Appendix
Proof of Theorem in Sect. 2.
From the Eqs. (3) and (4), we have a set of equations
$$\begin{aligned} \left\{ \begin{array}{l} \frac{B_1^2-\sigma _1^2}{\sigma _1 \bar{X}_1}= \frac{c_1 \bar{X}_1}{\sigma _1} + \frac{c_2 \bar{X}_2}{\sigma _2} + \cdots + \frac{c_k \bar{X}_k}{\sigma _k}\\ \frac{B_2^2-\sigma _2^2}{\sigma _2 \bar{X}_2}= \frac{c_1 \bar{X}_1}{\sigma _1} + \frac{c_2 \bar{X}_2}{\sigma _2} + \cdots + \frac{c_k \bar{X}_k}{\sigma _k}\\ \cdots \cdots \\ \frac{B_k^2-\sigma _k^2}{\sigma _k \bar{X}_k}= \frac{c_1 \bar{X}_1}{\sigma _1} + \frac{c_2 \bar{X}_2}{\sigma _2} + \cdots + \frac{c_k \bar{X}_k}{\sigma _k} \end{array} \right. \end{aligned}$$
(7)
whose equivalent equations are given as follows
$$\begin{aligned} \left\{ \begin{array}{l} \sigma _1 = \frac{1}{2} \left[ \sqrt{4 B_1^2\!+\!\bar{X}_1^2 \Bigg (\frac{c_1 \bar{X}_1}{\sigma _1} \!+\! \frac{c_2 \bar{X}_2}{\sigma _2} + \cdots + \frac{c_k \bar{X}_k}{\sigma _k} \Bigg )^2} \!-\! \bar{X}_1 \Bigg ( \frac{c_1 \bar{X}_1}{\sigma _1} \!+\! \frac{c_2 \bar{X}_2}{\sigma _2} + \cdots + \frac{c_k \bar{X}_k}{\sigma _k} \Bigg )\right] \\ \sigma _2 = \frac{1}{2} \left[ \sqrt{4 B_2^2\!+\!\bar{X}_2^2 \Bigg (\frac{c_1 \bar{X}_1}{\sigma _1} \!+\! \frac{c_2 \bar{X}_2}{\sigma _2} + \cdots + \frac{c_k \bar{X}_k}{\sigma _k} \Bigg )^2} \!-\! \bar{X}_2 \Bigg ( \frac{c_1 \bar{X}_1}{\sigma _1} \!+\! \frac{c_2 \bar{X}_2}{\sigma _2} + \cdots + \frac{c_k \bar{X}_k}{\sigma _k} \Bigg )\right] \\ \cdots \cdots \\ \sigma _k = \frac{1}{2} \left[ \sqrt{4 B_k^2\!+\!\bar{X}_k^2 \Bigg (\frac{c_1 \bar{X}_1}{\sigma _1} \!+\! \frac{c_2 \bar{X}_2}{\sigma _2} + \cdots + \frac{c_k \bar{X}_k}{\sigma _k} \Bigg )^2} \!-\! \bar{X}_k \Bigg ( \frac{c_1 \bar{X}_1}{\sigma _1} \!+\! \frac{c_2 \bar{X}_2}{\sigma _2} + \cdots + \frac{c_k \bar{X}_k}{\sigma _k} \Bigg )\right] . \end{array} \right. \end{aligned}$$
We can denote the above equations \({\varvec{\sigma }}={\varvec{\phi }}({\varvec{\sigma }})\), where \({\varvec{\sigma }}=(\sigma _1, \ldots , \sigma _k)'\), \({\varvec{\phi }}({\varvec{\sigma }}) = (\phi _1({\varvec{\sigma }}), \ldots , \phi _k({\varvec{\sigma }}))'\) with
$$\begin{aligned} \phi _i({\varvec{\sigma }})&= \frac{1}{2} \left[ \sqrt{4 B_i^2+\bar{X}_i^2 \Bigg (\frac{c_1 \bar{X}_1}{\sigma _1} + \frac{c_2 \bar{X}_2}{\sigma _2} + \cdots + \frac{c_k \bar{X}_k}{\sigma _k} \Bigg )^2}\right. \\&\left. - \bar{X}_i \Bigg ( \frac{c_1 \bar{X}_1}{\sigma _1} + \frac{c_2 \bar{X}_2}{\sigma _2} + \cdots + \frac{c_k \bar{X}_k}{\sigma _k} \Bigg )\right] . \end{aligned}$$
In the proposed iterative algorithm, we set the initial values \(\hat{\lambda }^{(0)}=\frac{c_1 \bar{X}_1}{s_1} + \frac{c_2 \bar{X}_2}{s_2} + \cdots + \frac{c_k \bar{X}_k}{s_k}\) and \(\hat{\sigma }_{i}^{(0)}=s_i, \; i=1, \ldots , k\), and we have the iterative formulae
$$\begin{aligned} \left\{ \begin{array}{l} \hat{\sigma }_1^{(l+1)} = \frac{1}{2} \left[ \sqrt{4 B_1^2\!+\!\bar{X}_1^2 \Bigg (\frac{c_1 \bar{X}_1}{\hat{\sigma }_1^{(l)}} \!+\! \frac{c_2 \bar{X}_2}{\hat{\sigma }_2^{(l)}} + \cdots + \frac{c_k \bar{X}_k}{\hat{\sigma }_k^{(l)}} \Bigg )^2} \!-\! \bar{X}_1 \Bigg ( \frac{c_1 \bar{X}_1}{\hat{\sigma }_1^{(l)}} \!+\! \frac{c_2 \bar{X}_2}{\hat{\sigma }_2^{(l)}} + \cdots + \frac{c_k \bar{X}_k}{\hat{\sigma }_k^{(l)}} \Bigg )\right] \\ \hat{\sigma }_2^{(l+1)} = \frac{1}{2} \left[ \sqrt{4 B_2^2\!+\!\bar{X}_2^2 \Bigg (\frac{c_1 \bar{X}_1}{\hat{\sigma }_1^{(l)}} \!+\! \frac{c_2 \bar{X}_2}{\hat{\sigma }_2^{(l)}} + \cdots + \frac{c_k \bar{X}_k}{\hat{\sigma }_k^{(l)}} \Bigg )^2} \!-\! \bar{X}_2 \Bigg ( \frac{c_1 \bar{X}_1}{\hat{\sigma }_1^{(l)}} \!+\! \frac{c_2 \bar{X}_2}{\hat{\sigma }_2^{(l)}} + \cdots + \frac{c_k \bar{X}_k}{\hat{\sigma }_k^{(l)}} \Bigg )\right] \\ \cdots \cdots \\ \hat{\sigma }_k^{(l+1)} = \frac{1}{2} \left[ \sqrt{4 B_k^2\!+\!\bar{X}_k^2 \Bigg (\frac{c_1 \bar{X}_1}{\hat{\sigma }_1^{(l)}} \!+\! \frac{c_2 \bar{X}_2}{\hat{\sigma }_2^{(l)}} + \cdots + \frac{c_k \bar{X}_k}{\hat{\sigma }_k^{(l)}} \Bigg )^2} \!-\! \bar{X}_k \Bigg ( \frac{c_1 \bar{X}_1}{\hat{\sigma }_1^{(l)}} \!+\! \frac{c_2 \bar{X}_2}{\hat{\sigma }_2^{(l)}} + \cdots + \frac{c_k \bar{X}_k}{\hat{\sigma }_k^{(l)}} \Bigg )\right] . \end{array} \right. \nonumber \\ \end{aligned}$$
(8)
Note that we have the Jacobian matrix of \({\varvec{\phi }}({\varvec{\sigma }})\) as \(J({\varvec{\phi }})= \partial {\varvec{\phi }}({\varvec{\sigma }})/\partial {\varvec{\sigma }}\) with the \((i,j)\)th element
$$\begin{aligned} J_{ij}=\frac{c_j \bar{X}_j}{2 \sigma _j^2} \left[ \bar{X}_i-\frac{\bar{X}_i^2 \lambda }{\sqrt{4B_i^2+(\bar{X}_i \lambda )^2}} \right] . \end{aligned}$$
If the following limits exist,
$$\begin{aligned} \lim _{n_i \rightarrow \infty } P \Bigg \{ \max _{1 \le i \le k} \Bigg [ \Bigg | \bar{X}_i-\frac{\bar{X}_i^2 \lambda }{\sqrt{4 B_i^2+(\bar{X}_i\lambda )^2}} \Bigg | \Bigg ] < \frac{2}{\sum \limits _{i=1}^{k} \frac{c_i |\bar{X}_i|}{\sigma _i^2}}\Bigg \}=1, \quad i=1, \ldots , k, \end{aligned}$$
the Jacobian matrix \(J({\varvec{\phi }})\) of \({\varvec{\phi }}({\varvec{\sigma }})\) has row norm
$$\begin{aligned} ||J({\varvec{\phi }})||_{\infty }&= \max _{1 \le i \le k} \Bigg \{ \Bigg |\frac{\partial \phi _i({\varvec{\sigma }})}{\partial \sigma _1} \Bigg | + \cdots + \Bigg |\frac{\partial \phi _i({\varvec{\sigma }})}{\partial \sigma _k} \Bigg |\Bigg \}\\&= \left| \sum \limits _{i=1}^{k} \frac{c_i \bar{X}_i}{\sigma _i^2}\right| \max _{1 \le i \le k} \left\{ \frac{|\bar{X}_i|}{2} \Bigg | 1-\frac{\bar{X}_i \lambda }{\sqrt{4 B_i^2+\bar{X}_i^2 \lambda ^2}}\Bigg |\right\} <1 \end{aligned}$$
in probability. By the discriminants of the iterative solution of nonlinear equations, the vector sequence \(\{\hat{{\varvec{\sigma }}}^{(l)},\; l=1, 2, \ldots \}=\{(\hat{\sigma }_1^{(l)}, \ldots , \hat{\sigma }_k^{(l)})',\; l=1, 2, \ldots \}\) obtained by the iterative formulae (8) converges to the solution of the nonlinear equations (7).
Note that under the null hypothesis \(H_0\), we have the log-likelihood function
$$\begin{aligned} l_0&= -\frac{n}{2} \log (2 \pi ) - \sum \limits _{i=1}^{k} n_i \log \sigma _i - \frac{1}{2} \sum \limits _{i=1}^{k} \Bigg [ \frac{n_i s_i^2}{\sigma _i^2} + n_i \Bigg ( \frac{\bar{X}_i}{\sigma _i}-\lambda \Bigg )^2\Bigg ], \end{aligned}$$
from which we derive the matrix \(\Phi \) of the second partial derivatives of \(l_0\) with respect to the parameters \(({\varvec{\sigma }}, \lambda )'\)
$$\begin{aligned} \Phi&= \left[ \begin{array}{l@{\quad }l@{\quad }l@{\quad }l@{\quad }l} \frac{\partial ^2 l_0}{\partial \sigma _1^2} &{}\frac{\partial ^2 l_0}{\partial \sigma _1 \partial \sigma _2}&{} \cdots &{} \frac{\partial ^2 l_0}{\partial \sigma _1 \partial \sigma _k}&{}\frac{\partial ^2 l_0}{\partial \sigma _1 \partial \lambda }\\ \frac{\partial ^2 l_0}{\partial \sigma _2 \sigma _1} &{}\frac{\partial ^2 l_0}{\partial \sigma _2^2}&{} \cdots &{} \frac{\partial ^2 l_0}{\partial \sigma _2 \partial \sigma _k}&{}\frac{\partial ^2 l_0}{\partial \sigma _2 \partial \lambda }\\ &{}&{}\cdots &{}&{}\\ \frac{\partial ^2 l_0}{\partial \sigma _k \partial \sigma _1} &{}\frac{\partial ^2 l_0}{\partial \sigma _k \partial \sigma _2}&{} \cdots &{} \frac{\partial ^2 l_0}{\partial \sigma _k^2}&{}\frac{\partial ^2 l_0}{\partial \sigma _k \partial \lambda }\\ \frac{\partial ^2 l_0}{\partial \lambda \partial \sigma _1} &{}\frac{\partial ^2 l_0}{\partial \lambda \partial \sigma _2}&{} \cdots &{} \frac{\partial ^2 l_0}{\partial \lambda \partial \sigma _k}&{}\frac{\partial ^2 l_0}{\partial \lambda ^2} \end{array} \right] \end{aligned}$$
where
$$\begin{aligned}&\frac{\partial ^2 l_0}{\partial \sigma _i \partial \sigma _j} = 0, \qquad i\ne j, \; i, \; j=1, \ldots , k,\\&\frac{\partial ^2 l_0}{\partial \sigma _i \partial \lambda } = \frac{\partial ^2 l_0}{\partial \lambda \partial \sigma _i}=-\frac{n_i \bar{X}_i}{\sigma _i^2}, \quad i=1, \ldots , k.\\&\frac{\partial ^2 l_0}{\partial \sigma _i^2} = \frac{n_i (\sigma _i^2+2 \lambda \bar{X}_i \sigma _i-3 s_i^2 - 3 \bar{X}_i^2)}{\sigma _i^4}, \quad i=1, \ldots , k,\\&\frac{\partial ^2 l_0}{\partial \lambda ^2} = -n. \end{aligned}$$
Thus the determinant of \(\Phi \) evaluated at the MLEs \((\hat{{\varvec{\sigma }}}_0, \hat{\lambda }_0)'\) of \(({\varvec{\sigma }}, \lambda )'\) is calculated as
$$\begin{aligned} |\Phi |_{(\hat{{\varvec{\sigma }}}_0, \hat{\lambda }_0)'}&= - \left[ \sum \limits _{i=1}^{k} \frac{n_i\left( \hat{\sigma }_{i0}^2 + 2 \hat{\lambda }_0 \bar{X}_i \hat{\sigma }_{i0} + \hat{\sigma }_{i0}^2 \bar{X}_i - 3 s_i^2 - 3 \bar{X}_i^2 \right) }{\hat{\sigma }_{i0}^2 + 2 \hat{\lambda }_0 \bar{X}_i \hat{\sigma }_{i0} -3 s_i^2 - 3 \bar{X}_i^2}\right] \times \left[ \,\prod _{i=1}^{k} \frac{n_i}{\hat{\sigma }_{i0}^4}\right] \\&\qquad \times \left[ \,\prod _{i=1}^{k}\left( \hat{\sigma }_{i0}^2 + 2 \hat{\lambda }_0 \bar{X}_i \hat{\sigma }_{i0} - 3 s_i^2 - 3 \bar{X}_i^2 \right) \right] . \end{aligned}$$
Since for any \(i\),
$$\begin{aligned} \lim _{n_i \rightarrow \infty } P\left( \hat{\sigma }_{i0}^2 + 2 \hat{\lambda }_0 \bar{X}_i \hat{\sigma }_{i0} - 3 s_i^2 - 3 \bar{X}_i^2 <0\right)&= 1, \qquad i=1, \dots , k, \end{aligned}$$
all the \(q\)th order leading principal minors of the matrix \(\Phi \) are negative in probability if \(q\) is odd and positive in probability if \(q\) is even. Therefore, the matrix \(\Phi \) is negative definite in probability. By the discriminants of multifunction extreme values, the solution \((\hat{\sigma }_{10}, \dots , \hat{\sigma }_{k0}, \hat{\lambda }_0)'\) is the maximizer of the log-likelihood function \(l_0\), and thus, it is the MLE of the parameter \((\sigma _1, \dots , \sigma _k, \lambda )'\) in the space \(\Theta _0\).
