Inference about a Common Mean Vector from Several Independent Multinormal Populations with Unequal and Unknown Dispersion Matrices

Abstract

This paper addresses the problem of making inferences about a common mean vector from several independent multivariate normal populations with unknown and unequal dispersion matrices. We propose an unbiased estimator of the common mean vector, along with its asymptotic estimated variance, which can be used to test hypotheses and construct confidence ellipsoids, both of which are valid for large samples. Additionally, we discuss an approximate method based on generalized p-values. The paper also presents exact test procedures and methods for constructing exact confidence sets for the common mean vector, with a comparison of the local power of these exact tests. The performance of the proposed methods is demonstrated through a simulation study and an application to data from the Current Population Survey (CPS) Annual Social and Economic (ASEC) Supplement 2021 conducted by the U.S. Census Bureau for the Bureau of Labor Statistics.

1. Introduction

The inferential problem of drawing an inference about a common mean vector $\mathbf{\mu }$ of several independent normal populations with unequal and unknown dispersion matrices is considered in this paper. We treat the problems of: (1) point estimation of $\mathbf{\mu }$; (2) test for $H_0:\mathbf{\mu }={\mathbf{\mu }}_0$ versus $H_1:\mathbf{\mu }\ne {\mathbf{\mu }}_0$; and (3) construction of confidence sets for $\mathbf{\mu }$.

Suppose there are $k(k\ge 2)$ p-variate normal populations with common mean vector $\mathbf{\mu }$ and unknown covariance matrices ${\sum }_1,\dots ,{\sum }_k$. Let ${\mathit{X}}_{i1},\dots ,{\mathit{X}}_{i{n}_i}$ be independent p-variate vector sample observations from the i-th population ($i=1,\dots ,k$) and ${\mathit{X}}_{ij}\sim N_p(\mathbf{\mu },{\sum }_i)$, $j=1,\cdots ,n_i$. For the i-th population, let

$${\overline{\mathit{X}}}_i={\displaystyle \frac{1}{n_i}}\sum _{j=1}^{n_i}{\mathit{X}}_{ij}\mathrm{and}{\mathit{S}}_i=\sum _{j=1}^{n_i}({\mathit{X}}_{ij}-{\overline{\mathit{X}}}_i){({\mathit{X}}_{ij}-{\overline{\mathit{X}}}_i)}^t$$

(1)

be the sample mean vector and sample sum of squares and products matrix. Jointly, $\{{\overline{\mathit{X}}}_i,{\mathit{S}}_i,i=1,\cdots ,k\}$ provide minimal sufficient statistics for the unknown parameters $\mathbf{\mu }$ and ${\sum }_i$ ($i=1,\cdots ,k$). It is well-known that one can use the familiar Hotelling’s $T^2$ test for $H_0$ and reject the null based on the i-th dataset when $T_i^2=n_i{({\overline{\mathit{X}}}_i-{\mathbf{\mu }}_0)}^t{\mathit{S}}_i^{-1}({\overline{\mathit{X}}}_i-{\mathbf{\mu }}_0)$ is large.

A confidence set for $\mathbf{\mu }$ based on the i-th dataset is also readily obtained as $Pr\left\{\mathbf{\mu }:\left[{\displaystyle \frac{n_i(n_i-p)}{p}}\right]{({\overline{\mathit{X}}}_i-\mathbf{\mu })}^t{\mathit{S}}_i^{-1}({\overline{\mathit{X}}}_i-\mathbf{\mu })\le F_{\alpha ,p,n_i-p}\right\}=1-\alpha ,i=1,\dots ,k.$

In Section 2, we provide an unbiased estimate of $\mathbf{\mu }$ based on the minimal sufficient statistics and provide an expression for its estimated asymptotic variance. A test procedure for $H_0$ and a confidence ellipsoid for $\mathbf{\mu }$ then readily follow. These results are asymptotic in nature.

An approximate procedure for a test set and a confidence set for $\mathbf{\mu }$ in our context was suggested by [1] based on the notion of generalized p-values [2,3,4]. This is briefly mentioned in Section 3. The authors clearly presented relevant algorithms to carry out the suggested procedures and also discussed their performance in terms of coverage probabilities and expected volumes in comparison with some existing methods. Notwithstanding their claim of anticipated better performance over existing procedures, the fact remains that the generalized p-value-based procedure is indeed approximate and may not work well in some situations in view of the unknown and unequal nature of the population dispersion matrices (see Tables 5 and 6 in [1]).

Exact tests for $H_0$ and exact confidence sets for $\mathbf{\mu }$ can be derived by efficiently combining the k independent Hotelling’s $T^2$ statistics. Fortunately, several well-known exact procedures exist in the literature [5,6]. In Section 4, we provide a review of these exact test procedures and in Section 5, we provide a review of exact confidence sets. Local powers of the exact tests are discussed in Section 6 based on a Taylor expansion of the power, and these are compared in Section 7. In Section 8, two applications are provided. First, we reproduce a dataset from [5] for $p=2$, $k=2$, and $n_1=n_2=12$, each from $N_2(\mathbf{\mu },{\sum }_{\mathbf{1}})$, $N_2(\mathbf{\mu },{\sum }_{\mathbf{2}})$, and use it to construct a confidence set for the bivariate common mean vector based on the above exact procedures. Plots showing the confidence sets appear in Figure 1. Our second example is based on data arising from Current Population Survey (CPS) conducted by the Bureau of the Census for the Bureau of Labor Statistics. It turns out that the sample sizes are large for the CPS data, thus enabling us to also include the large sample procedure described in Section 2. Plots showing the confidence sets appear in Figure 2, Figure 3 and Figure 4. For the sake of completeness, we have also plotted the confidence set derived from the generalized p-value-based method [1] in both the applications. We conclude the paper with some conclusions in Section 9.

2. A Large Sample Procedure

In this section, we propose an unbiased estimate of $\mathbf{\mu }$, which is essentially a generalization of the familiar Graybill–Deal estimate [7] of $\mathrm{\mu }$ in the case of univariate normal populations. This estimate and its estimated asymptotic variance in the univariate case are given by:

$${\widehat{\mathrm{\mu }}}_{GD}={\left[\sum _{i=1}^k{\displaystyle \frac{n_i}{S_i^2}}\right]}^{-1}\left[\sum _{i=1}^k{\displaystyle \frac{n_i}{S_i^2}}{\overline{X}}_i\right]\mathrm{with}\widehat{Var}\left({\widehat{\mathrm{\mu }}}_{GD}\right)={\left[\sum _{i=1}^k{\displaystyle \frac{n_i}{S_i^2}}\right]}^{-1}.$$

(2)

A test for $H_0:\mu ={\mu }_0$ versus $H_1:\mu \ne {\mu }_0$ is based on the standard normal Z statistic defined as $Z=({\widehat{\mathrm{\mu }}}_{GD}-{\mu }_0)/\sqrt{\widehat{Var}\left({\widehat{\mathrm{\mu }}}_{GD}\right)}$. An asymptotic confidence interval for $\mathrm{\mu }$ is given by ${\widehat{\mathrm{\mu }}}_{GD}\mp Z_{\alpha /2}\sqrt{\widehat{Var}\left({\widehat{\mathrm{\mu }}}_{GD}\right)}$.

As a generalization to the multivariate case, we propose:

$${\tilde{\mathbf{\mu }}}_{GD}={\left[\sum _{i=1}^k{n}_i{\mathit{S}}_i^{-1}\right]}^{-1}\left[\sum _{i=1}^k{n}_i{\mathit{S}}_i^{-1}{\overline{\mathit{X}}}_i\right]\mathrm{with}\widehat{Var}\left({\tilde{\mathbf{\mu }}}_{GD}\right)={\left[\sum _{i=1}^k{n}_i{\mathit{S}}_i^{-1}.\right]}^{-1}$$

(3)

An asymptotic test for $H_0:\mathbf{\mu }={\mathbf{\mu }}_0$ versus $H_1:\mathbf{\mu }\ne {\mathbf{\mu }}_0$ can then be based on the ${\chi }_p^2$ statistic:

$${\chi }_p^2={({\tilde{\mathbf{\mu }}}_{GD}-\mathbf{\mu })}^t\left[\sum _{i=1}^k{n}_i{\mathit{S}}_i^{-1}\right]({\tilde{\mathbf{\mu }}}_{GD}-\mathbf{\mu }).$$

(4)

The $(1-\alpha )100\%$ asymptotic ellipsoidal confidence set for $\mathbf{\mu }$ is provided by:

$$Pr\left\{{({\tilde{\mathbf{\mu }}}_{GD}-\mathbf{\mu })}^t\left[\sum _{i=1}^k{n}_i{\mathit{S}}_i^{-1}\right]({\tilde{\mathbf{\mu }}}_{GD}-\mathbf{\mu })\le {\chi }_{p,\alpha }^2\right\}=1-\alpha .$$

(5)

Our simulation studies (see Appendix A) demonstrate the robustness of the ${\chi }^2$ cut-off point for variations in the unknown dispersion matrices. Applications of (5) for a real ASEC dataset appear in Section 8.

3. Confidence Set Based on Generalized p-Value

Tsui and Weerahandi, 1989 [2] came up with a novel idea to deal with uncommon inference problems. Examples include the ANOVA problem under variance heteroscedasticity, the test for treatment of variance components in a one-way random effects model, the test for reliability parameter $Pr(X>Y)=1-\mathsf{\Phi }\left[{\displaystyle \frac{{\mu }_y-{\mu }_x}{\sqrt{{\sigma }_x^2+{\sigma }_y^2}}}\right]$ when $X\sim N({\mu }_x,{\sigma }_x^2)$, independent of $Y\sim N({\mu }_y,{\sigma }_y^2)$, and so on. The method is based on a function $h(X;x,\theta ,\eta )$ of underlying random variable X, its observed value x, and the parameter of interest $\theta$, with $\eta$ being a nuisance parameter. Under certain conditions of $h(·)$, a test for $\theta$ and a confidence set for $\theta$ can be derived. Their method is referred to as the generalized p-value-based approach.

In our context, following [1], Ref. [2] suggested the following algorithm to construct a confidence ellipsoid for $\mathbf{\mu }$. An extension of the generalized p-value method from a common univariate normal mean with unknown variances to the case of a common mean vector with unknown dispersion matrices is highly nontrivial, and the authors deserve a lot of credit for providing a solution. Starting with the basic ingredients, namely, $(n_1,\cdots ,n_k;{\overline{\mathit{X}}}_i,\dots ,{\overline{\mathit{X}}}_k;$${\mathit{S}}_1,\dots ,{\mathit{S}}_k)$, define ${\mathit{u}}_i=n_i^{-1}{\mathit{S}}_i$, ${\mathit{W}}_i={\left[{\mathit{u}}_i^{1/2}{\tilde{\mathit{R}}}_i^{-1}{\mathit{u}}_i^{1/2}\right]}^{-1}$, ${\mathit{T}}_i^{\ast }={\overline{\mathit{X}}}_i-{\left[{\mathit{u}}_i^{1/2}{\mathit{R}}_i^{-1}{\mathit{u}}_i^{1/2}\right]}^{1/2}{\mathit{Z}}_i$, and $\mathit{W}={\sum }_{i=1}^k{\mathit{W}}_i$.

• For $j=1,\dots ,m$:
• Generate ${\mathit{Z}}_1,\dots ,{\mathit{Z}}_k$ from $N_p(\mathbf{0},{\mathit{I}}_p)$.
• Generate independent ${\mathit{R}}_i$ and ${\tilde{\mathit{R}}}_i$ from $W_p(n_i-1,{\mathit{I}}_p)$, $i=1,\dots ,k$.
• Compute ${\mathit{W}}_1,\dots ,{\mathit{W}}_k$ and $\mathit{W}$.
• Compute ${\mathit{T}}_j={\mathit{W}}^{-1}{\sum }_{i=1}^k{\mathit{W}}_i{\mathit{T}}_i^{\ast }$.
• (End j loop)

Compute ${\widehat{\mathbf{\mu }}}_T=1/m{\sum }_{j=1}^m{\mathit{T}}_j$, ${\widehat{\sum }}_T=1/(m-1){\sum }_{j=1}^m({\mathit{T}}_j-{\widehat{\mathbf{\mu }}}_T){({\mathit{T}}_j-{\widehat{\mathbf{\mu }}}_T)}^t$, and $||{\widetilde{\widehat{\mathit{T}}}}_j||$, where ${\widetilde{\widehat{\mathit{T}}}}_j={\widehat{\sum }}_T^{-1/2}({\mathit{T}}_j-{\widehat{\mathbf{\mu }}}_T)$, $j=1,\dots ,m$.

Let $q_{\{||\widetilde{\widehat{\mathit{T}}}||;1-\alpha \}}$ be the $100(1-\alpha )$-th percentile of $||{\widetilde{\widehat{\mathit{T}}}}_j||$, $j=1,\dots ,m$; then, the confidence ellipsoid of $\mathbf{\mu }$ can be obtained from the inequality

$$\left\{\mathbf{\mu }:{(\mathbf{\mu }-{\widehat{\mathbf{\mu }}}_T)}^t{\widehat{\sum }}_T^{-1}(\mathbf{\mu }-{\widehat{\mathbf{\mu }}}_T)\le q_{\{||\widetilde{\widehat{\mathit{T}}}||;1-\alpha \}}^2\right\}.$$

(6)

We use Equation (6) in Section 8.

4. Exact Tests for ${\mathit{H}}_{\mathbf{0}}:\mathbf{\mu }={\mathit{\mu }}_{\mathbf{0}}$ versus ${\mathit{H}}_{\mathbf{1}}:\mathbf{\mu }\ne {\mathbf{\mu }}_{\mathbf{0}}$

To develop the exact test for testing $H_0:\mathbf{\mu }={\mathbf{\mu }}_0$ versus $H_1:\mathbf{\mu }\ne {\mathbf{\mu }}_0$ based on all the datasets, we proceed as follows. Recall that $T^2=n{(\overline{\mathit{X}}-\mathbf{\mu })}^t{\mathit{S}}^{-1}(\overline{\mathit{X}}-\mathbf{\mu })$ satisfies $T^2=\left({\displaystyle \frac{p}{n-p}}\right)F_{p,n-p}$, where $F_{{\nu }_1,{\nu }_2}$ follows an F-distribution with ${\nu }_1$ and ${\nu }_2$ degrees of freedom. Our test procedure rejects $H_0$ when $F_{obs}>F_{{\nu }_1,{\nu }_2;\alpha }$, with $\alpha$ being the Type I error level and $F_{obs}$ being the observed value of F under $\mu ={\mu }_0$. A test for $H_0$ based on a p-value, on the other hand, is based on $P_{obs}=P[F_{{\nu }_1,{\nu }_2}>F_{obs}]$, and we reject $H_0$ at the $\alpha$ level if $P_{obs}<\alpha$. It is easy to check that the two approaches are obviously equivalent.

A random p-value which follows a uniform $[0,1]$ distribution under the null hypothesis is defined as $P_{ran}=P[F_{{\nu }_1,{\nu }_2}>F_{ran}]$, where $F_{ran}=\left[{\displaystyle \frac{n(n-p)}{p}}\right]{(\overline{\mathit{X}}-{\mathit{\mu }}_{\mathbf{0}})}^t{\mathit{S}}^{-1}(\overline{\mathit{X}}-{\mathit{\mu }}_{\mathbf{0}})$. All suggested exact tests for $H_0$ are based on $P_{obs}$ and $F_{obs}$ values, and their properties, including size and power, are studied under $P_{ran}$ and $F_{ran}$. To simplify notations, we denote $P_{obs}$ by small p and $P_{ran}$ by large P. Four exact tests based on p values and one exact test based on $F_{obs}$, as available in the literature, are listed below.

4.1. Tippett’s Test

This minimum p-value test was proposed by [8], who noted that, if $P_1,\cdots ,P_k$ are independent p-values from continuous test statistics, then each has a uniform distribution under $H_0$. According to this method, the null hypothesis $H_0:\mathbf{\mu }={\mathit{\mu }}_{\mathbf{0}}$ is rejected at the $\alpha$ level of significance if $P_{\left(1\right)}<\left[1-{(1-\alpha )}^{{\displaystyle \frac{1}{k}}}\right]$, where $P_{\left(1\right)}=min\{P_1,\cdots ,P_k\}$. Incidentally, this test is equivalent to the test based on $M_t=ma{x}_{1\le i\le k}\left\{T_i^2\right\}$ as suggested by [9].

4.2. Wilkinson’s Test

This test statistic proposed by [10] is a generalization of Tippett’s test that uses the r-th smallest p-value ($P_{\left(r\right)}$) as a test statistic. The null hypothesis $H_0:\mathbf{\mu }={\mathit{\mu }}_{\mathbf{0}}$ will be rejected if $P_{\left(r\right)}<d_{r,\alpha }$, where $P_{\left(r\right)}$ follows a Beta distribution with parameters r and $(k-r+1)$ under $H_0$ and where $d_{r,\alpha }$ satisfies $Pr\{P_{\left(r\right)}<d_{r,\alpha }|H_0\}=\alpha$. Obviously, this procedure generates a sequence of tests for different values of $r=1,2,\cdots ,k$, and an attempt has been made to identify the best choice of r.

4.3. Inverse Normal Test

This exact test procedure, which involves transforming each p-value to the corresponding normal score, was proposed independently by [11,12]. Using this inverse normal method, the null hypothesis $H_0$ will be rejected at the $\alpha$ level of significance if $\left[{\sum }_{i=1}^k{\mathsf{\Phi }}^{-1}\left(P_i\right)\right]{\left[\sqrt{k}\right]}^{-1}<-z_{\alpha }$, where ${\mathsf{\Phi }}^{-1}$ denotes the inverse of the cumulative distribution function (CDF) of a standard normal distribution, and $z_{\alpha }$ stands for the upper $\alpha$-level cut-off point of a standard normal distribution.

4.4. Fisher’s Inverse ${\chi }^2$-Test

This inverse ${\chi }^2$-test is one of the most widely used exact test procedures for combining k independent p-values [13]. This procedure uses the ${\prod }_{i=1}^k{P}_i$ to combine the k independent p-values. Then, using the connection between uniform and ${\chi }^2$ distributions, the null hypothesis $H_0$ is rejected if $-2{\sum }_{i=1}^{k}ln\left(P_i\right)>{\chi }_{2k,\alpha }^2$, where ${\chi }_{2k,\alpha }^2$ denotes the upper $\alpha$ critical value of a ${\chi }^2$-distribution with $2k$ degrees of freedom.

4.5. Jordan–Kris Test

Jordan and Krishnamoorthy, 1995 [5] considered a weighted linear combination of Hotelling’s $T^2$ statistic, namely $T={\sum }_{i=1}^k{C}_i{T}_i^2$, where $T_i^2=n_i{({\overline{\mathit{X}}}_i-{\mathbf{\mu }}_0)}^t{\mathit{S}}_i^{-1}({\overline{\mathit{X}}}_i-{\mathbf{\mu }}_0)$, and $C_i={\displaystyle \frac{{\left[Var\left(T_i^2\right)\right]}^{-1}}{{\sum }_{j=1}^k{\left[Var\left(T_j^2\right)\right]}^{-1}}}$ with $Var\left(T_i^2\right)={\displaystyle \frac{2p{m}_i^2(m_i-1)}{{(m_i-p-1)}^2(m_i-p-3)}},m_i=n_i-1,n_i>p+4,{\forall }_i,i=1,\cdots ,k$. The null hypothesis $H_0:\mathbf{\mu }={\mathit{\mu }}_{\mathbf{0}}$ will be rejected if $T>a$, where $Pr\{T>a|H_0\}=\alpha$. In applications, a is computed by using the approximation $T\approx d{F}_{kp,\nu }$, where $\nu ={\displaystyle \frac{4M_{2}kp-2M_1^2(kp+2)}{M_{2}kp-M_1^2(kp+2)}}$, $d=M_1\left({\displaystyle \frac{\nu -2}{\nu }}\right)$, $M_1=p{\sum }_{i=1}^k{\displaystyle \frac{C_i{m}_i}{m_i-p-1}}$, and $M_2=p(p+2){\sum }_{i=1}^k{\displaystyle \frac{C_i^2{m}_i^2}{(m_i-p-1)(m_i-p-3)}}+2p^2{\sum }_{i>j}{\displaystyle \frac{C_i{C}_j{m}_i{m}_j}{(m_i-p-1)(m_j-p-1)}}$.

5. Exact Confidence Sets for the Mean Vector $\mathbf{\mu }$

In this section, we present some exact confidence sets for $\mathbf{\mu }$, essentially based upon inverting the acceptance sets resulting from the discussion in Section 4.

5.1. Confidence Set Based on Jordan–Kris Method

Following the method proposed in [5], which is presented in Section 4.5, a $100(1-\alpha )\%$ confidence ellipsoid for $\mathbf{\mu }$ is a set of values $\mathbf{\mu }$ satisfying the following inequality:

$${(\mathbf{\mu }-\widehat{\mathbf{\mu }})}^t\mathit{V}(\mathbf{\mu }-\widehat{\mathbf{\mu }})\le a-\sum _{i=1}^k\left(\sum _{j\ne i}^k{({\overline{\mathit{X}}}_i-{\overline{\mathit{X}}}_j)}^t{\mathit{W}}_j^{-1}\right){\mathit{V}}^{-1}{\mathit{W}}_i^{-1}{\mathit{V}}^{-1}\left(\sum _{j\ne i}^k{\mathit{W}}_j^{-1}({\overline{\mathit{X}}}_i-{\overline{\mathit{X}}}_j)\right),$$

(7)

where $\widehat{\mathbf{\mu }}={\mathit{V}}^{-1}{\sum }_{j=1}^k{\mathit{W}}_j^{-1}{\overline{\mathit{X}}}_j$, ${\mathit{W}}_i^{-1}=c_i{n}_i{\mathit{S}}_i^{-1}$, and $\mathit{V}={\sum }_{i=1}^k{\mathit{W}}_i^{-1}$.

5.2. p-Value-Based Confidence Sets

All the p-value-based confidence sets are obtained by inverting the corresponding acceptance sets; the following are the results. We define $P_i\left(\mathbf{\mu }\right)=Pr\left\{F_{p,n_i-p}>\left[{\displaystyle \frac{n_i(n_i-p)}{p}}\right]{({\overline{\mathit{X}}}_i-\mathbf{\mu })}^t{\mathit{S}}_i^{-1}({\overline{\mathit{X}}}_i-\mathbf{\mu })\right\},i=1,\dots ,k$.

5.2.1. Confidence Set Based on Tippett’s Method

A $100(1-\alpha )\%$ Tippett’s confidence set for $\mathbf{\mu }$ is a set of values $\mathbf{\mu }$ satisfying $\left\{\mathbf{\mu }:P_{\left(1\right)}\left(\mathbf{\mu }\right)>1-{[1-\alpha ]}^{1/k}\right\}$.

5.2.2. Confidence Set Based on Wilkinson’s Method

A $100(1-\alpha )\%$ Wilkinson’s (order r) confidence set for $\mathbf{\mu }$ is a set of values $\mathbf{\mu }$ satisfying $\left\{\mathbf{\mu }:P_{\left(r\right)}\left(\mathbf{\mu }\right)>d_{r,\alpha }\right\}$.

5.2.3. Confidence Set Based on the INN Method

A $100(1-\alpha )\%$ confidence set for $\mathbf{\mu }$ based on inverse normal (INN) method is a set of values $\mathbf{\mu }$ satisfying $\left\{\mathbf{\mu }:{\sum }_{i=1}^k{\displaystyle \frac{{\mathsf{\Phi }}^{-1}\left(P_i\left(\mathbf{\mu }\right)\right)}{\sqrt{k}}}>-Z_{\alpha }\right\}$.

5.2.4. Confidence Set Based on Fisher’s Method

A $100(1-\alpha )\%$ confidence set for $\mathbf{\mu }$ based on Fisher’s inverse ${\chi }^2$-test is a set of values $\mathbf{\mu }$ satisfying $\left\{\mathbf{\mu }:-2{\sum }_{i=1}^{k}ln\left(P_i\left(\mathbf{\mu }\right)\right)<{\chi }_{2k,\alpha }^2\right\}$.

Remark 1.

Unlike the large sample-based confidence ellipsoid presented in Section 2, the generalized p-value-based confidence ellipsoid presented in Section 3, and the Jordan–Kris confidence ellipsoid presented in Section 4, the p-value-based confidence sets described above may not always lead to confidence ellipsoids!

6. Expressions of Local Powers of Proposed Exact Tests

In this section, we provide the expressions of local powers of the suggested exact tests. A common premise is that we derive an expression of the power of a test under ${\Delta }^2>0$, carry out its Taylor expansion around ${\Delta }^2=0$, where ${\Delta }^2=n{(\mathbf{\mu }-{\mathbf{\mu }}_0)}^t{\sum }^{-1}(\mathbf{\mu }-{\mathbf{\mu }}_0)$, and retain terms of order $O\left({\Delta }^2\right)$.

The probability density functions (PDFs) of the F statistic under the null and alternative hypotheses that will be required in the sequel are given below. Below, ${\Delta }^2=n{({\mu }_{\mathbf{1}}-{\mathbf{\mu }}_0)}^t{\sum }^{-1}({\mathit{\mu }}_{\mathbf{1}}-{\mathbf{\mu }}_0)$ stands for the non-centrality parameter when ${\mathit{\mu }}_{\mathbf{1}}$ is chosen as an alternative value.

$$f\left(F_{{\nu }_1,{\nu }_2}\right)={\displaystyle \frac{{\left[\frac{{\nu }_1}{{\nu }_2}\right]}^{{\nu }_1/2}}{B(\frac{{\nu }_1}{2},\frac{{\nu }_2}{2})}}{F}^{{\nu }_1/2-1}{\left[1+{\displaystyle \frac{{\nu }_1}{{\nu }_2}}F\right]}^{-({\nu }_1+{\nu }_2)/2}$$

(8)

$$f_{{\Delta }^2}\left(F_{{\nu }_1,{\nu }_2}\right)=\sum _{k=0}^{\infty }{\displaystyle \frac{e^{-{\Delta }^2/2}{\left[\frac{{\Delta }^2}{2}\right]}^k}{B(\frac{{\nu }_2}{2},\frac{{\nu }_1}{2}+k)k!}}{\left({\displaystyle \frac{{\nu }_1}{{\nu }_2}}\right)}^{{\displaystyle \frac{{\nu }_1}{2}}+k}{\left[{\displaystyle \frac{{\nu }_2}{{\nu }_2+{\nu }_{1}F}}\right]}^{{\displaystyle \frac{{\nu }_1+{\nu }_2}{2}}+k}{F}^{{\nu }_1/2-1+k}$$

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$$\begin{array}{ccc}\hfill f_{{\Delta }^2}\left(F_{{\nu }_1,{\nu }_2}\right)& \approx & f_{{\Delta }^2=0}\left(F_{{\nu }_1,{\nu }_2}\right)+{\displaystyle \frac{{\Delta }^2}{2}}{f}_{{\Delta }^2=0}\left(F\right)\left[{\displaystyle \frac{F-1}{1+\frac{{\nu }_1}{{\nu }_2}F}}\right]\hfill \\ & =& f_{{\Delta }^2=0}\left(F_{{\nu }_1,{\nu }_2}\right)\left[1+{\displaystyle \frac{{\Delta }^2}{2}}\left\{{\displaystyle \frac{F-1}{1+\frac{{\nu }_1}{{\nu }_2}F}}\right\}\right]\hfill \end{array}$$

(10)

The final expressions of the local powers of the proposed tests are given below in the general case and also in the special case when $n_1=\cdots =n_k=n$. For detailed proofs of all technical results, we refer to Appendix B of this paper.

6.1. Local Power of Tippett’s Test $\left[LP\right(T\left)\right]$

$$\begin{array}{ccc}\hfill LP\left(T\right)& \approx & \alpha +{(1-\alpha )}^{{\displaystyle \frac{k-1}{k}}}\sum _{i=1}^k{\displaystyle \frac{{\Delta }_i^2}{2}}{\xi }_{F_{c_{\alpha ;{\nu }_1,{\nu }_{2i}}}}\hfill \\ & =& \alpha +{(1-\alpha )}^{{\displaystyle \frac{k-1}{k}}}{\xi }_{F_{c_{\alpha ;{\nu }_1,{\nu }_2}}}\left\{\sum _{i=1}^k{\displaystyle \frac{{\Delta }_i^2}{2}}\right\}\left[\mathrm{special}\mathrm{case}\right],\hfill \end{array}$$

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where ${\xi }_{F_{c_{\alpha ;{\nu }_1,{\nu }_{2i}}}}={\int }_{c_{\alpha };{\nu }_1,{\nu }_{2i}}^{\infty }{f}_0\left(F_{{\nu }_1,{\nu }_{2i}}\right)\left[{\displaystyle \frac{F-1}{1+\frac{{\nu }_1}{{\nu }_{2i}}F}}\right]dF$.

6.2. Local Power of Wilkinson’s Test $\left[LP\left(W_r\right)\right]$

$$\begin{array}{ccc}\hfill LP\left(W_r\right)& \approx & \alpha +\left({\displaystyle \genfrac{}{}{0pt}{}{k-1}{r-1}}\right)d_{r;\alpha }^{r-1}{(1-d_{r;\alpha })}^{k-r}\left[\sum _{i=1}^k{\displaystyle \frac{{\Delta }_i^2}{2}}{\xi }_{F_{d_{r,\alpha ;{\nu }_1,{\nu }_{2i}}}}\right]\hfill \\ & =& \alpha +\left({\displaystyle \genfrac{}{}{0pt}{}{k-1}{r-1}}\right)d_{r;\alpha }^{r-1}{(1-d_{r;\alpha })}^{k-r}{\xi }_{F_{d_{r,\alpha },{\nu }_1,{\nu }_2}}\left\{\sum _{i=1}^k{\displaystyle \frac{{\Delta }_i^2}{2}}\right\}\left[\mathrm{special}\mathrm{case}\right],\hfill \end{array}$$

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where ${\xi }_{F_{d_{r,\alpha },{\nu }_1,{\nu }_2}}$ is equivalent to ${\xi }_{F_{c_{\alpha ;{\nu }_1,{\nu }_2}}}$, with $c_{\alpha }=d_{r;\alpha }$.

Remark 2.

For the special case $r=1$, $LP\left(W_r\right)=LP\left(T\right)$, as expected, because $d_{1;\alpha }=[1-{(1-\alpha )}^{{\displaystyle \frac{1}{k}}}]$, implying ${(1-d_{1;\alpha })}^{k-1}={(1-\alpha )}^{{\displaystyle \frac{k-1}{k}}}$.

6.3. Local Power of Inverse Normal Test $\left[LP\right(INN\left)\right]$

$$\begin{array}{ccc}\hfill LP\left(INN\right)& \approx & \alpha +{\displaystyle \frac{\varphi \left(z_{\alpha }\right)}{2\sqrt{k}}}\sum _{i=1}^k{\Delta }_i^2\left[{\displaystyle \frac{z_{\alpha }}{2\sqrt{k}}}{B}_{{\nu }_{1i},{\nu }_{2i}}-A_{{\nu }_{1i},{\nu }_{2i}}\right]\hfill \\ & =& \alpha +{\displaystyle \frac{\varphi \left(z_{\alpha }\right)}{\sqrt{k}}}\left[{\displaystyle \frac{z_{\alpha }}{2\sqrt{k}}}{B}_{{\nu }_1,{\nu }_2}-A_{{\nu }_1,{\nu }_2}\right]\left\{\sum _{i=1}^k{\displaystyle \frac{{\Delta }_i^2}{2}}\right\}\left[\mathrm{special}\mathrm{case}\right],\hfill \end{array}$$

(13)

where $A_{{\nu }_1,{\nu }_2}={\int }_{-\infty }^{\infty }u\varphi \left(u\right)Q_{{\nu }_1,{\nu }_2}\left(u\right)du$, $Q_{{\nu }_1,{\nu }_2}\left(u\right)={\left[{\displaystyle \frac{F-1}{1+\frac{{\nu }_1}{{\nu }_2}F}}\right]}_{F=F_{\mathsf{\Phi }\left(u\right);{\nu }_1,{\nu }_2}}$, $B_{{\nu }_1,{\nu }_2}={\int }_{-\infty }^{\infty }{u}^2\varphi \left(u\right)Q_{{\nu }_1,{\nu }_2}^{\ast }\left(u\right)du$, $Q_{{\nu }_1,{\nu }_2}^{\ast }\left(u\right)=\left\{Q_{{\nu }_1,{\nu }_2}\left(u\right)-E\left[Q_{{\nu }_1,{\nu }_2}\left(u\right)\right]\right\}$, $\varphi \left(u\right)$ is standard normal PDF, and $\mathsf{\Phi }\left(u\right)$ is standard normal CDF.

6.4. Local Power of Fisher’s Test $\left[LP\right(F\left)\right]$

$$\begin{array}{ccc}\hfill LP\left(F\right)& \approx & \alpha +\sum _{i=1}^k{\displaystyle \frac{{\Delta }_i^2}{4}}{D}_{{\nu }_{1i},{\nu }_{2i}}\left[E{\left\{\left\{ln(T/2)\right\}{I}_{\{T\ge {\chi }_{2k;\alpha }^2\}}\right\}}_{T\sim {\chi }_{2k}^2}-\alpha D_0\right]\hfill \\ & =& \alpha +{\displaystyle \frac{D_{{\nu }_1,{\nu }_2}}{2}}\left[E{\left\{\left\{ln(T/2)\right\}{I}_{\{T\ge {\chi }_{2k;\alpha }^2\}}\right\}}_{T\sim {\chi }_{2k}^2}-\alpha D_0\right]\left\{\sum _{i=1}^k{\displaystyle \frac{{\Delta }_i^2}{2}}\right\}\left[\mathrm{special}\mathrm{case}\right],\hfill \end{array}$$

(14)

where $D_0=E\left[log\left(q\right)\right]$; $D_{{\nu }_1,{\nu }_2}=E\left[V{Q}_{{\nu }_1,{\nu }_2}^{\ast }\left(v\right)\right]$; $V\sim$ exp[2]; $q\sim$ gamma[1,k]; $T\sim gamma[2,k]$; $Q_{{\nu }_1,{\nu }_2}\left(v\right)={\left[{\displaystyle \frac{F-1}{1+\frac{{\nu }_1}{{\nu }_2}F}}\right]}_{F=F_{\mathsf{\Phi }\left(v\right);{\nu }_1,{\nu }_2}}$; $Q_{{\nu }_1,{\nu }_2}^{\ast }\left(u\right)=\left\{Q_{{\nu }_1,{\nu }_2}\left(u\right)-E\left[Q_{{\nu }_1,{\nu }_2}\left(u\right)\right]\right\}$.

6.5. Local Power of Jordan–Kris Test $\left[LP\right(JK\left)\right]$

$$\begin{array}{ccc}\hfill LP\left(JK\right)& \approx & \alpha +\sum _{i=1}^k{\displaystyle \frac{{\Delta }_i^2}{2}}{E}_{H_0}\left[\left\{{\displaystyle \frac{F_i-1}{1+\frac{p}{n_i-p}{F}_i}}\right\}{I}_{\{{\sum }_{i=1}^k{C}_i^{\ast }{F}_i>ak\}}\right]\hfill \\ & =& \alpha +E_{H_0}\left[\left\{{\displaystyle \frac{F_i-1}{1+\frac{p}{n-p}{F}_i}}\right\}{I}_{\{{\sum }_{i=1}^k{F}_i>ak\}}\right]\left\{\sum _{i=1}^k{\displaystyle \frac{{\Delta }_i^2}{2}}\right\}\left[\mathrm{special}\mathrm{case}\right],\hfill \end{array}$$

(15)

where $E_{H_0}[·]$ stands for expectation with respect to $F_1,\dots ,F_k$ under $H_0[F_i\sim F({\nu }_1,{\nu }_{2i})]$.

7. Comparison of Local Powers

It is interesting to observe from the above expressions that, in the special case of equal sample size, local powers can be readily compared, irrespective of the values of the unknown dispersion matrices ${\sum }_i$, $i=1,\cdots ,k$.

Table 1. Comparison of the second term of local power (without ${\sum }_{i=1}^k{\Delta }_i^2/2$) of five exact tests for different values of k, p, and n (equal sample size).

Exact Test	k = 2
	n = 15			n = 25			n = 40
	p = 2	p = 3	p = 4	p = 2	p = 3	p = 4	p = 2	p = 3	p = 4
Tippett	0.0693	0.0490	0.0375	0.0777	0.0571	0.0455	0.0825	0.0618	0.0502
Wilkinson	0.0669	0.0514	0.0417	0.0777	0.0571	0.0463	0.0825	0.0618	0.0502
Inverse Normal	0.0778	0.0585	0.0471	0.0833	0.0648	0.0529	0.0862	0.0672	0.0569
Fisher	0.0596	0.0458	0.0355	0.0635	0.0493	0.0416	0.0667	0.0517	0.0445
Jordan–Kris	0.0795	0.0598	0.0486	0.0863	0.0664	0.0552	0.0899	0.0697	0.0591
Exact Test	k = 2
	n = 15			n = 25			n = 40
	p = 2	p = 3	p = 4	p = 2	p = 3	p = 4	p = 2	p = 3	p = 4
Tippett	0.0496	0.0348	0.0264	0.0562	0.0410	0.0325	0.0601	0.0447	0.0361
Wilkinson	0.0545	0.0408	0.0325	0.0582	0.0449	0.0369	0.0601	0.0471	0.0393
Inverse Normal	0.0615	0.0463	0.0372	0.0647	0.0509	0.0421	0.0681	0.0535	0.0455
Fisher	0.0487	0.0375	0.0315	0.0526	0.0404	0.0345	0.0534	0.0426	0.0352
Jordan–Kris	0.0621	0.0472	0.0392	0.0668	0.0521	0.0433	0.0701	0.0547	0.0462
Exact Test	k = 2
	n = 15			n = 25			n = 40
	p = 2	p = 3	p = 4	p = 2	p = 3	p = 4	p = 2	p = 3	p = 4
Tippett	0.0322	0.0223	0.0168	0.0370	0.0268	0.0211	0.0399	0.0294	0.0236
Wilkinson	0.0393	0.0299	0.0241	0.0424	0.0324	0.0270	0.0443	0.0340	0.0285
Inverse Normal	0.0459	0.0349	0.0283	0.0487	0.0376	0.0314	0.0495	0.0395	0.0336
Fisher	0.0373	0.0299	0.0242	0.0418	0.0334	0.0239	0.0403	0.0331	0.0283
Jordan–Kris	0.0469	0.0351	0.0295	0.0499	0.0397	0.0331	0.0516	0.0419	0.0344

represents values of the second term of local power in the case of equal sample size n given above in (11)–(15), apart from the common term $\left[{\sum }_{i=1}^k{\Delta }_i^2/2\right]$, for different values of k, p, and n. A comparison of the second term of local power of Wilkinson’s test for different values of r$(\le k)$ is provided in

Table 2. Comparison of the second term of local power (without ${\sum }_{i=1}^k{\Delta }_i^2/2$) of Wilkinson’s exact test for $n=15$ (equal sample size) and different values of k, p, and $r(\le k)$.

r	k = 2			k = 3			k = 5			k = 9			k = 10
	p = 2	p = 3	p = 4	p = 2	p = 3	p = 4	p = 2	p = 3	p = 4	p = 2	p = 3	p = 4	p = 2	p = 3	p = 4
1	0.0693	0.0490	0.0375	0.0496	0.0348	0.0264	0.0322	0.0223	0.0168	0.0194	0.0133	0.0099	0.0177	0.0121	0.009
2	0.0669	0.0514	0.0417	0.0545	0.0408	0.0325	0.0391	0.0286	0.0224	0.0253	0.0182	0.014	0.0234	0.0167	0.0129
3				0.0463	0.0369	0.0306	0.0393	0.0299	0.0241	0.0276	0.0204	0.0161	0.0257	0.0189	0.0148
4							0.0358	0.0283	0.0234	0.0281	0.0212	0.0170	0.0264	0.0199	0.0159
5							0.0286	0.0238	0.0203	0.0275	0.0213	0.0173	0.0262	0.0201	0.0163
6										0.026	0.0206	0.0171	0.0253	0.0198	0.0163
7										0.0239	0.0194	0.0163	0.0238	0.019	0.0159
8										0.0208	0.0174	0.0149	0.0217	0.0178	0.015
9										0.0162	0.0141	0.0123	0.0188	0.0158	0.0136
10													0.0146	0.0128	0.0113

for $n=15$, $k\in \{2,3,5,9,10\}$, and $p\in \{2,3,4\}$. Throughout, we have used $\alpha =5\%$. It turns out that the exact tests based on the inverse normal and Jordan–Kris methods perform the best. It also appears from Table 2 that an optimal choice of r for Wilkinson’s method is nearly $\sqrt{k}$.

8. Applications

8.1. Confidence Set Comparison Using Simulated Data

In this section, we follow the framework in [5], who simulated bivariate samples of 12 vectors (equal sample size $n_1=n_2=12$) each from two bivariate normal distributions, $N_2(\mathbf{\mu },{\sum }_1)$ and $N_2(\mathbf{\mu },{\sum }_2)$, with ${\mathbf{\mu }}^{\prime }=(5,8)$, ${\sum }_1=\left[\begin{array}{cc}4.5& 3.0\\ 3.0& 7.9\end{array}\right]$, and ${\sum }_2=\left[\begin{array}{cc}5.5& 3.8\\ 3.8& 6.3\end{array}\right]$. Summary statistics based on these two simulated datasets are: ${\overline{\mathit{X}}}_1^{\prime }=(4.73,7.93)$, ${\overline{\mathit{X}}}_2^{\prime }=(5.21,8.89)$, ${\mathit{S}}_1=\left[\begin{array}{cc}2.71& 4.46\\ 4.46& 10.91\end{array}\right]$, ${\mathit{S}}_2=\left[\begin{array}{cc}5.39& 1.37\\ 1.37& 2.84\end{array}\right]$, and $\mathit{V}=\left[\begin{array}{cc}8.07& -3.39\\ -3.39& 4.10\end{array}\right]$.

Based on the above simulated data, we present below the 95% confidence sets for $\mathbf{\mu }$ resulting from the five exact methods (Figure 1) and the method based on the generalized p-value. It turns out that the INN method followed by the Jordan–Kris and Fisher methods yield smaller observed confidence sets than the Tippett and Wilkinson ($r=2$) methods. As remarked earlier, although the confidence ellipsoid based on the generalized p-value method seems to have a smaller volume, its coverage probability cannot be guaranteed.

Simulated Coverage Probabilities

In this section, we present the simulated coverage probabilities (CPs) for various confidence sets discussed in Section 3 and Section 5. These CPs have been estimated through simulation studies based on a bivariate normal distribution, with parameters outlined in Section 8.1.

Table 3. 95% simulated coverage probabilities based on different methods and equal sample sizes ($n_1=n_2=n$) for the bivariate normal distribution.

Confidence Set Based on	n = 5	n = 12	n = 20	n = 30	n = 40
Generalized p-value	88.00	91.10	92.70	93.50	94.50
Jordan–Kris method	91.40	92.30	93.30	94.10	93.05
Tippett’s method	93.60	94.30	94.40	94.94	95.06
Wilkinson’s method (r = 2)	62.30	62.60	60.10	62.70	63.10
INN method	94.20	94.40	94.70	94.90	95.10
Fisher’s method	93.70	94.30	94.50	95.00	95.20

provides the 95% CP values for an equal sample size case in the six methods, including the generalized p-value method. As expected, the CP increases with the sample size, indicating improved reliability of the confidence sets as the sample size increases.

To evaluate the robustness of these methods under distributional misspecification, we performed a simulation study using a multivariate Laplace distribution, applying the same parameters as described previously. We calculated the 95% coverage probabilities for the six confidence sets. The results, presented in

Table 4. 95% simulated coverage probabilities based on different methods and equal sample sizes ($n_1=n_2=n$) for the multivariate Laplace distribution.

Confidence Set Based on	n = 5	n = 12	n = 20	n = 30	n = 40
Generalized p-value	87.33	91.30	97.30	94.00	94.67
Jordan–Kris method	86.78	88.88	89.44	90.32	91.10
Tippett’s method	93.00	94.00	94.20	95.00	95.10
Wilkinson’s method (r = 2)	62.00	61.80	59.10	58.30	57.40
INN method	93.40	94.00	94.40	94.70	95.60
Fisher’s method	93.00	94.00	94.20	94.80	95.00

, demonstrate that most methods remain robust despite the distributional misspecification. However, Wilkinson’s method exhibits a decline in coverage probabilities as the sample size increases, indicating some sensitivity to the underlying distribution. In general, these methods generally maintain their reliability even when the assumed distribution differs from the true one.

8.2. Data Analysis: Current Population Survey (CPS) Annual Social and Economic (ASEC) Supplement

In this section, we provide a statistical analysis of data arising from the Current Population Survey (CPS) Annual Social and Economic (ASEC) Supplement 2021, conducted by the Bureau of the Census for the Bureau of Labor Statistics.

Under CPS, typically some 70,000 housing units or other living quarters are assigned for interview each month; about 50,000 of them that contain approximately 100,000 persons 15 years old and over are interviewed. The universe in this survey is the civilian non-institutional population of the United States living in housing units and members of the armed forces living off post or living with their families on post. Sampling units are scientifically selected (based on a probability sample) on the basis of area of residence to represent the nation as a whole, individual states, and other specified areas.

Although the main purpose of the survey is to collect information on the employment situation, a very important secondary purpose is to collect information on the demographic status of the population, such as age, sex, race, marital status, educational attainment, and family structure. The statistics resulting from these questions serve to update similar information collected once every 10 years through the decennial census and are used by government policymakers and legislators as important indicators of our nation’s economic situation and for planning and evaluating many government programs. CPS is the only source of monthly estimates of total employment (both farm and non-farm); non-farm self-employed persons, domestics, and unpaid workers in non-farm family enterprises; wage and salary employees; and, finally, estimates of total unemployment.

The Annual Social and Economic (ASEC) Supplement contains the basic monthly demographic and labor force data described above, plus additional data on work experience, income, non-cash benefits, health insurance coverage, and migration. Since 1976, the survey has been supplemented with about 6000 Hispanic households from which at least 4500 are interviewed. However, in 2002, another sample expansion occurred to help improve states’ estimates of the Children’s Health Insurance Program (CHIP), resulting in the addition of 19,000 households, which raised the total sample size for the ASEC to about 95,000 households. All current population reports are available online at https://www.census.gov/library/publications.html (accessed on 19 November 2022).

For the purpose of our data analysis, we mainly consider two variables out of a wide range of available data: total income and income components covering nine non-cash income sources: food stamps, school lunch program, employer-provided group health insurance plan, employer-provided pension plan, personal health insurance, Medicaid, Medicare, or military health care, and energy assistance. Characteristics such as age, sex, race, household relationship, and Hispanic origin are shown for each person in the enumerated household. Although the above type of data is available for all 50 U.S. states, we focus on the data from California (CA), as it is argued that this data source is the most reliable, and the sample sizes are fairly large. There are 58 counties in CA, and the table below shows a summary of the bivariate sample mean vectors and 2 × 2 sample variance–covariance matrices for 13 selected counties in CA, divided into three groups (

Table 5. Summary of the bivariate sample mean vectors and $2\times 2$ sample variance–covariance matrices for 13 selected counties in CA, divided into three groups.

Group-I [K = 5 Counties]
Butte County	Kings County	Shasta County	Tulare County	Stanislaus County
n= 32	n = 49	n = 45	n = 57	n = 67
${\overline{\mathit{X}}}_{11}=\left[\begin{array}{c}46.8048\\ 63.8037\end{array}\right]$	${\overline{\mathit{X}}}_{12}=\left[\begin{array}{c}45.6735\\ 64.3907\end{array}\right]$	${\overline{\mathit{X}}}_{13}=\left[\begin{array}{c}48.9864\\ 71.6568\end{array}\right]$	${\overline{\mathit{X}}}_{14}=\left[\begin{array}{c}53.5000\\ 73.6172\end{array}\right]$	${\overline{\mathit{X}}}_{15}=\left[\begin{array}{c}49.2051\\ 79.8356\end{array}\right]$
${\mathit{S}}_{11}=\left[\begin{array}{c}2255.2631820.256\\ 1820.2561851.509\end{array}\right]$	${\mathit{S}}_{12}=\left[\begin{array}{c}2249.2001958.085\\ 1958.0852223.194\end{array}\right]$	${\mathit{S}}_{13}=\left[\begin{array}{c}2290.2331670.224\\ 1670.2242172.094\end{array}\right]$	${\mathit{S}}_{14}=\left[\begin{array}{c}4702.4514576.456\\ 4576.4565330.648\end{array}\right]$	${\mathit{S}}_{15}=\left[\begin{array}{c}3298.9612527.469\\ 2527.4692828.913\end{array}\right]$
Group-II [K = 5 Counties]
Monterey  County	Los Angeles County	Sacramento County	Santa Cruz County	San Luis Obispo County
n = 62	n = 1548	n = 216	n = 50	n = 43
${\overline{\mathit{X}}}_{21}=\left[\begin{array}{c}80.5238\\ 96.1007\end{array}\right]$	${\overline{\mathit{X}}}_{22}=\left[\begin{array}{c}80.8304\\ 101.0140\end{array}\right]$	${\overline{\mathit{X}}}_{23}=\left[\begin{array}{c}75.0044\\ 99.6372\end{array}\right]$	${\overline{\mathit{X}}}_{24}=\left[\begin{array}{c}77.1391\\ 100.2768\end{array}\right]$	${\overline{\mathit{X}}}_{25}=\left[\begin{array}{c}76.0411\\ 103.1578\end{array}\right]$
${\mathit{S}}_{21}=\left[\begin{array}{c}7888.2088107.861\\ 8107.8618989.313\end{array}\right]$	${\mathit{S}}_{22}=\left[\begin{array}{c}10992.7111338.02\\ 11338.0212834.65\end{array}\right]$	${\mathit{S}}_{23}=\left[\begin{array}{c}6417.0466005.736\\ 6005.7367266.494\end{array}\right]$	${\mathit{S}}_{24}=\left[\begin{array}{c}7108.7537022.462\\ 7022.4627855.648\end{array}\right]$	${\mathit{S}}_{25}=\left[\begin{array}{c}5310.2684276.638\\ 4276.6384593.337\end{array}\right]$
Group-III [K = 3 Counties]
Alameda County	San Francisco County	Sonoma County
n = 247	n = 90	n = 50
${\overline{\mathit{X}}}_{31}=\left[\begin{array}{c}126.3072\\ 147.3478\end{array}\right]$	${\overline{\mathit{X}}}_{32}=\left[\begin{array}{c}127.4039\\ 155.6503\end{array}\right]$	${\overline{\mathit{X}}}_{33}=\left[\begin{array}{c}122.2838\\ 166.5137\end{array}\right]$
${\mathit{S}}_{31}=\left[\begin{array}{c}17,857.0717442.44\\ 17,442.4419,388.81\end{array}\right]$	${\mathit{S}}_{32}=\left[\begin{array}{c}19,639.9118,455.13\\ 18,455.1319,352.74\end{array}\right]$	${\mathit{S}}_{33}=\left[\begin{array}{c}16,522.6817,558.55\\ 17,558.5523,443.13\end{array}\right]$

). We observe that the simple mean vectors in each group are fairly close, suggesting a common population mean vector within the chosen counties of a group. The 95% confidence sets of the common mean vector based on the methods discussed in this paper appear in Figure 2, Figure 3 and Figure 4. The location of the well-known Graybill–Deal estimate of the common mean vector is shown in each ellipsoid. As an illustration, we have also added a figure (Figure 5) depicting three confidence sets under the Fisher method for the three groups for the sake of comparison.

9. Conclusions

This paper presents a comprehensive approach to drawing inferences about a common mean vector from several independent multi-normal populations with unequal and unknown dispersion matrices. By developing both asymptotic and exact methods, we offer robust tools for statistical analysis that can be applied to a variety of practical situations, including statistical meta-analysis. The procedures based on the standardized Graybill–Deal estimate of the common mean vector prove to be effective and easy to implement for large sample sizes. In small samples, however, several exact procedures with good frequentist properties exist.

A notable limitation of our approach is its reliance on the assumption of multivariate normality, which may not be applicable in all practical settings. Although our simulations under distributional misspecification, such as using a multivariate Laplace distribution, indicate robustness, this robustness could be sensitive to more extreme deviations from normality or in the presence of heavy tails. Future research could explore the extension of these methods to broader classes of distributions, enhancing their applicability in real-world data analysis. We anticipate that the methods discussed will find practical use and inspire further developments in the field of multivariate analysis.