, P_{i_{l}}< d_{r, \alpha } < P_{i_{l+1}} , \ldots , P_{i_{k}} |H_1 \} \nonumber \\\approx & {} \bigg \{ \prod _{j=1}^{l} \big ( d_{r, \alpha } - \frac{n_{i_j} \Delta ^2}{2 \sigma _{i_j}^2} \xi _{i_j W}(d_{r, \alpha }) \big ) \bigg \} \bigg \{ \prod _{j=l+1}^{k} \big (1- d_{r, \alpha } + \frac{n_{i_j} \Delta ^2}{2 \sigma _{i_j}^2} \xi _{i_j W}({d_{r, \alpha } }) \big ) \bigg \} \nonumber \\\approx & {} \bigg \{ d_{r, \alpha }^{l} - d_{r, \alpha }^{l-1} \frac{\Delta ^2}{2} \bigg (\sum _{j=1}^l \frac{n_{i_j}}{\sigma _{i_j}^2}\xi _{i_j W}({d_{r, \alpha } }) \bigg ) \bigg \} \times \nonumber \\&\bigg \{(1- d_{r, \alpha })^{k-l} + (1- d_{r, \alpha })^{k-l-1} \frac{\Delta ^2}{2} \bigg (\sum _{j=l+1}^k \frac{n_{i_j}}{\sigma _{i_j}^2}\xi _{i_j W}({d_{r, \alpha } }) \bigg ) \bigg \} \nonumber \\\approx & {} d_{r, \alpha }^{l}(1-d_{r, \alpha })^{k-l} + \frac{\Delta ^2}{2} \bigg \{ d_{r, \alpha }^{l} (1 - d_{r, \alpha })^{k-l-1} \bigg (\sum _{j=l+1}^k \frac{n_{i_j}}{\sigma _{i_j}^2}\xi _{i_j W}({d_{r, \alpha } }) \bigg ) \nonumber \\&- d_{r, \alpha }^{l-1} (1 - d_{r, \alpha })^{k-l} \bigg (\sum _{j=1}^l \frac{n_{i_j}}{\sigma _{i_j}^2}\xi _{i_j W}(a_{d_{r, \alpha } }) \bigg ) \bigg \}. \nonumber \end{aligned}$$
Permuting \((i_1, \ldots , i_k)\) over \((1, \ldots , k)\), we get for any fixed l \((r\le l \le k)\),
$$\begin{aligned} \text {1st term}&= {{k} \atopwithdelims (){l}} d_{r, \alpha }^{l} (1 - d_{r, \alpha })^{k-l} \nonumber \\ \text {2nd term}&= \frac{\Delta ^2}{2} d_{r, \alpha }^{l} (1 - d_{r, \alpha })^{k-l-1} \bigg \{ {{k-1} \atopwithdelims (){k-l-1}} \bigg (\sum _{i=1}^k \frac{n_i}{\sigma _{i}^2}\xi _{i W}({d_{r, \alpha } }) \bigg ) \bigg \} \nonumber \\ \text {3rd term}&= - \frac{\Delta ^2}{2} d_{r, \alpha }^{l-1} (1 - d_{r, \alpha })^{k- l} \bigg \{ {{k-1} \atopwithdelims (){l-1}} \bigg (\sum _{i=1}^k \frac{n_i}{\sigma _{i}^2}\xi _{i W}({d_{r, \alpha } }) \bigg ) \bigg \}. \nonumber \end{aligned}$$
The second term above follows upon noting that when \(\big [\sum _{j=l+1}^k \frac{n_{i_j}}{\sigma _{i_j}^2}\xi _{i_j W}({d_{r, \alpha } }) \big ]\) is permuted over \((i_{l+1}< \cdots <i_{k})\) \(\subset (1, \ldots , k)\), each term \(\frac{n_{i}}{\sigma _{i}^2}\xi _{i W}({d_{r, \alpha } })\) appears exactly \({{k-1} \atopwithdelims (){k- l -1}}\) times, for each \(i=1,\cdots , k\). The 3rd term, likewise, follows upon noting that when \(\big [\sum _{j=1}^l \frac{n_{i_j}}{\sigma _{i_j}^2}\xi _{i_j W}({d_{r, \alpha } }) \big ]\) is permuted over \((i_{1}< \cdots <i_{l})\) \(\subset (1, \ldots , k)\), each term \(\frac{n_{i}}{\sigma _{i}^2}\xi _{i W}({d_{r, \alpha } })\) appears exactly \({{k-1} \atopwithdelims (){l -1}}\) times, for each \(i=1,\cdots , k\).
Adding the above three terms and simplifying, we get
$$\begin{aligned} LP(W_r) \approx \alpha + {{k-1} \atopwithdelims (){r-1}} d_{r;\alpha }^{r-1} (1 - d_{r;\alpha })^{k-r} \frac{\Delta ^2}{2} \bigg [\sum _{i=1}^k \frac{n_{i}}{\sigma _{i}^2} |\xi _{i W}({d_{r, \alpha } })| \bigg ]. \nonumber \end{aligned}$$
For the special case \(n_1=\cdots =n_k=n\); \(\nu _1=\cdots =\nu _k=\nu =n-1\) and \( \xi _{\nu _1 W}(d_{r;\alpha })=\cdots = \xi _{\nu _k W}(d_{r;\alpha }) = \xi _{\nu W}(d_{r;\alpha })\), the local power of Wilkinson’s test reduces to
$$\begin{aligned} LP(W_r)\approx & {} \alpha + {{k-1} \atopwithdelims (){r-1}} d_{r;\alpha }^{r-1} (1 - d_{r;\alpha })^{k-r} \frac{n \Delta ^2}{2} |\xi _{i W}({d_{r, \alpha } })| \bigg (\sum _{i=1}^k \frac{1}{\sigma _{i}^2} \bigg ) \nonumber \\= & {} \alpha + \bigg [\frac{n \Delta ^2}{2} \Psi \bigg ] {{k-1} \atopwithdelims (){r-1}} |\xi _{\nu W}(d_{r;\alpha })| d_{r;\alpha }^{r-1} (1 - d_{r;\alpha })^{k-r} \quad \text {where} \quad \Psi =\sum _{i=1}^k \frac{1}{\sigma _i^2}. \nonumber \end{aligned}$$
1.3 III. Local Power of Inverse Normal Test [LP(INN)]
Under this test, the null hypothesis will be rejected if \( \frac{1}{\sqrt{k}}\sum _{i=1}^k U_i < - z_{\alpha } \), where \(U_i=\Phi ^{-1} (P_i)\), \(\Phi ^{-1}\) is the inverse cdf and \(z_{\alpha }\) is the upper \(\alpha \) level critical value of a standard normal distribution. This leads to
$$\begin{aligned} \text {Power} = Pr\bigg \{ \frac{1}{\sqrt{k}}\sum _{i=1}^k U_i < - z_{\alpha } |H_1 \bigg \}. \nonumber \end{aligned}$$
First, let us determine the pdf of U under \(H_1\), \(f_{H_1}(u)\), via its cdf \(F_{H_1}(u)= Pr \{ U \le u |H_1\}\).
$$\begin{aligned} Pr \{ U \le u |H_1 \}= & {} Pr \{ \Phi (U) \le \Phi (u) |H_1 \} \nonumber \\= & {} Pr \{ P \le \Phi (u) |H_1\} \quad \big [ U=\Phi ^{-1} (P) \implies P=\Phi (U) \big ]\nonumber \\= & {} 1 - Pr \{ P > \Phi (u) |H_1\} \nonumber \\\approx & {} 1- \bigg [ [1 - \Phi (u)] + \frac{n\Delta ^2}{2 \sigma ^2} \big [\xi _{\nu }(c) \big ]_{c=\Phi (u)} \bigg ] \quad \quad \big [\text {upon applying Lemma 1}\big ]\nonumber \\\approx & {} \Phi (u) - \frac{n\Delta ^2}{2 \sigma ^2} \big [\xi _{\nu }(c) \big ]_{c=\Phi (u)} . \nonumber \\ \text {This implies} \nonumber \\ f_{H_1}(u)\approx & {} \frac{d}{du} \bigg [\Phi (u) - \frac{n\Delta ^2}{2 \sigma ^2} \big [\xi _{\nu }(c) \big ]_{c=\Phi (u)} \bigg ]\nonumber \\\approx & {} \phi (u)\bigg [1 - \frac{n\Delta ^2}{2 \sigma ^2} \bigg ( \frac{d}{dc} \big [\xi _{\nu }(c) \big ]_{c=\Phi (u)} \bigg ) \bigg ]\nonumber \\\approx & {} \frac{\phi (u)\big [1 + \frac{n \nu \Delta ^2}{2 \sigma ^2} Q_\nu (u) \big ]}{{1 + \frac{n \nu \Delta ^2}{2 \sigma ^2} \int _{-\infty }^{\infty } \phi (u)Q_\nu (u) du}}, \quad Q_\nu (u)=\bigg [\frac{x^2 - 1}{x^2 +\nu } \bigg ]_{x=t_\nu (\frac{c}{2}), \quad c=\Phi (u)}. \nonumber \end{aligned}$$
Here we have used the fact that \(\frac{d}{du}[\xi _\nu (c)]=\frac{d}{dc}[\xi _\nu (c)]\frac{dc}{du}\), \(\frac{d}{dc}[\xi _\nu (c)]=-\nu Q_\nu (\cdot )\) given below in Eq. (10), upon simplification, and \(\frac{dc}{du}=\phi (u)\). The denominator in the last expression is a normalizing constant.
$$\begin{aligned} \frac{d}{dc} \xi _{\nu }(c)= & {} \frac{d}{dc} \bigg [\int _{-t_{\nu } (\frac{c}{2})}^{t_{\nu } (\frac{c}{2})} f^*(x) dx \bigg ] \quad \bigg [f^*(x)= \frac{\partial ^2 f(x |\nu , \delta )}{\partial \delta ^2} \Bigr |_{\delta = 0}=\frac{\Gamma {(\frac{\nu +1}{2})}}{\Gamma {(\frac{\nu }{2})}\sqrt{\nu \pi }} \bigg (\frac{x^2 - 1}{[\frac{x^2}{\nu }+1]^{\frac{\nu +3}{2}}}\bigg ) \bigg ]\nonumber \\= & {} \frac{d}{dc} \big [ F^*\big ({t_{\nu } (c/2)}\big ) - F^*\big ({-t_{\nu } (c/2)}\big ) \big ] \nonumber \\= & {} f^*\big ({t_{\nu } (c/2)}\big )\big [\frac{d}{dc} {t_{\nu } (c/2)} \big ] + f^*\big ({-t_{\nu } (c/2)}\big )\big [\frac{d}{dc} {t_{\nu } (c/2)} \big ]\nonumber \\= & {} \frac{d}{dc} {t_{\nu } (c/2)} \big [f^*\big ({t_{\nu } (c/2)}\big ) + f^*\big ({-t_{\nu } (c/2)}\big ) \big ] \quad f^*(x) \text { is a symmetric distribution} \nonumber \\= & {} 2f^*\big ({t_{\nu } (c/2)}\big ) \big [ \frac{d}{dc} {t_{\nu } (c/2)} \big ] . \end{aligned}$$
(8)
Further \( \big [ \frac{d}{dc} {t_{\nu } (c/2)} \big ]\) can be expressed in terms of \(f\big ({t_{\nu } (c/2)}\big )\) as follows.
$$\begin{aligned} \frac{c}{2}= & {} Pr\big [ t_{\nu } \ge t_{\nu }(c/2) \big ] \nonumber \\= & {} \int _{t_{\nu }(c/2)}^{\infty } f_{\nu }(x) dx = 1- F\big ({t_{\nu } (c/2)}\big ) \quad \bigg [ f_{\nu }(x) = \frac{\Gamma {(\frac{\nu +1}{2})}}{\sqrt{\nu \pi } \Gamma {(\frac{\nu }{2})}} \bigg ( 1 + \frac{x^2}{\nu } \bigg )^{-\frac{\nu +1}{2}} \bigg ] \nonumber \\ \frac{d}{dc}\big [ \frac{c}{2}\big ]= & {} \frac{d}{dc}\big [ 1- F\big ({t_{\nu } (c/2)}\big ) \big ]\nonumber \\= & {} - f\big ({t_{\nu } (c/2)}\big )\big [\frac{d}{dc} {t_{\nu } (c/2)} \big ] \nonumber \\ \implies \frac{d}{dc} {t_{\nu } (c/2)}= & {} \frac{-1}{2f\big ({t_{\nu } (c/2)}\big )}. \end{aligned}$$
(9)
Replacing Eq. (9) in (8) results in:
$$\begin{aligned} \frac{d}{dc} \xi _{\nu }(c)= & {} 2 f^*\big ({t_{\nu } (c/2)}\big )\bigg [ \frac{-1}{2 f\big ({t_{\nu } (c/2)}\big )} \bigg ] = -\frac{f^*\big ({t_{\nu } (c/2)}\big )}{f\big ({t_{\nu } (c/2)}\big )} \nonumber \\= & {} -\nu \bigg [\frac{x^2 - 1}{x^2 +\nu } \bigg ]_{x=t_\nu (\frac{c}{2}), \quad c=\Phi (u)}. \end{aligned}$$
(10)
Let us define \(A_\nu \), \(B_\nu \) and \(C_\nu \) as \(A_\nu =\int _{-\infty }^{\infty } u \phi (u) Q_\nu (u) du\), \(B_\nu =\int _{-\infty }^{\infty } u^2 \phi (u) Q_\nu (u) du\) and \(C_\nu =\int _{-\infty }^{\infty } \phi (u) Q_\nu (u) du\). Using these three quantities, we now approximate the distribution of U as
$$\begin{aligned} U\sim & {} N[E(U), Var(U)] \quad \text {where} \quad E(U)=\int _{-\infty }^{\infty } u f_{H_1}(u) du \approx \frac{n \nu \Delta ^2}{2 \sigma ^2} A_\nu \quad \text {and} \nonumber \\&Var(U)= \int _{-\infty }^{\infty } u^2 f_{H_1}(u) du \approx 1 + \frac{n \nu \Delta ^2}{2 \sigma ^2} [B_\nu -C_\nu ] . \nonumber \end{aligned}$$
This leads to
$$\begin{aligned} \frac{1}{\sqrt{k}}\sum _{i=1}^k U_i\sim & {} N\bigg [\frac{1}{\sqrt{k}} \sum _{i=1}^k E(U_i), \frac{1}{k} \sum _{i=1}^k Var(U_i) \bigg ] \nonumber \\\sim & {} N\bigg [\frac{\Delta ^2}{\sqrt{k}} \delta _1, 1+\frac{\Delta ^2}{k} \delta _2\bigg ] \nonumber \\ \text {where}&\delta _1&=\sum _{i=1}^k {\frac{n_i \nu _i }{2 \sigma _i^2}}A_{\nu _i} \quad \text {and} \quad \delta _2=\sum _{i=1}^k {\frac{n_i \nu _i }{2 \sigma _i^2}}[B_{\nu _i}-C_{\nu _i}]. \nonumber \end{aligned}$$
Using the above result, the local power of inverse normal test is obtained by approximating its \(Power = Pr\bigg \{ \frac{1}{\sqrt{k}}\sum _{i=1}^k U_i < - z_{\alpha } |H_1 \bigg \}\) as
$$\begin{aligned} \text {Local power (INN)}\approx & {} \Phi \bigg [ \frac{- z_\alpha - \frac{\Delta ^2}{\sqrt{k}} \delta _1}{\sqrt{1+\frac{\Delta ^2}{k} \delta _2}}\bigg ]\nonumber \\\approx & {} \Phi \bigg [- z_\alpha - \frac{\Delta ^2}{\sqrt{k}} \delta _1 + \frac{z_\alpha }{2}\frac{\Delta ^2}{k} \delta _2\bigg ]\nonumber \\\approx & {} \Phi \bigg [- z_\alpha + \frac{\Delta ^2}{\sqrt{k}}\bigg (\frac{z_\alpha }{2\sqrt{k}} \delta _2 - \delta _1 \bigg ) \bigg ]\nonumber \\\approx & {} \Phi (- z_\alpha ) + \frac{\Delta ^2}{\sqrt{k}}\phi (z_\alpha )\bigg [\frac{z_\alpha }{2\sqrt{k}} \delta _2 - \delta _1 \bigg ] \nonumber \\\approx & {} \alpha + \frac{\Delta ^2}{\sqrt{k}} \phi (z_\alpha ) \bigg [\frac{z_\alpha }{2\sqrt{k}} \delta _2 - \delta _1 \bigg ] . \nonumber \end{aligned}$$
Substituting back the expressions for \(\delta _1\) and \(\delta _2\) results in
$$\begin{aligned} LP(INN)\approx & {} \alpha + \frac{\Delta ^2}{2 \sqrt{k}} \phi (z_\alpha ) \sum _{i=1}^k \frac{n_i \nu _i}{\sigma _i^2} \bigg [\frac{z_\alpha [B_{\nu _i}-C_{\nu _i}]}{2\sqrt{k}} - A_{\nu _i} \bigg ]. \nonumber \end{aligned}$$
For the special case \(n_1=\cdots =n_k=n\) and \(\nu _1=\cdots =\nu _k=\nu =n-1\), the local power of Inverse Normal test reduces to
$$\begin{aligned} LP(INN)\approx & {} \alpha + \frac{n \nu \Delta ^2}{2 \sqrt{k}} \phi (z_\alpha ) \bigg (\sum _{i=1}^k \frac{1}{\sigma _i^2} \bigg )\bigg [ \frac{z_\alpha [B_{\nu }-C_{\nu }]}{2\sqrt{k}} - A_{\nu } \bigg ] \nonumber \\= & {} \alpha + \bigg [\frac{n \Delta ^2}{2} \Psi \bigg ] \frac{\nu }{\sqrt{k}} \phi {(z_{\alpha })}\bigg [\frac{z_{\alpha } [B_{\nu } - C_{\nu }]}{2 \sqrt{k}} - A_{\nu }\bigg ] \quad \text {where} \quad \Psi =\sum _{i=1}^k \frac{1}{\sigma _i^2}. \nonumber \end{aligned}$$
1.4 IV. Local Power of Fisher’s Test [LP(F)]
According to Fisher’s exact test, the null hypothesis will be rejected if \( \sum _{i=1}^k U_i > \chi _{2k; \alpha }^2\), where \(U_i= -2 \ln {(P_i)} \), and \(\chi _{2k; \alpha }^2\) is the upper \(\alpha \) level critical value of a \(\chi ^2\)-distribution with 2k degrees of freedom. This leads to
$$\begin{aligned} \text {Power} = Pr\bigg \{\sum _{i=1}^k U_i > \chi _{2k; \alpha }^2 |H_1 \bigg \}. \nonumber \end{aligned}$$
In a similar way to the inverse normal test in Appendix III, first let us determine the pdf of U under \(H_1\), \(g_{H_1}(u)\), via its cdf \(G_{H_1}(u)= Pr \{ U \le u |H_1\}\).
$$\begin{aligned} Pr \{ U \le u |H_1 \}= & {} Pr \{ -2 \ln {(P)} \le u |H_1\} \nonumber \\= & {} Pr \{ \ln {(P)}> -u/2 |H_1\} \nonumber \\= & {} Pr \{ P > \exp {(-u/2)} |H_1\} \nonumber \\\approx & {} [1 - \exp {(-u/2)} ] + \frac{n\Delta ^2}{2 \sigma ^2} \big [\xi _{\nu }(c) \big ]_{c=\exp {(-u/2)}} \quad \big [\text {upon applying Lemma 1}\big ]. \nonumber \\ \text {This implies} \nonumber \\ g_{H_1}(u)\approx & {} \frac{d}{du} \bigg [1 - \exp {(-u/2)} + \frac{n\Delta ^2}{2 \sigma ^2} \big [\xi _{\nu }(c) \big ]_{c=\exp {(-u/2)}} \bigg ]\nonumber \\\approx & {} \frac{1}{2} \exp {(-u/2)} + \big [\frac{n\Delta ^2}{2 \sigma ^2}\big ] \frac{d}{du} \big [\xi _{\nu }(c) \big ]_{c=\exp {(-u/2)}} \nonumber \\\approx & {} \frac{1}{2} \exp {(-u/2)} - \frac{1}{2}\exp {(-u/2)} \big [\frac{n\Delta ^2}{2 \sigma ^2}\big ] \frac{d}{dc} \big [\xi _{\nu }(c) \big ]_{c=\exp {(-u/2)}} \nonumber \\\approx & {} \frac{\frac{1}{2} \exp {(-u/2)} \big [1 + \frac{n \nu \Delta ^2}{2 \sigma ^2} \Psi _\nu (u) \big ]}{{1 + \frac{n \nu \Delta ^2}{2 \sigma ^2} \big [ \int _{0}^{\infty } \frac{1}{2}\exp {(-u/2)} \Psi _\nu (u) du\big ]}}, \quad \Psi _\nu (u){=}\bigg [\frac{x^2 - 1}{x^2 +\nu } \bigg ]_{x=t_\nu (\frac{c}{2}), \quad c=\exp {(-u/2)}} .\nonumber \end{aligned}$$
Here we have used the fact that \(\frac{d}{du}[\xi _\nu (c)]=\frac{d}{dc}[\xi _\nu (c)]\frac{dc}{du}\), \(\frac{d}{dc}[\xi _\nu (c)]=-\nu \Psi _\nu (\cdot )\) given in Eq. (10), upon simplification, and \(\frac{dc}{du}= - \frac{1}{2}\exp {(-u/2)}\). The denominator in the last expression is a normalizing constant.
Define \(D_0=\int _{0}^{\infty } \frac{1}{\Gamma {(k)}}\exp {(-u)} u^{k-1} \ln {(u)} du\) and \(D_\nu =\int _{0}^{\infty } \frac{1}{2}\exp {(-u/2)} (u-2)\Psi _\nu (u) du\). Using these quantities, we can now approximate the distribution of U as
$$\begin{aligned} U\sim & {} Gamma[\beta =2, \gamma _\nu ] \quad \text {where} \quad \gamma _\nu = \big [1 + \frac{n \nu \Delta ^2}{4\sigma ^2} D_\nu \big ]. \nonumber \end{aligned}$$
Here Gamma\([\beta , \gamma _\nu ]\) stands for a Gamma random variable with scale parameter \(\beta \) and shape parameter \(\gamma _\nu \) with the pdf \(f(x)=[e^{-x/\beta }x^{\gamma _\nu -1}]/[\beta ^{\gamma _\nu }\Gamma (\gamma _\nu )]\). By the additive property of independent \(Gamma[\beta =2, \gamma _{\nu _1}], \cdots , Gamma[\beta =2, \gamma _{\nu _k}]\) corresponding to \(U_1, \cdots , U_k\), we readily get the approximate distribution of \( (U_1+\cdots +U_k)\) as
$$\begin{aligned} \sum _{i=1}^k U_i\sim & {} Gamma\big [\beta =2, k + \Delta ^2 A \big ] \quad \text {where} \quad A= \frac{1}{4} \sum _{i=1}^k{\frac{n_i \nu _i}{\sigma _i^2}}D_{\nu _i}. \nonumber \end{aligned}$$
The local power of Fisher’s test under \(H_1\) is then obtained as follows:
$$\begin{aligned} \text {Local power (F)}\approx & {} \int _{\chi _{2k; \alpha }^2}^{\infty } \frac{\exp {(-t/2)} t^{k+A\Delta ^2 -1}}{2^{k+A\Delta ^2} \Gamma {(k+A\Delta ^2)}}dt \quad \bigg [\text {since} \quad \sum _{i=1}^k U_i {\sim } Gamma\big [\beta =2, k + \Delta ^2 A \big ] \bigg ]\nonumber \\= & {} Q(\Delta ^2). \nonumber \end{aligned}$$
We now expand \(Q(\Delta ^2)\) around \(\Delta ^2=0\) to get
$$\begin{aligned} \text {Local power (F)}\approx & {} \alpha + \Delta ^2 \int _{\chi _{2k; \alpha }^2}^{\infty } \frac{\exp {(-t/2)} t^{k-1}}{2^k} \bigg [\frac{\partial }{\partial \Delta ^2} \bigg (\frac{(t/2)^{A\Delta ^2}}{\Gamma {(k+A\Delta ^2)}} \bigg )_{\Delta ^2=0} \bigg ]dt \nonumber \\\approx & {} \alpha + \Delta ^2 \int _{\chi _{2k; \alpha }^2}^{\infty } \frac{\exp {(-t/2)} t^{k-1}}{2^k} \bigg [\frac{A \ln {(t/2})}{\Gamma {(k)}} - \frac{A \int _0^\infty \exp {(-u)} u^{k-1} \ln {(u)} du}{\Gamma ^2{(k)}} \bigg ]dt \nonumber \\\approx & {} \alpha + \Delta ^2 A \int _{\chi _{2k; \alpha }^2}^{\infty } \frac{\exp {(-t/2)} t^{k-1}}{2^k \Gamma {(k)}} \bigg [\ln {(t/2}) - \frac{ \int _0^\infty \exp {(-u)} u^{k-1} \ln {(u)} du}{\Gamma {(k)}} \bigg ]dt \nonumber \\\approx & {} \alpha + \Delta ^2 A \bigg [ E\bigg \{ \big \{\ln (T/2) \big \} I_{\{T \ge \chi _{2k; \alpha }^2\}}\bigg \}_{T\sim \chi ^2_{2k}} - \alpha D_0\bigg ]. \nonumber \end{aligned}$$
Substituting back the expressions for A results in
$$\begin{aligned} LP (F) \approx \alpha + \frac{\Delta ^2}{2} \bigg [\sum _{i=1}^k{\frac{n_i \nu _i}{2\sigma _i^2}}D_{\nu _i} \bigg ] \bigg [ E\bigg \{ \big \{\ln (T/2) \big \} I_{\{T \ge \chi _{2k; \alpha }^2\}}\bigg \}_{T\sim \chi ^2_{2k}} - \alpha D_0\bigg ] . \nonumber \end{aligned}$$
For the special case \(n_1=\cdots =n_k=n\) and \(\nu _1=\cdots =\nu _k=\nu =n-1\), the local power of Fisher’s test reduces to
$$\begin{aligned} LP(F)\approx & {} \alpha + \frac{n\Delta ^2}{2} \nu D_\nu \bigg [\sum _{i=1}^k{\frac{1}{2\sigma _i^2}}\bigg ] \bigg [ E\bigg \{ \big \{\ln (T/2) \big \} I_{\{T \ge \chi _{2k; \alpha }^2\}}\bigg \}_{T\sim \chi ^2_{2k}} - \alpha D_0\bigg ] \nonumber \\= & {} \alpha + \bigg [\frac{n \Delta ^2}{2} \Psi \bigg ] \frac{\nu D_\nu }{2} \bigg [ E\bigg \{ \big \{\ln (T/2) \big \} I_{\{T \ge \chi _{2k; \alpha }^2\}}\bigg \}_{T\sim \chi ^2_{2k}} - \alpha D_0\bigg ] \quad \text {where} \quad \Psi =\sum _{i=1}^k \frac{1}{\sigma _i^2}. \nonumber \end{aligned}$$
1.5 V. Local Power of a Modified t Test \([LP(T_1)]\)
Using this exact test based on a modified t, the null hypothesis \(H_0:\mu =\mu _0\) will be rejected if \(T_1 > d_{1\alpha }\), where \(T_1= \sum _{i=1}^k{w_{1i}} |t_i|\), \(w_{1i} \propto [Var(|t_i|)]^{-1}, Var(|t_i|)= [\nu _i (\nu _i -2)^{-1}] - \big ([\Gamma (\frac{\nu _i - 1}{2}) \sqrt{\nu _i}][\Gamma (\frac{\nu _i}{2})\sqrt{\pi }]^{-1}\big )^2\), and \(Pr\{T_1 > d_{1\alpha } | H_0 \}=\alpha \). In applications \(d_{1\alpha }\) is computed by simulation. This leads to
$$\begin{aligned} \text {Power of }T_{1}= & {} Pr\bigg \{ \sum _{i=1}^k w_{1i} |t_i|> d_{1\alpha } |H_1 \bigg \} \nonumber \\= & {} \idotsint \limits _{\sum _{i=1}^k w_{1i} |t_i|> d_{1\alpha }} \prod _{i=1}^k \big [ f_{\nu _i, \delta _i}{(t_i)} \big ] \mathrm {d} t_i \quad \big [\delta _i=\frac{\sqrt{n_i} \Delta }{\sigma _i} \big ] \nonumber \\\approx & {} \idotsint \limits _{\sum _{i=1}^k w_{1i}|t_i|> d_{1\alpha }} \prod _{i=1}^k \bigg [ f_{\nu _i}(t_i) + \delta _i \frac{\partial f_{\nu _i, \delta _i}{(t_i)}}{\partial \delta _i}\Bigr |_{\delta _i= 0} + \frac{\delta _i^2}{2} \frac{\partial ^2 f_{\nu _i, \delta _i}{(t_i)}}{\partial \delta _i^2}\Bigr |_{\delta _i= 0}\bigg ] \mathrm {d} t_i \nonumber \\\approx & {} \alpha + \sum _{j=1}^k \frac{\delta _j^2 }{2} \bigg [\idotsint \limits _{\sum _{i=1}^k w_{1i}|t_i|> d_{1\alpha }} \bigg \{\prod _{i=1}^k f_{\nu _i}(t_i)\bigg \} \bigg \{ \frac{\frac{\partial ^2 f_{\nu _j, \delta _j}{(t_j)}}{\partial \delta ^2}\big |_{\delta = 0}}{f_{\nu _j}(t_j)} \bigg \}\bigg ] \prod _{i=1}^k\mathrm {d} t_i \nonumber \\\approx & {} \alpha + \sum _{j=1}^k \frac{\delta _j^2 }{2} \bigg [E_{H_0}\bigg [ \bigg \{\frac{\frac{\partial ^2 f_{\nu _j, \delta _j}(t_j)}{\partial \delta _j^2}\Bigr |_{\delta _j= 0}}{f_{\nu _j}(t_j)} \bigg \} I_{\{\sum _{i=1}^k w_{1i}|t_i|> d_{1\alpha }\}}\bigg ] \bigg ] \nonumber \\\approx & {} \alpha + \sum _{j=1}^k \frac{\delta _j^2 }{2} \bigg [E_{H_0}\bigg [ \bigg \{\frac{(t_j^2 - 1)\nu _j}{t_j^2 + \nu _j} \bigg \} I_{\{\sum _{i=1}^k w_{1i}|t_i|> d_{1\alpha }\}} |H_0 \bigg ] \bigg ] \nonumber \\\approx & {} \alpha + \frac{\Delta ^2}{2}\bigg (\sum _{j=1}^k \frac{n_j }{\sigma ^2_j} E_{H_0}\bigg [ \bigg \{\frac{(t_j^2 - 1)\nu _j}{t_j^2 + \nu _j} \bigg \} I_{\{\sum _{i=1}^k w_{1i}|t_i| > d_1\alpha \}}\bigg ] \bigg ) \quad \text {using} \quad \bigg [ \delta _j=\frac{\sqrt{n_j} \Delta }{\sigma _j} \bigg ]. \nonumber \end{aligned}$$
\(E_{H_0}[\cdot ]\) above is computed by simulation. It is easy to verify from Sect. 3 that the product terms \(\bigg \{ \frac{\partial f_{\nu _i, \delta _i}(t_i) }{\partial \delta _i} \Bigr |_{\delta _i= 0}\bigg \} \times \bigg \{ \frac{\partial f_{\nu _j, \delta _j}(t_j) }{\partial \delta _j} \Bigr |_{\delta _j= 0}\bigg \}\) involve \((t_i t_j)\), apart from \(t_i^2\) and \(t_j^2\), whose integral over \(\{\sum _{i=1}^k w_{1i}|t_i| > d_{1\alpha }\}\) under \(H_0\) is zero.
For the special case \(n_1=\cdots =n_k=n\) and \(\nu _1=\cdots =\nu _k=\nu =n-1\) which implies \(w_{11}=\cdots =w_{1k}=1\), the local power of this exact test based on modified t reduces to
$$\begin{aligned} LP(T_1)\approx & {} \alpha + \frac{n \Delta ^2}{2} \bigg (\sum _{j=1}^k \frac{1 }{\sigma ^2_j} \bigg ) E_{H_0}\bigg [ \bigg \{\frac{(t_1^2 - 1)\nu }{t_1^2 + \nu } \bigg \} I_{\{\sum _{i=1}^k |t_i|> d_1\alpha \}}\bigg ] \nonumber \\= & {} \alpha + \bigg [\frac{n \Delta ^2}{2} \Psi \bigg ] E_{H_0} \bigg [ \bigg \{\frac{(t_1^2-1)\nu }{t_1^2+\nu } \bigg \} I_{\{\sum _{i=1}^{k} |t_i| > d_{1\alpha }\}} \bigg ] \quad \text {where} \quad \Psi =\sum _{j=1}^k \frac{1}{\sigma _j^2}. \nonumber \end{aligned}$$
1.6 VI. Local Power of a Modified F Test \([LP(T_2)]\)
According to this exact test based on a modified F, the null hypothesis \(H_0:\mu =\mu _0\) will be rejected if \(T_2 > d_{2\alpha }\), where \(T_2=\sum _{i=1}^k{w_{2i}}F_i\), \(F_i \sim F(1, \nu _i)\), \(w_{2i} \propto [Var(F_i)]^{-1}=[2\nu _i^2 (\nu _i-1)]^{-1}[(\nu _i - 2)^2 (\nu _i - 4)]\), and \(Pr\{T_2 > d_{2\alpha } | H_0 \}=\alpha \). In applications \(d_{2\alpha }\) is computed by simulation. This leads to
$$\begin{aligned} \text {Power of }T_{2}= & {} Pr\bigg \{ \sum _{i=1}^k w_{2i}F_i> d_{2\alpha } |H_1 \bigg \} \nonumber \\= & {} \idotsint \limits _{ \sum _{i=1}^k w_{2i}F_i > d_{2\alpha }} \prod _{i=1}^k \big [ f_{\nu _i, \lambda _i}{(F_i)} \big ] \mathrm {d} F_i \quad \bigg [f_{\nu , \lambda }{(F)} \sim \text {non-central}\quad F_{1,\nu }\bigg (\lambda =\frac{n\Delta ^2}{\sigma ^2}\bigg ) \bigg ]. \nonumber \end{aligned}$$
Note that \(f_{\nu , \lambda }(F)\) and its local expansion around \(\lambda =0\) are give by
$$\begin{aligned} f_{\nu ,\lambda }(F)= & {} \exp {(-\frac{\lambda }{2})} \sum _{j=0}^\infty \frac{(\frac{\lambda }{2})^j}{j!}\bigg [\frac{(\frac{\nu _1}{\nu _2})^{\frac{\nu _1+2j}{2}} \Gamma {(\frac{\nu _1+\nu _2+2j}{2})}}{\Gamma {(\frac{\nu _1+2j}{2})}\Gamma {(\frac{\nu _2}{2})}} \bigg ]\bigg [\frac{ F^{\frac{\nu _1+2j}{2}-1}}{\big (1+F\frac{\nu _1}{nu_2} \big )^{\frac{\nu _1+\nu _2+2j}{2}}} \bigg ] \nonumber \\\approx & {} f_\nu (F) \big (1-\frac{\lambda }{2}\big ) + \bigg [\frac{ (\frac{\lambda }{2}) (\frac{\nu _1}{\nu _2})^{\frac{\nu _1+2}{2}} \Gamma {(\frac{\nu _1+\nu _2+2}{2})}}{\Gamma {(\frac{\nu _1+2}{2})}\Gamma {(\frac{\nu _2}{2})}} \bigg ]\bigg [\frac{ F^{\nu _1}}{\big (1+F\frac{\nu _1}{\nu _2} \big )^{\frac{\nu _1+\nu _2+2}{2}}} \bigg ] \nonumber \\= & {} f_\nu (F) + \frac{\lambda }{2} \big [f_\nu ^*(F) - f_\nu (F) \big ], \quad \text {where} \quad f_{\nu }^*(F)=\bigg (\frac{1}{\nu }\bigg )^{\frac{3}{2}} \bigg [\frac{F}{(1+\frac{F}{\nu })^{\frac{\nu +3}{2}} B[\frac{3}{2}, \frac{\nu }{2}]} \bigg ]. \nonumber \end{aligned}$$
Using the above first-order expansion of \(f_{\nu ,\lambda }(F)\) leads to the following local power of \(T_2\).
$$\begin{aligned} LP(T_2)\approx & {} \idotsint \limits _{ \sum _{i=1}^k w_{2i}F_i> d_{2\alpha }}\bigg [\prod _{i=1}^k f_{\nu _i}(F_i) + \sum _{j=1}^k \frac{\lambda _j}{2} \bigg ( f_{\nu _j}^*(F_j) - f_{\nu _j}(F_j) \bigg ) \bigg \{\prod _{i \ne j} \big [f_{\nu _i}(F_i) \big ]\bigg \} \bigg ] \prod _{i=1}^k\mathrm {d} F_i \nonumber \\\approx & {} \alpha + \bigg (\sum _{j=1}^k \frac{\lambda _j}{2}E_{H_0}\bigg [\bigg \{\frac{f_{\nu _j}^*(F_j) - f_{\nu _j}(F_j)}{f_{\nu _j}(F_j)}\bigg \}I_{\{\sum _{i=1}^k w_{2i}F_i> d_{2\alpha }\}} \bigg ]\bigg ) \nonumber \\&E_{H_0}[\cdot ]\quad \text {stands for expectation w.r.t} \quad F_1, \ldots , F_k \quad \text {under} \quad H_0 [F_i \sim F(1, \nu _i)]. \nonumber \\\approx & {} \alpha + \bigg (\sum _{j=1}^k \frac{\lambda _j}{2} E_{H_0}\bigg [\bigg \{\frac{F_j - 1}{\frac{F_j}{\nu _j}+1}\bigg \}I_{\{\sum _{i=1}^k w_{2i}F_i> d_{2\alpha }\}} \bigg ] \bigg )\nonumber \\\approx & {} \alpha + \frac{\Delta ^2}{2}\bigg (\sum _{j=1}^k \frac{n_j}{\sigma _j^2} E_{H_0}\bigg [\bigg \{\frac{[F_j - 1]\nu _j}{F_j+\nu _j}\bigg \}I_{\{\sum _{i=1}^k w_{2i}F_i > d_{2\alpha }\}} \bigg ] \bigg ) \quad \text {using} \quad \bigg [ \lambda _j=\frac{n_j\Delta ^2}{\sigma _j^2} \bigg ] .\nonumber \\&E_{H_0}[\cdot ] \quad \text {is obtained by simulation.}\nonumber \end{aligned}$$
For the special case \(n_1=\cdots =n_k=n\) and \(\nu _1=\cdots =\nu _k=\nu =n-1\) which implies \(w_{21}=\cdots =w_{2k}=1\), the local power of this exact test based on modified F reduces to
$$\begin{aligned} LP(T_2)\approx & {} \alpha + \frac{n\Delta ^2}{2}\bigg (\sum _{j=1}^k \frac{1}{\sigma _j^2}\bigg ) E_{H_0}\bigg [\bigg \{\frac{[F_1 - 1]\nu }{F_1+\nu }\bigg \}I_{\{\sum _{i=1}^k F_i> d_{2\alpha }\}} \bigg ] \nonumber \\= & {} \alpha + \bigg [\frac{n \Delta ^2}{2} \Psi \bigg ] E_{H_0} \bigg [ \bigg \{\frac{[F_1 - 1]\nu }{F_1 + \nu } \bigg \} I_{\{\sum _{i=1}^{k} F_i > d_{2\alpha }\}} \bigg ] \quad \text {where} \quad \Psi =\sum _{j=1}^k \frac{1}{\sigma _j^2}.\nonumber \end{aligned}$$
