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On the asymptotic distribution of likelihood ratio test when parameters lie on the boundary

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Abstract

The paper discusses statistical inference dealing with the asymptotic theory of likelihood ratio tests when some parameters may lie on boundary of the parameter space. We derive a closed form solution for the case when one parameter of interest and one nuisance parameter lie on the boundary. The asymptotic distribution is not always a mixture of several chi-square distributions. For the cases when one parameter of interest and two nuisance parameters or two parameters of interest and one nuisance parameter are on the boundary, we provide an explicit solution which can be easily computed by simulation. These results can be used in many applications, e.g. testing for random effects in genetics. Contrary to the claim of some authors in the applied literature that use of chi-square distribution with degrees of freedom as in case of interior parameters will be too conservative when some parameters are on the boundary, we show that when nuisance parameters are on the boundary, that approach may often be anti-conservative.

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Acknowledgements

The views expressed in this article are those of the authors and do not necessarily reflect the views or policies of the U.S. Environmental Protection Agency. Research of the second author (Sinha) was supported under a NCEA/ORD/EPA visiting faculty fellowship program. The authors thank John Fox and Paul White of NCEA/ORD/EPA for encouragement and anonymous reviewers for helpful comments.

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Correspondence to Bimal Sinha.

Appendices

Appendix A

In the following, matrix R is defined as

$$ {\bf R} = \left( \begin{array}{ccc} 1 & \rho_{12} & \rho_{13} \\ \rho_{12} & 1 & \rho_{23}\\ \rho_{13} & \rho_{23} & 1 \end{array} \right). $$

Case I

MLE = (Z 1, Z 2, Z 3) over Region R I : Z 1 ≥ 0, Z 2 ≥ 0, Z 3 ≥ 0.

Obviously, the minimum value of the quadratic here is Q I  = 0 and the region R I can be reexpressed in terms of U 1, U 2, U 3 as:

$$ R_{I}: U_1>0, U_2+\frac{U_1\rho_{12}}{\sqrt{1-\rho^2_{12}}}>0, U_3+\frac{U_1\rho_{13}}{\sqrt{1-\rho^2_{13}}}>0. $$
(A.1)

Case II

MLE = (0, Z 2.1, Z 3.1) over Region R II : Z 1 < 0, Z 2.1 ≥ 0, Z 3.1 ≥ 0.

Obviously, the minimum value of the quadratic here is Q II given by the following expression

$$\begin{array}{rll} Q_{II} & = & (Z_1, Z_2 - Z_{2.1}, Z_3 - Z_{3.1})'[\Sigma]^{-1} \big(Z_1,Z_2 - Z_{2.1}, Z_3 - Z_{3.1}\big)\\ & =& U^2_1 \big(1, \rho_{12}, \rho_{13}\big)'{\bf R}^{-1}\big(1, \rho_{12}, \rho_{13}\big) \end{array}$$
(A.2)

and the region R II can be reexpressed in terms of U 1, U 2, U 3 as:

$$ R_{II}: U_1<0, U_2 >0, U_3 >0. $$
(A.3)

Case III

\(\mathit{MLE} = (Z_{1.2},0,Z_{3.2})\) over region \(R_{\it III}: Z_2 < 0, Z_{1.2} \geq 0\), Z 3.2 ≥ 0.

Obviously, the minimum value of the quadratic here is Q III given by the following expression

$$\begin{array}{rll} Q_{III} &= & \big(Z_1-Z_{1.2}, Z_2, Z_3 - Z_{3.2}\big)'[\Sigma]^{-1} \big(Z_1-Z_{1.2}, Z_2, Z_3 - Z_{3.2}\big)\\ &= & \left(Z_1\rho^2_{12}+\rho_{12}\sigma_1\frac{Y_2}{\sigma_2}, Y_2+\rho_{12}\sigma_2\frac{Z_1}{\sigma_1},\rho_{12}\rho_{23}\frac{Z_1}{\sigma_1}\right)' [\Sigma]^{-1}\\ && \times\left(Z_1\rho^2_{12}+\rho_{12}\sigma_1\frac{Y_2}{\sigma_2}, Y_2+\rho_{12}\sigma_2\frac{Z_1}{\sigma_1},\rho_{12}\rho_{23}\frac{Z_1}{\sigma_1}\right)\\ &= & \left[\rho_{12}\frac{Z_1}{\sigma_1}+\frac{Y_2}{\sigma_2}\right]^2\big(\rho_{12}, 1, \rho_{23}\big)'{\bf R}^{-1}\big(\rho_{12}, 1, \rho_{23}\big)\\ &= & \left[U_2+\frac{\rho_{12}U_1}{\sqrt{1-\rho_{12}^2}}\right]^2\big(1-\rho^2_{12}\big) \big(\rho_{12}, 1, \rho_{23}\big)'{\bf R}^{-1}\big(\rho_{12}, 1, \rho_{23}\big) \end{array} $$
(A.4)

and the region R III can be reexpressed in terms of U 1, U 2, U 3 as

$$\begin{array}{rll} R_{III} & : & Z_2<0, Z_{1.2} >0, Z_{3.2}>0\\ & \sim & Y_2+\rho_{12}\sigma_2\frac{Z_1}{\sigma_1}<0, Z_1-\frac{\rho_{12}}{1-\rho^2_{12}}\frac{\sigma_1}{\sigma_2}Y_2>0, \\ & & \big(\rho_{13}-\rho_{12}\rho_{23}\big)\frac{Z_1}{\sigma_1} - \rho_{23}\frac{Y_2}{\sigma_2}+\frac{Y_3}{\sigma_3} >0\\ & \sim & \rho_{12}\frac{Z_1}{\sigma_1}+\frac{Y_2}{\sigma_2}<0, \frac{Z_1}{\sigma_1}-\frac{\rho_{12}}{1-\rho^2_{12}} \times \frac{Y_2}{\sigma_2}>0, \\ & & \big(\rho_{13}-\rho_{12}\rho_{23}\big)\frac{Z_1}{\sigma_1} > \rho_{23}\frac{Y_2}{\sigma_2}-\frac{Y_3}{\sigma_3} >0 \\ & \sim & U_2+\frac{\rho_{12}}{\sqrt{1-\rho^2_{12}}}U_1<0, U_1>\frac{\rho_{12}}{\sqrt{1-\rho^2_{12}}}U_2, \\ & & c_{13}U_1 >\frac{\rho_{23}}{\sqrt{1-\rho^2_{13}}}U_2 - \frac{U_3}{\sqrt{1-\rho^2_{12}}}, \end{array} $$
(A.5)

where \(c_{13} = \frac{ \rho_{13} - \rho_{12}\rho_{23}}{ \sqrt{(1-\rho^2_{12})(1-\rho^2_{13})}} \).

Case IV

\(\mathit{MLE} = (Z_{1.3},Z_{2.3}, 0)\) over region R VI : Z 3 < 0, Z 1.3 ≥ 0, Z 2.3 ≥ 0.

Similarly to the previous case

$$\begin{array}{rll}Q_{IV} &= & \big(Z_1-Z_{1.3}, Z_2 - Z_{2.3}, Z_3\big)'[\Sigma]^{-1} \big(Z_1-Z_{1.3}, Z_3 - Z_{2.3}, Z_3\big)\\ &= & \left(Z_1\rho^2_{13}+\rho_{13}\sigma_1\frac{Y_3}{\sigma_3}, \rho_{13}\rho_{23}\sigma_2\frac{Z_1}{\sigma_1}+\rho_{23}\sigma_2\frac{Y_3}{\sigma_3}, Y_3+\rho_{13}\sigma_3\frac{Z_1}{\sigma_1}\right)'\\ &&\times\, [\Sigma]^{-1}\left(Z_1\rho^2_{13}+\rho_{13}\sigma_1\frac{Y_3}{\sigma_3}, \rho_{13}\rho_{23}\sigma_2\frac{Z_1}{\sigma_1}+\rho_{23}\sigma_2\frac{Y_3}{\sigma_3},Y_3+\rho_{13}\sigma_3\frac{Z_1}{\sigma_1}\right)\\ &=& \left(\frac{Z_1}{\sigma_1}\rho^2_{13}+\rho_{13}\frac{Y_3}{\sigma_3}, \rho_{13}\rho_{23}\frac{Z_1}{\sigma_1}+\rho_{23}\frac{Y_3}{\sigma_3}, \frac{Y_3}{\sigma_3}+\rho_{13}\frac{Z_1}{\sigma_1}\right)' {\bf{R}^{-1}} \\ & &\times\, \left(\frac{Z_1}{\sigma_1}\rho^2_{13}+\rho_{13}\frac{Y_3}{\sigma_3}, \rho_{13}\rho_{23}\frac{Z_1}{\sigma_1}+\rho_{23}\frac{Y_3}{\sigma_3}, \frac{Y_3}{\sigma_3}+\rho_{13}\frac{Z_1}{\sigma_1}\right) \\ &= & \left[U_3+\frac{\rho_{13}U_1}{\sqrt{1-\rho_{13}^2}}\right]^2\big(1-\rho^2_{13}\big) \big(\rho_{13}, \rho_{23},1\big)'{\bf R}^{-1}(\rho_{13}, \rho_{23}, 1) \end{array} $$
(A.6)
$$\begin{array}{rll}R_{IV} & : & Z_3<0, Z_{1.3} >0, Z_{2.3} >0 \\ & \sim & Y_3+\rho_{13}\sigma_3\frac{Z_1}{\sigma_1}<0, Z_1\big(1-\rho^2_{13}\big)-\rho_{13}\sigma_1\frac{Y_3}{\sigma_3}>0,\\ & & \rho_{23}\frac{Y_3}{\sigma_3}+\frac{Y_2}{\sigma_2} >0 \\ & \sim & \rho_{13}\frac{Z_1}{\sigma_1}+\frac{Y_3}{\sigma_3}<0, \frac{Z_1}{\sigma_1}-\frac{\rho_{13}}{1-\rho^2_{13}} \times \frac{Y_3}{\sigma_3},\\ & & \big(\rho_{12}-\rho_{13}\rho_{23}\big)\frac{Z_1}{\sigma_1} > \rho_{23}\frac{Y_3}{\sigma_3}-\frac{Y_2}{\sigma_2} >0 \\ & \sim & U_3+\frac{\rho_{13}}{\sqrt{1-\rho^2_{13}}}U_1<0, U_1>\frac{\rho_{13}}{\sqrt{1-\rho^2_{13}}}U_3, \\ & & c_{12}U_1 >\frac{\rho_{23}}{\sqrt{1-\rho^2_{12}}}U_3 - \frac{U_2}{\sqrt{1-\rho^2_{13}}}, \end{array} $$
(A.7)

where \(c_{12} =\frac{\rho_{12} - \rho_{13}\rho_{23}}{\sqrt{(1-\rho^2_{12})(1-\rho^2_{13})}}\).

Case V

MLE = (0, 0, Z 3.12) over Region R V : Z 1.2 < 0, Z 2.1 < 0, Z 3.12 > 0.

$$\begin{array}{rll} Q_{V} &= & \big(Z_1, Z_2, Z_3 - Z_{3.12}\big)'[\Sigma]^{-1}\big(Z_1, Z_2, Z_3-Z_{3.12}\big) \\ & = & \left(Z_1, Y_2+\rho_{12}\sigma_2\frac{Z_1}{\sigma_1}, \rho_{13}\sigma_3\frac{Z_1}{\sigma_1} +\sigma_3\frac{Y_2(\rho_{23}-\rho_{12}\rho_{13})}{\sigma_2(1-\rho^2_{12})}\right)' \Sigma^{-1} \\ && \times \left(Z_1, Y_2+\rho_{12}\sigma_2\frac{Z_1}{\sigma_1}, \rho_{13}\sigma_3\frac{Z_1}{\sigma_1} +\sigma_3\frac{Y_2\big(\rho_{23}-\rho_{12}\rho_{13}\big)}{\sigma_2\big(1-\rho^2_{12}\big)}\right)\\ &= & \left[\frac{Z_1}{\sigma_1}\big(1, \rho_{12}, \rho_{13}\big)+\frac{Y_2}{\sigma_2}\left(0, 1,\frac{\rho_{23}-\rho_{12} \rho_{13}}{1-\rho^2_{12}}\right)\right]'{\bf R}^{-1} \\ &&\times \left[\frac{Z_1}{\sigma_1}\big(1, \rho_{12}, \rho_{13}\big)+\frac{Y_2}{\sigma_2}\left(0, 1,\frac{\rho_{23}-\rho_{12} \rho_{13}}{1-\rho^2_{12}}\right)\right] \\ &=& \left[U_1\big(1, \rho_{12}, \rho_{13}\big)+U_2\left(0, \sqrt{1-\rho^2_{12}},\frac{\rho_{23}-\rho_{12} \rho_{13}}{\sqrt{1-\rho^2_{12}}}\right)\right]' {\bf R}^{-1} \\ &&\times \left[U_1(1, \rho_{12}, \rho_{13})+U_2\left(0, \sqrt{1-\rho^2_{12}},\frac{\rho_{23}-\rho_{12} \rho_{13}}{\sqrt{1-\rho^2_{12}}}\right)\right] \end{array} $$
(A.8)
$$\begin{array}{rll} R_{V}& : & Z_{1.2}<0, Z_{2.1} <0, Z_{3.12} >0\\ &\sim & \frac{Z_1}{\sigma_1} < \frac{Y_2}{\sigma_2} \frac{\rho_{12}}{1-\rho^2_{12}}, Y_2<0, \frac{Y_3}{\sigma_3} > \frac{Y_2}{\sigma_2} \frac{\big(\rho_{23}-\rho_{12}\rho_{13}\big)}{1-\rho^2_{12}} \\ & \sim & U_1 < \frac{\rho_{12}}{\sqrt{1-\rho^2_{12}}}U_2, U_2<0, U_{3.2}>0 \end{array} $$
(A.9)

Case VI

\(\mathit{MLE} (0, Z_{2.13}, 0)\) over the region \(R_{\it VI}\!:\! Z_{1.3}<, Z_{3.1}<, Z_{2.13}>\).

$$\begin{array}{rll} Q_{VI}({\bf Z}|\hat {\boldsymbol \theta}) & = & \big(Z_1, Z_2 - Z_{2.13}, Z_3\big)'\Sigma^{-1}\big(Z_1, Z_2 - Z_{2.13}, Z_3\big)\\ & = & \left(Z_1, \rho_{12}\sigma_2\frac{Z_1}{\sigma_1}+\sigma_2 \left[\frac{\rho_{23}-\rho_{12}\rho_{13}}{1-\rho^2_{13}}\right]\frac{Y_3}{\sigma_{3}}, Y_3+\rho_{13}\sigma_{3}\frac{Z_1}{\sigma_1}\right)' \Sigma^{-1} \\ & & \times\left(Z_1, \rho_{12}\sigma_{2}\frac{Z_1}{\sigma_1}+\sigma_2 \left[\frac{\rho_{23}-\rho_{12}\rho_{13}}{1-\rho^2_{13}}\right]\frac{Y_3}{\sigma_{3}}, Y_3+\rho_{13}\sigma_{3}\frac{Z_1}{\sigma_1}\right)\\ & = & \left(\frac{Z_1}{\sigma_{1}}, \rho_{12}\frac{Z_1}{\sigma_1}+ \left[\frac{\rho_{23}-\rho_{12}\rho_{13}}{1-\rho^2_{13}}\right]\frac{Y_3}{\sigma_{3}}, \frac{Y_3}{\sigma_{3}}+\rho_{13}\frac{Z_1}{\sigma_1}\right)' {\bf R}^{-1} \\ & & \times\left(\frac{Z_1}{\sigma_{1}}, \rho_{12}\frac{Z_1}{\sigma_1}+ \left[\frac{\rho_{23}-\rho_{12}\rho_{13}}{1-\rho^2_{13}}\right]\frac{Y_3}{\sigma_{3}}, \frac{Y_3}{\sigma_{3}}+\rho_{13}\frac{Z_1}{\sigma_1}\right)\\ & = & \left[\frac{Z_1}{\sigma_{1}}\big(1, \rho_{12}, \rho_{13}\big) + \frac{Y_3}{\sigma_3}\left(0, \frac{\rho_{23}-\rho_{12}\rho_{13}}{1-\rho^2_{13}}, 1\right)\right]' {\bf R}^{-1} \\ & & \times\left[\frac{Z_1}{\sigma_{1}}(1, \rho_{12}, \rho_{13}) + \frac{Y_3}{\sigma_3}\left(0, \frac{\rho_{23}-\rho_{12}\rho_{13}}{1-\rho^2_{13}}, 1\right)\right]\\ & = & \left[U_1\big(1, \rho_{12}, \rho_{13}\big) + U_3\left(0, \frac{\rho_{23}-\rho_{12}\rho_{13}}{\sqrt{1-\rho^2_{13}}}, \sqrt{1-\rho^2_{13}}\right)\right]'{\bf R}^{-1} \\ && \times\left[U_1(1, \rho_{12}, \rho_{13}) + U_3\left(0, \frac{\rho_{23}-\rho_{12}\rho_{13}}{\sqrt{1-\rho^2_{13}}}, \sqrt{1-\rho^2_{13}}\right)\right] \end{array} $$
(A.10)
$$ R_{VI}: U_1 < \frac{\rho_{13}}{\sqrt{1-\rho^2_{13}}}U_3, U_3<0, U_{2.3}>0. $$
(A.11)

Case VII

\(\mathit{MLE}\! =\! (Z_{1.23}, 0, 0)\) over the region \(R_{\it VII}\!:\! Z_{2.3}<, Z_{3.2}<, Z_{1.23}>\).

To compute the quadratic \(Q_{VII}({\bf Z}|\hat {\boldsymbol \theta})\), note from Eq. B.6 that

$$\begin{array}{rll} \frac{Z_1 - Z_{1.23}}{\sigma_1} & = & \frac{Z_1}{\sigma_1}\frac{\rho^2_{12}+\rho^2_{13}-2\rho_{12}\rho_{13}\rho_{23}} {1-\rho^2_{23}} \\ & & +\, \frac{Y_2}{\sigma_2}\frac{\rho_{12}-\rho_{13}\rho_{23}}{1-\rho^2_{23}} +\frac{Y_3}{\sigma_3}\frac{\rho_{13}-\rho_{12}\rho_{23}}{1-\rho^2_{23}} \end{array} $$
(A.12)

This yields

$$\begin{array}{rll} &&{\kern-8.5pt} \left[\frac{Z_1-Z_{1.23}}{\sigma_1}, \frac{Z_2}{\sigma_2},\frac{Z_3}{\sigma_3}\right] \\ [9pt] &&{\kern4pt} = \left[\frac{Z_1}{\sigma_1}\frac{\rho^2_{12}+\rho^2_{13}-2\rho_{12}\rho_{13}\rho_{23}} {1-\rho^2_{23}} + \frac{Y_2}{\sigma_2}\frac{\rho_{12}-\rho_{13}\rho_{23}}{1-\rho^2_{23}}\right.\\ [9pt] &&{\kern1.5pt}\qquad \left.+\,\frac{Y_3}{\sigma_3}\frac{\rho_{13}-\rho_{12}\rho_{23}}{1-\rho^2_{23}}, \frac{Y_2}{\sigma_2}+\rho_{12}\frac{Z_1}{\sigma_1}, \frac{Y_3}{\sigma_3}+\rho_{13}\frac{Z_1}{\sigma_1}\right] \\ [9pt] &&{\kern4pt}= \left[\frac{Z_1}{\sigma_1}\left(\frac{\rho^2_{12}+\rho^2_{13}-2\rho_{12}\rho_{13}\rho_{23}}{1-\rho^2_{23}}, \rho_{12}, \rho_{13}\right)\right. \\ [9pt] &&{\kern1.5pt}\qquad\left. + \frac{Y_2}{\sigma_{2}}\left(\frac{\rho_{12}-\rho_{13}\rho_{23}}{1-\rho^2_{23}},1,0\right) + \frac{Y_3}{\sigma_{3}}\left(\frac{\rho_{13}-\rho_{12}\rho_{23}}{1-\rho^2_{23}}, 0, 1\right)\right] \\ [9pt] &&{\kern4pt}= \left[U_1\left(\frac{\rho^2_{12}+\rho^2_{13}-2\rho_{12}\rho_{13}\rho_{23}}{1-\rho^2_{23}}, \rho_{12}, \rho_{13}\right)\right. \\ [9pt] && {\kern1pt}\qquad+\left. U_2\sqrt{1-\rho^2_{12}}\left(\frac{\rho_{12}-\rho_{13}\rho_{23}}{1-\rho^2_{23}}, 1,0\right) + U_3\sqrt{1-\rho^2_{13}}\left(\frac{\rho_{13}-\rho_{12}\rho_{23}}{1-\rho^2_{23}}, 0, 1\right)\right] \\ [9pt] &&{\kern4pt}= {\bf V} (say). \end{array} $$
(A.13)

Hence \(Q_{VII}({\bf Z}|\hat {\boldsymbol \theta}) = {\bf V}'{\bf R}^{-1}{\bf V}\).

From Eq. B.3, note that

$$\begin{array}{rll} Z_{2.3} <0 & \sim& \frac{Z_1}{\sigma_1}(\rho_{12}-\rho_{13}\rho_{23}) + \frac{Y_2}{\sigma_2} -\rho_{23}\frac{Y_3}{\sigma_3} < 0 \\& \sim& U_1 (\rho_{12}-\rho_{13}\rho_{23})+U_2\sqrt{1-\rho^2_{12}}- U_3\rho_{23}\sqrt{1-\rho^2_{13}} < 0. \end{array} $$
(A.14)

Similarly, from Eq. B.4, we get

$$\begin{array}{rll}Z_{3.2} <0 & \sim& \frac{Z_1}{\sigma_1}(\rho_{13}-\rho_{12}\rho_{23}) - \rho_{23}\frac{Y_2}{\sigma_2} + \frac{Y_3}{\sigma_3} < 0 \\ & \sim& U_1(\rho_{13}-\rho_{12}\rho_{23})-U_2\rho_{23}\sqrt{1-\rho^2_{12}}+ U_3\sqrt{1-\rho^2_{13}} < 0. \end{array} $$
(A.15)

Finally, from Eq. B.6, we get

$$\begin{array}{rll} Z_{1.23}>0 & \sim & \frac{Z_1}{\sigma_1}\big(1-\rho^2_{12}-\rho^2_{13}-\rho^2_{23}+2\rho_{12}\rho_{13}\rho_{23}\big) \\ && > \frac{Y_2}{\sigma_2}(\rho_{12}- \rho_{13}\rho_{23}) + \frac{Y_3}{\sigma_3}(\rho_{13}- \rho_{12}\rho_{23}) \\ & \sim & U_1\frac{\big(1-\rho^2_{12}-\rho^2_{13}-\rho^2_{23}+2\rho_{12}\rho_{13}\rho_{23}\big)}{\sqrt{\big(1-\rho^2_{12}\big)\big(1-\rho^2_{13}\big)}} \\ & & > U_2\frac{\rho_{12}- \rho_{13}\rho_{23}}{\sqrt{1-\rho^2_{13}}} + U_3 \frac{\rho_{13}- \rho_{12}\rho_{23}}{\sqrt{1-\rho^2_{12}}}. \end{array} $$
(A.16)

Case VIII

MLE = (0, 0, 0) over the region R VIII : Z 1.23 < 0, Z 2.13 < 0, Z 3.12 < 0.

$$ Q_{VIII}({\bf Z}|\hat {\boldsymbol \theta}) = \mbox{full quadratic} = U^2_1 +\frac{U^2_2+U^2_3-2c_{23}U_2U_3}{1-c^2_{23}}. $$
(A.17)

To describe R VIII , from Eq. B.6

$$\begin{array}{rll} Z_{1.23} <0 & \sim & \frac{U_1\big( 1-\rho^2_{12}-\rho^2_{13}-\rho^2_{23}+2\rho_{12}\rho_{13}\rho_{23}\big)}{\sqrt{1-\rho^2_{12}}\sqrt{1-\rho^2_{13}}}\\ & < & \frac{U_2(\rho_{12}-\rho_{13}\rho_{23})}{\sqrt{1-\rho^2_{12}}} + \frac{U_3(\rho_{13}-\rho_{12}\rho_{23})}{\sqrt{1-\rho^2_{13}}}. \end{array} $$
(A.18)

Also, it is easy to verify that Z 2.13 = U 2.3 and Z 3.12 = U 3.2. Hence R VIII can be expressed as

$$\begin{array}{rll} R_{VIII} & : & U_{2.3}<0, U_{3.2}<0, \\ & & \frac{U_1\big(1-\rho^2_{12}-\rho^2_{13}-\rho^2_{23}+2\rho_{12}\rho_{13}\rho_{23}\big)}{\sqrt{1-\rho^2_{12}}\sqrt{1-\rho^2_{13}}} \\ && < \frac{U_2(\rho_{12}-\rho_{13}\rho_{23})}{\sqrt{1-\rho^2_{12}}} + \frac{U_3(\rho_{13}-\rho_{12}\rho_{23})}{\sqrt{1-\rho^2_{13}}}. \end{array} $$
(A.19)

Appendix B

Here we express some standard residuals which are used in Appendix A in terms of Z 1, Y 2 and Y 3.

  • Z 1.2

    $$\begin{array}{rll} Z_{1.2} = Z_1 -\sigma_1\rho_{12}\frac{Z_2}{\sigma_2} & =& Z_1 -\sigma_1\rho_{12}\left[\frac{Y_2}{\sigma_2}+\rho_{12}\frac{Z_1}{\sigma_1}\right] \\ & =& Z_1\big(1-\rho^2_{12}\big) - \sigma_1\rho_{12}\frac{Y_2}{\sigma_2}. \end{array} $$
    (B.1)
  • Z 1.3 By symmetry with Z 1.2, we readily get

    $$ Z_{1.3} = Z_1\big(1-\rho^2_{13}\big) - \sigma_1\rho_{13}\frac{Y_3}{\sigma_3}. $$
    (B.2)
  • Z 2.3

    $$\begin{array}{rll} Z_{2.3} = Z_{2} - \sigma_2\rho_{23}\frac{Z_3}{\sigma_3} & =& \left[Y_2 + \sigma_2\rho_{12}\frac{Z_1}{\sigma_1}\right] - \sigma_2\rho_{23}\left[\rho_{13}\frac{Z_1}{\sigma_1}+\frac{Y_3}{\sigma_3}\right]\\ & =& \sigma_2(\rho_{12}-\rho_{13}\rho_{23})\frac{Z_1}{\sigma_1}+Y_2-\sigma_2\rho_{23}\frac{Y_3}{\sigma_3}. \end{array} $$
    (B.3)
  • Z 3.2 By symmetry with Z 2.3, we readily get

    $$ Z_{3.2} = \sigma_3(\rho_{13}-\rho_{12}\rho_{23})\frac{Z_1}{\sigma_1}-\sigma_3\rho_{23}\frac{Y_2}{\sigma_2}+Y_3. $$
    (B.4)
  • Z 1.23 This term which is the residual of Z 1, given Z 2 and Z 3, is defined as Z 1 − E(Z 1|Z 2, Z 3). Recalling that Z ~N[0, Σ] and writing \({\bf B}_{2 \times 2}= \mbox{dispersion matrix} \) of (Z 2, Z 3), we get

    $$\begin{array}{rll} E(Z_1|Z_2,Z_3) & = & (\sigma_{12}, \sigma_{13}){\bf B}^{-1}(Z_2, Z_3)' \\ & = & \frac{(\sigma_{12}\sigma_{33}-\sigma_{13}\sigma_{23}, -\sigma_{12}\sigma_{23}+\sigma_{13}\sigma_{22})(Z_2, Z_3)'}{\sigma_{22}\sigma_{33}-\sigma^2_{23}} \\ & = & \frac{(\sigma_{12}\sigma_{33}-\sigma_{13}\sigma_{23})Z_2 + (-\sigma_{12}\sigma_{23}+\sigma_{13}\sigma_{22})Z_3 }{\sigma_{22}\sigma_{33}-\sigma^2_{23}} \\ & = & \sigma_1\frac{\rho_{12}-\rho_{13}\rho_{23}}{1-\rho^2_{23}}\frac{Z_2}{\sigma_2} + \sigma_1\frac{\rho_{13} - \rho_{12}\rho_{23}}{1-\rho^2_{23}} . \end{array} $$
    (B.5)

    Now replacing Z 2 by \(Y_2+\sigma_2\rho_{12}\frac{Z_1}{\sigma_1}\) and Z 3 by \(Y_3+\sigma_3\rho_{13}\frac{Z_1}{\sigma_1}\) and simplifying, we get

    $$\begin{array}{rll} Z_{1.23} &= & Z_1\frac{1-\rho^2_{12}-\rho^2_{13}-\rho^2_{23}+2\rho_{12}\rho_{13}\rho_{23}}{1-\rho^2_{23}} \\ && -\sigma_1\frac{\rho_{12}-\rho_{13}\rho_{23}}{1-\rho^2_{23}}\frac{Y_2}{\sigma_2} -\sigma_1\frac{\rho_{13}-\rho_{12}\rho_{23}}{1-\rho^2_{23}}\frac{Y_3}{\sigma_3}. \end{array} $$
    (B.6)
  • Z 2.13 This term which is the residual of Z 2, given Z 1 and Z 3, is defined as Z 2 − E(Z 2|Z 1, Z 3). Using symmetry with E(Z 1|Z 2, Z 3), we readily get

    $$ E(Z_2|Z_1,Z_3) = \sigma_2\frac{\rho_{12}-\rho_{13}\rho_{23}}{1-\rho^2_{13}}\frac{Z_1}{\sigma_1} + \sigma_2\frac{\rho_{23} - \rho_{12}\rho_{13}}{1-\rho^2_{13}}\frac{Z_3}{\sigma_3}$$
    (B.7)

    Now replacing Z 2 by \(Y_2+\sigma_2\rho_{12}\frac{Z_1}{\sigma_1}\) and Z 3 by \(Y_3+\sigma_3\rho_{13}\frac{Z_1}{\sigma_1}\) and noting that the coefficient of Z 1 is 0, we get

    $$ Z_{2.13} = Y_2 - \sigma_2 \times \frac{\rho_{23}-\rho_{12}\rho_{13}}{1-\rho^2_{13}}\times \frac{Y_3}{\sigma_3}. $$
    (B.8)
  • Z 3.12 By symmetry with Z 2.13, we readily get

    $$\label{58} Z_{3.12} = Y_3 - \sigma_3 \times \frac{\rho_{23}-\rho_{12}\rho_{13}}{1-\rho^2_{12}}\times \frac{Y_2}{\sigma_2}. $$
    (B.9)

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Kopylev, L., Sinha, B. On the asymptotic distribution of likelihood ratio test when parameters lie on the boundary. Sankhya B 73, 20–41 (2011). https://doi.org/10.1007/s13571-011-0022-z

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