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Asymptotic Inferences in a Multinomial Logit Mixed Model for Spatial Categorical Data

Abstract

There exist many studies on regression analysis for spatial binary data, espsecially in ecological, environmental and socio-economic setups, where spatial responses from neighboring locations within a given threshold distance are correlated. However, in some of these studies, it could be more natural to consider a spatial regression analysis for categorical response data with more than two categories, as an improvement over the spatial binary analysis. But, this type of regression analysis for spatial categorical/multinomial data is not adequately addressed in the literature. One of the main reasons is the difficulty of modeling the spatial familial correlations for categorical data, where a spatial family is generated within the threshold distance for each of the two selected neighboring locations. Also, some of the locations from two families may be pair-wise correlated. Unlike the existing studies, in this paper we propose a familial random effects based multinomial logits mixed (MLM) effects model which accommodates both within and between familial correlations for spatial multinomial data. In this context, the proposed spatial multinomial correlations are contrasted with existing longitudinal multinomial correlations so that the longitudinal correlation models are avoided for spatial multinomial data. Both regression effects and the random effects influence parameters are estimated using the generalized quasi-likelihood approach, whereas the random effects variance and correlation parameters are estimated by the well known method of moments. The large sample properties such as consistency of the proposed estimators are studied analytically. The asymptotic normality of the regression estimators is also studied for the convenience of constructing the confidence intervals when needed. The devirations and proofs are given in details, as opposed to conducting a limited simulation study, to justify the validity and convergence properties of the proposed estimators. The estimating equations those produced consistent estimates are clearly formulated for the computational benefit to the practitioners.

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Acknowledgements

The authors would like to thank two reviewers for their valuable comments and suggestions that led to the improvement of the paper. Thanks are also due to the Editor-in-Chief, the Editor, and the Associate Editor for their helpful suggestions.

Funding

This research was partially supported by a grant (RGPIN-04503-2015) from the Natural Sciences and the Engineering Research Council of Canada, awarded to the first author.

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Correspondence to Brajendra C. Sutradhar.

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Appendices

Appendix: Unconditional Mean (2.8) and Variance (2.13) Computation Using Binomial Approximation to Normal Integration

Mean Computation:

By Eq. 2.10 (see also Assumption 1 and Lemma 2.1) γiγi(ni) has the multivariate normal distribution as

$$ {\gamma}_{i}(n_{i}) \sim N_{n_{i}}(0,{\Phi}_{i}(\sigma^{2}_{\gamma},\phi)), \text{with} {\Phi}_{i}(\sigma^{2}_{\gamma},\phi)=\sigma^{2}_{\gamma} {C}_{ii}(\phi). $$
(a.1)

Consequently, by using a non-singular transformation \({g}_{i}={\Phi }^{-\frac {1}{2}}_{i}(\sigma ^{2}_{\gamma },\phi ){\gamma }_{i}\) so that \({g}_{i} \sim N_{n_{i}}(0,I),\) we first replace γi in Eq. 2.8 with \({\gamma }_{i}={\Phi }^{\frac {1}{2}}_{i}{g}_{i},\) and re-express the unconditional multinomial mean as

$$ \begin{array}{@{}rcl@{}} &&\!\!\!\!\!\!\pi_{i,c}(\beta,\tau_{\gamma},\sigma_{\gamma},\phi)=Pr[Y_{s_{i}}=y^{(c)}_{s_{i}}]\\ &\!\!\!\!\!\!\!\!\!=&\!\!\!\!\!\!\!\left\{\!\!\!\begin{array}{ll} {\int}^{\infty}_{-\infty}\exp(x^{\prime}_{i}\beta_{c} + \tau_{\gamma,c}a'_{i}{\Phi}^{\frac{1}{2}}_{i}{g}_{i})/ [1 + {\sum}^{C-1}_{u=1}\exp(x^{\prime}_{i}\beta_{u} +\tau_{\gamma,u}{a}'_{i}{\Phi}^{\frac{1}{2}}_{i}{g}_{i})] \\ g^{*}_{N}(g_{i}) {\Pi}^{n_{i}}_{u_{i}=1}dg_{iu_{i}} & \text{for} c=1,\ldots,C-1 \\ {\int}^{\infty}_{-\infty}1/ [1+{\sum}^{C-1}_{u=1}\exp(x^{\prime}_{i}\beta_{u} + \tau_{\gamma,u}a'_{i}{\Phi}^{\frac{1}{2}}_{i}{g}_{i})]g^{*}_{N}(g_{i}) {\Pi}^{n_{i}}_{u_{i}=1}dg_{iu_{i}} & \text{for} c = C \end{array} \right. \\ &\!\!\!\!\!\!\!\!\!=&\!\!\!\!\!\!\!\!\left\{\!\!\!\begin{array}{ll} {\int}^{\infty}_{-\infty}\exp(x^{\prime}_{i}\beta_{c} + \tau_{\gamma,c}{\sum}^{n_{i}}_{u_{i}=1} {a^{*}}_{iu_{i}}(\sigma^{2}_{\gamma},\phi)g_{iu_{i}})/ [1 + {\sum}^{C-1}_{u=1}\exp(x^{\prime}_{i}\beta_{u} + \tau_{\gamma,u}{\sum}^{n_{i}}_{u_{i}=1} {a^{*}}_{iu_{i}}(\sigma^{2}_{\gamma},\phi)g_{iu_{i}})]\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \\ {\Pi}^{n_{i}}_{u_{i}=1}g^{*}_{N}(g_{iu_{i}}) dg_{iu_{i}} \text{for} c=1,\ldots,C-1 &\\ {\int}^{\infty}_{-\infty}1/ [1+{\sum}^{C-1}_{u=1}\exp(x^{\prime}_{i}\beta_{u} +\tau_{\gamma,u}{\sum}^{n_{i}}_{u_{i}=1} {a^{*}}_{iu_{i}}(\sigma^{2}_{\gamma},\phi)g_{iu_{i}})]{\Pi}^{n_{i}}_{u_{i}=1}g^{*}_{N}(g_{iu_{i}}) dg_{iu_{i}} \text{for} c = C \end{array} \right. \\ &\!\!\!\!\!\!\!\!\!=&\!\!\!\!\!\!\!\left\{\!\!\!\begin{array}{ll} {\int}^{\infty}_{-\infty}\frac{\tilde{\pi}^{*}_{(i,c)N}({a^{*}}_{i1},\ldots, {a^{*}}_{iu_{i}},\ldots,{a^{*}}_{in_{i}};g_{i1},\ldots,g_{iu_{i}},\ldots, g_{in_{i}})}{\tilde{\pi}^{*}_{(i,c)D}({a^{*}}_{i1},\ldots, {a^{*}}_{iu_{i}},\ldots,{a^{*}}_{in_{i}};g_{i1},\ldots,g_{iu_{i}},\ldots, g_{in_{i}})}{\Pi}^{n_{i}}_{u_{i}=1}g^{*}_{N}(g_{iu_{i}}) dg_{iu_{i}} & \text{for} c=1,\ldots,C-1\!\!\! \\ {\int}^{\infty}_{-\infty}\frac{1}{\tilde{\pi}^{*}_{(i,c)D}({a^{*}}_{i1},\ldots, {a^{*}}_{iu_{i}},\ldots,{a^{*}}_{in_{i}};g_{i1},\ldots,g_{iu_{i}},\ldots, g_{in_{i}})}{\Pi}^{n_{i}}_{u_{i}=1}g^{*}_{N}(g_{iu_{i}}) dg_{iu_{i}} & \text{for} c=C, \end{array} \right. \end{array} $$
(a.2)

where \({{{a}}}^{*'}_{i}={a}'_{i}{\Phi }^{\frac {1}{2}}_{i}=({a^{*}}_{i1},\ldots , {a^{*}}_{iu_{i}},\ldots ,{a^{*}}_{in_{i}})',\) and \(g^{*}_{N}(g_{iu_{i}}) \equiv N(0,1)\) for all ui = 1,…,ni.

Next we use a Binomial approximation (Ten Have and Morabia (1999, Eqn. (7)), Sutradhar (2014, Eqns. (5.48)-(5.50))) to perform the standard normal integration in Eq. a.2. More specifically, suppose that \(v_{iu_{i}}\) is binomial variable which ranges from 0 to V, V being a positive integer. Now replace the standard normal variable \(g_{iu_{i}}\) in Eq. a.2 with a standardized binomial variable

$$ g_{iu_{i}}\equiv [v_{iu_{i}}-V(\frac{1}{2})]/\sqrt{V\frac{1}{2}\frac{1}{2}}, $$
(a.3)

with large V such as V = 10, and re-express the functions involved under the integration as

$$ \begin{array}{@{}rcl@{}} &&{}\tilde{\pi}^{*}_{(i,c)N}(\beta,\tau_{\gamma},\sigma_{\gamma},\phi; {a^{*}}_{i1},\ldots, {a^{*}}_{iu_{i}},\ldots,{a^{*}}_{in_{i}};v_{i1},\ldots,v_{iu_{i}},\ldots,v_{in_{i}}) \\ &&{}=\exp(x^{\prime}_{i}\beta_{c}+\tau_{\gamma,c}{\sum}^{n_{i}}_{u_{i}=1} {a^{*}}_{iu_{i}}(\sigma^{2}_{\gamma},\phi)(v_{iu_{i}}-V(\frac{1}{2})) /\sqrt{V(\frac{1}{2})(\frac{1}{2})}), \end{array} $$
(a.4)
$$ \begin{array}{@{}rcl@{}} &&{}\tilde{\pi}^{*}_{(i,c)D}(\beta,\tau_{\gamma},\sigma_{\gamma},\phi; {a^{*}}_{i1},\ldots, {a^{*}}_{iu_{i}},\ldots,{a^{*}}_{in_{i}};v_{i1},\ldots,v_{iu_{i}},\ldots,v_{in_{i}})\\ &&{}=[1+{\sum}^{C-1}_{u=1}\exp(x^{\prime}_{i}\beta_{u} +\tau_{\gamma,u}{\sum}^{n_{i}}_{u_{i}=1} {a^{*}}_{iu_{i}}(\sigma^{2}_{\gamma},\phi)(v_{iu_{i}}-V(\frac{1}{2}))/\sqrt{V(\frac{1}{2})(\frac{1}{2})} )], \end{array} $$
(a.5)

and

$$ \begin{array}{@{}rcl@{}} &&\tilde{\pi}^{*}_{i,c}(\beta,\tau_{\gamma},\sigma_{\gamma},\phi; {a^{*}}_{i1},\ldots, {a^{*}}_{iu_{i}},\ldots,{a^{*}}_{in_{i}};v_{i1},\ldots,v_{iu_{i}},\ldots,v_{in_{i}})\\ &=&\frac{\tilde{\pi}^{*}_{(i,c)N}(\beta,\tau_{\gamma},\sigma_{\gamma},\phi; {a^{*}}_{i1},\ldots, {a^{*}}_{iu_{i}},\ldots,{a^{*}}_{in_{i}};v_{i1},\ldots,v_{iu_{i}},\ldots,v_{in_{i}})} {\tilde{\pi}^{*}_{(i,c)D}(\beta,\tau_{\gamma},\sigma_{\gamma}, \phi;{a^{*}}_{i1},\ldots, {a^{*}}_{iu_{i}},\ldots,{a^{*}}_{in_{i}};v_{i1},\ldots,v_{iu_{i}},\ldots,v_{in_{i}})}. \end{array} $$
(a.6)

It then follows that the unconditional spatial multinomial mean in Eq. a.2 may be computed using the above binomial approximation (a.4)–(a.6), as

$$ \begin{array}{@{}rcl@{}} &&\pi_{i,c}(\beta,\tau_{\gamma},\sigma_{\gamma},\phi)=Pr[y_{s_{i}}=y^{(c)}_{s_{i}}]\\ &\!\!\!\!\!=&\!\!\!\!\!\left\{\!\!\begin{array}{ll} {\sum}^{V}_{v_{i1}=0}\ldots {\sum}^{V}_{v_{iu_{i}}=0}{\ldots} {\sum}^{V}_{v_{in_{i}}=0} \tilde{\pi}^{*}_{i,c}(\beta,\tau_{\gamma},\sigma_{\gamma},\phi; {a^{*}}_{i1},\ldots, {a^{*}}_{iu_{i}},\ldots,\\~{a^{*}}_{in_{i}};v_{i1},\ldots,v_{iu_{i}},\ldots,v_{in_{i}}) \\ {\Pi}^{n_{i}}_{u_{i}=1}\left\{\left( \begin{array}{cc}V \\ v_{iu_{i}} \end{array}\right)(1/2)^{v_{iu_{i}}}(1/2)^{V-v_{iu_{i}}}\right\} \text{for} c=1,\ldots,C-1 &\\ {\sum}^{V}_{v_{i1}=0}{\ldots} {\sum}^{V}_{v_{iu_{i}}=0}\ldots {\sum}^{V}_{v_{in_{i}}=0} \left[\tilde{\pi}^{*}_{(i,c)D}(\beta,\tau_{\gamma},\sigma_{\gamma},\phi; {a^{*}}_{i1},\ldots, {a^{*}}_{iu_{i}},\ldots,\right.\!\!\!\!\\~\left.{a^{*}}_{in_{i}};v_{i1},\ldots,v_{iu_{i}}, \ldots,v_{in_{i}})\right]^{-1}\\ {\Pi}^{n_{i}}_{u_{i}=1}\left\{\left( \begin{array}{cc}V \\ v_{iu_{i}} \end{array}\right)(1/2)^{v_{iu_{i}}}(1/2)^{V-v_{iu_{i}}}\right\} \text{for} c = C, & \end{array} \right. \end{array} $$
(a.7)

Variance Computation by Eqs. 2.13 and 2.18:

Recall from Section 2.4.1 that

$$ \begin{array}{@{}rcl@{}} &&g^{*}_{N}(\gamma_{i}, \bar{\gamma}_{j}) \equiv N(0,\sigma^{2}_{\gamma} {{C}}^{\dag}_{ij}(\phi) ), \end{array} $$
(a.8)

where \({{C}}^{\dag }_{ij}(\phi ) ): (n_{i}+n^{*}_{j}) \times (n_{i}+n^{*}_{j})\) is the correlation matrix as in Eq. 2.26, where \(n^{*}_{j}=n_{j}-n_{ij}\). This is equivalent to write

$$ \begin{array}{@{}rcl@{}} {\gamma}^{\dag}_{ij}=\left( \begin{array}{cc}\gamma_{i} \\ \bar{\gamma}_{j} \end{array}\right)=\left( \begin{array}{cc}\bar{\gamma}_{i} \\ {\gamma}_{ij} \\ \bar{\gamma}_{j} \end{array}\right) \sim N(0,\sigma^{2}_{\gamma} {{C}}^{\dag}_{ij}(\phi) ) \equiv N(0,\tilde{\Phi}_{ij}(\sigma^{2}_{\gamma},\phi)) \end{array} $$
(a.9)

It then follows that

$$ \begin{array}{@{}rcl@{}} &&{g}^{\dagger}_{ij}=\tilde{\Phi}^{-\frac{1}{2}}_{ij}(\sigma^{2}_{\gamma},\phi) {\gamma}^{\dagger}_{ij} \sim N (0,I_{n_{i}+n^{*}_{j}}). \end{array} $$
(a.10)

Next write

$$ \begin{array}{@{}rcl@{}} &&\left( \begin{array}{cc}\gamma_{i} \\ \bar{\gamma}_{j} \end{array}\right)=\left( \begin{array}{cc}\bar{\gamma}_{i} \\ {\gamma}_{ij} \\\bar{\gamma}_{j} \end{array}\right) \\ &=&\tilde{\Phi}^{\frac{1}{2}}_{ij}(\sigma^{2}_{\gamma},\phi) {g}^{\dagger}_{ij} = \left( \begin{array}{c} {A}(\sigma^{2}_{\gamma},\phi) \\ {B}(\sigma^{2}_{\gamma},\phi) \end{array} \right) {g}^{\dagger}_{ij} \end{array} $$
(a.11)
$$ \begin{array}{@{}rcl@{}} &=& \left( \begin{array}{c} {A}_{1}(\sigma^{2}_{\gamma},\phi):n^{*}_{i} \times (n_{i}+n^{*}_{j}) \\ {A}_{2}(\sigma^{2}_{\gamma},\phi):n_{ij} \times (n_{i}+n^{*}_{j}) \\ {B}(\sigma^{2}_{\gamma},\phi):n^{*}_{j} \times (n_{i}+n^{*}_{j}) \end{array} \right) {g}^{\dagger}_{ij}. \end{array} $$
(a.12)

Using Eq. a.12, write

$$ \begin{array}{@{}rcl@{}} &&\left( \begin{array}{cc}\bar{\gamma}_{j} \\ \gamma_{ij} \end{array} \right)= \left( \begin{array}{c} {B}(\sigma^{2}_{\gamma},\phi) \\ {A_{2}}(\sigma^{2}_{\gamma},\phi) \end{array} \right) {g}^{\dagger}_{ij}=B^{*}(\sigma^{2}_{\gamma},\phi){g}^{\dagger}_{ij}. \end{array} $$
(a.13)

Thus by Eqs. a.11 and a.13, we write

$$ \gamma_{i}=A(\sigma^{2}_{\gamma},\phi){g}^{\dagger}_{ij}, \text{and} \gamma_{j}=B^{*}(\sigma^{2}_{\gamma},\phi){g}^{\dagger}_{ij}. $$
(a.14)

Use them in Eq. 2.13 and follow the notations from Eq. a.2 to compute the joint probability as follows:

$$ \begin{array}{@{}rcl@{}} &&\lambda_{ij,cr}(\beta,\tau_{\gamma},\sigma_{\gamma},\phi) =Pr[y_{s_{i}}=y^{(c)}_{s_{i}},y_{s_{j}}=y^{(r)}_{s_{j}}]\\ &\!\!\!\!=&\!\!\!\!\left\{\!\!\!\begin{array}{ll} {\int}^{\infty}_{-\infty}\left[\frac{\tilde{\pi}^{*}_{(i,c)N}({a^{\dagger}}_{i1},\ldots, {a^{\dagger}}_{ik},\ldots,{a^{\dagger}}_{i,n_{i}+n^{*}_{j}};g^{\dagger}_{ij,1},\ldots, g^{\dagger}_{ij,k},\ldots, g^{\dagger}_{ij,n_{i}+n^{*}_{j}})}{\tilde{\pi}^{*}_{(i,c)D}({a^{\dagger}}_{i1},\ldots, {a^{\dagger}}_{ik},\ldots,{a^{\dagger}}_{i,n_{i}+n^{*}_{j}};g^{\dagger}_{ij,1},\ldots, g^{\dagger}_{ij,k},\ldots, g^{\dagger}_{ij,n_{i}+n^{*}_{j}} )}\right. \\ \times \left.\frac{\tilde{\pi}^{*}_{(j,r)N}({b^{\dagger}}_{j1},\ldots, {b^{\dagger}}_{jk},\ldots,{b^{\dagger}}_{j,n_{i}+n^{*}_{j}}; g^{\dagger}_{ij,1},\ldots, g^{\dagger}_{ij,k},\ldots, g^{\dagger}_{ij,n_{i}+n^{*}_{j}})} {\tilde{\pi}^{*}_{(j,r)D}({b^{\dagger}}_{j1},\ldots, {b^{\dagger}}_{jk},\ldots,{b^{\dagger}}_{j,n_{i}+n^{*}_{j}}; g^{\dagger}_{ij,1},\ldots, g^{\dagger}_{ij,k},\ldots, g^{\dagger}_{ij,n_{i}+n^{*}_{j}})}\right]{\Pi}^{n_{i}+n^{*}_{j}}_{k=1}g^{*}_{N}(g^{\dagger}_{ij,k}) dg^{\dagger}_{ij,k}\!\!\!\! \\~~~ \text{for} c,r=1,\ldots,C-1 \\ {\int}^{\infty}_{-\infty} \left[\frac{\tilde{\pi}^{*}_{(i,c)N}({a^{\dagger}}_{i1},\ldots, {a^{\dagger}}_{ik},\ldots,{a^{\dagger}}_{i,n_{i}+n^{*}_{j}}; g^{\dagger}_{ij,1},\ldots, g^{\dagger}_{ij,k},\ldots, g^{\dagger}_{ij,n_{i}+n^{*}_{j}})}{\tilde{\pi}^{*}_{(i,c)D}({a^{\dagger}}_{i1},\ldots, {a^{\dagger}}_{ik},\ldots,{a^{\dagger}}_{i,n_{i}+n^{*}_{j}}; g^{\dagger}_{ij,1},\ldots, g^{\dagger}_{ij,k},\ldots, g^{\dagger}_{ij,n_{i}+n^{*}_{j}})}\right. \\ \times \left. \frac{1}{\tilde{\pi}^{*}_{(j,r)D}({b^{\dagger}}_{j1},\ldots, {b^{\dagger}}_{jk},\ldots,{b^{\dagger}}_{j,n_{i}+n^{*}_{j}}; g^{\dagger}_{ij,1},\ldots, g^{\dagger}_{ij,k},\ldots, g^{\dagger}_{ij,n_{i}+n^{*}_{j}})}\right]{\Pi}^{n_{i}+n^{*}_{j}}_{k=1}g^{*}_{N}(g^{\dagger}_{ij,k}) dg^{\dagger}_{ij,k} \!\!\!\! \\~~~\text{for} c=1,\ldots,C-1; r=C\\ {\int}^{\infty}_{-\infty} \left[\frac{1} {\tilde{\pi}^{*}_{(i,c)D}({a^{\dagger}}_{i1},\ldots, {a^{\dagger}}_{ik},\ldots,{a^{\dagger}}_{i,n_{i}+n^{*}_{j}}; g^{\dagger}_{ij,1},\ldots, g^{\dagger}_{ij,k},\ldots, g^{\dagger}_{ij,n_{i}+n^{*}_{j}})}\right. \\ \times \left. \frac{\tilde{\pi}^{*}_{(j,r)N}({b^{\dagger}}_{j1},\ldots, {b^{\dagger}}_{jk},\ldots,{b^{\dagger}}_{j,n_{i}+n^{*}_{j}}; g^{\dagger}_{ij,1},\ldots, g^{\dagger}_{ij,k},\ldots, g^{\dagger}_{ij,n_{i}+n^{*}_{j}})} {\tilde{\pi}^{*}_{(j,r)D}({b^{\dagger}}_{j1},\ldots, {b^{\dagger}}_{jk},\ldots,{b^{\dagger}}_{j,n_{i}+n^{*}_{j}}; g^{\dagger}_{ij,1},\ldots, g^{\dagger}_{ij,k},\ldots, g^{\dagger}_{ij,n_{i}+n^{*}_{j}})}\right]{\Pi}^{n_{i}+n^{*}_{j}}_{k=1}g^{*}_{N}(g^{\dagger}_{ij,k}) dg^{\dagger}_{ij,k}\!\!\!\! \\~~~ \text{for} c=C; r=1,\ldots,C-1 \\ {\int}^{\infty}_{-\infty} \left[\frac{1} {\tilde{\pi}^{*}_{(i,c)D}({a^{\dagger}}_{i1},\ldots, {a^{\dagger}}_{ik},\ldots,{a^{\dagger}}_{i,n_{i}+n^{*}_{j}}; g^{\dagger}_{ij,1},\ldots, g^{\dagger}_{ij,k},\ldots, g^{\dagger}_{ij,n_{i}+n^{*}_{j}})}\right. \\ \times \left. \frac{1}{\tilde{\pi}^{*}_{(j,r)D}({b^{\dagger}}_{j1},\ldots, {b^{\dagger}}_{jk},\ldots,{b^{\dagger}}_{j,n_{i}+n^{*}_{j}}; g^{\dagger}_{ij,1},\ldots, g^{\dagger}_{ij,k},\ldots, g^{\dagger}_{ij,n_{i}+n^{*}_{j}})}\right]{\Pi}^{n_{i}+n^{*}_{j}}_{k=1}g^{*}_{N}(g^{\dagger}_{ij,k}) dg^{\dagger}_{ij,k} \!\!\!\!\\~~~ \text{for} c=C; r=C, \end{array} \right. \end{array} $$
(a.15)

where

$$ \begin{array}{@{}rcl@{}} {{{a}}}^{\dagger'}_{i}&=&{a}'_{i}A(\sigma^{2}_{\gamma},\phi)=({a^{\dagger}}_{i1},\ldots, {a^{\dagger}}_{ik},\ldots,{a^{\dagger}}_{i,n_{i}+n^{*}_{j}})', \\ {{{b}}}^{\dagger'}_{j}&=&{a}'_{j}B^{*}(\sigma^{2}_{\gamma},\phi)=({b^{\dagger}}_{j1},\ldots, {b^{\dagger}}_{jk},\ldots,{b^{\dagger}}_{j,n_{i}+n^{*}_{j}})', \end{array} $$

and \(g^{*}_{N}(g^{\dagger }_{ij,k}) \equiv N(0,1)\) for all \(k=1,\ldots ,n_{i}+n^{*}_{j}.\)

Next, these four multiple integrations in Eq, a.15 may be computed by using the Binomial approximation in the same as done through Eqs. a.3a.7, with a difference that instead of Eq. a.3, we now use

$$ g^{\dagger}_{ij,k}\equiv [v_{ij,k}-V(\frac{1}{2})]/\sqrt{V\frac{1}{2}\frac{1}{2}}, $$
(a.16)

for \(k=1,\ldots ,n_{i}+n^{*}_{j}.\)

Appendix B. Proof for Asymptotic Normality of the GQL Regression Estimator

Asymptotic Normality:

Recall from Eq. 4.2 that \(\hat {\beta }_{GQL}\) is obtained by solving \(\frac {\partial Q(\beta |\tau _{\gamma },\sigma _{\gamma },\phi )}{\partial \beta }=0.\) More specifically, by using a first order Taylor series approximation, it can be shown that \(\hat {{\beta }}_{GQL}\) from Eq. 4.2 satisfies

$$ \begin{array}{@{}rcl@{}} &&{}\hat{{\beta}}_{GQL}-{\beta} \approx -\left[\frac{\partial {\pi}'({\beta}|\cdot)}{\partial {\beta}}{\Sigma}^{-1}(\beta,\tau_{\gamma},\sigma_{\gamma},\phi) \frac{\partial {\pi}({\beta}|\cdot)}{\partial {\beta}'}\right]^{-1} \\ &&{}\times \frac{\partial {\pi}'({\beta}|\cdot)}{\partial {\beta}}{\Sigma}^{-1}(\beta,\tau_{\gamma},\sigma_{\gamma},\phi) (y-\pi(\beta|\tau_{\gamma},\sigma_{\gamma},\phi)) +o_{p}(1/\sqrt{K}), \end{array} $$
(b.1)

where \({y}=({y}'_{s_{1}},\ldots ,{y}'_{s_{i}},\ldots ,{y}'_{s_{K}})'\) is the complete spatial multinomial vector response of dimension K(C − 1) × 1, as defined in Eq. 3.1. Use

$$ \begin{array}{@{}rcl@{}} &&{z}={\Sigma}^{-\frac{1}{2}}({\beta},\tau_{\gamma}, \sigma_{\gamma},\phi) {y}, \text{and} \tilde{{\pi}}({\beta}|\cdot)={\Sigma}^{-\frac{1}{2}} ({\beta},\tau_{\gamma},\sigma_{\gamma},\phi){\pi}({\beta}|\cdot), \end{array} $$
(b.2)

and for large K, using Eq. 4.7, re-express (b.1) as

$$ \begin{array}{@{}rcl@{}} &&[\hat{{\beta}}_{GQL}-{\beta}] \approx -{V}_{K}({\beta},\tau_{\gamma}, \sigma_{\gamma},\phi){M}({\beta},\tau_{\gamma},\sigma_{\gamma},\phi) ({z}-\tilde{{\pi}}({\beta}|\cdot)), \end{array} $$
(b.3)

where \({M}({\beta },\tau _{\gamma },\sigma _{\gamma },\phi ) =\frac {\partial {\pi }'({\beta }|\cdot )}{\partial {\beta }}{\Sigma }^{-\frac {1}{2}}({\beta },\tau _{\gamma },\sigma _{\gamma },\phi ).\) Notice that

$$ \begin{array}{@{}rcl@{}} &&{z} \sim (\tilde{{\pi}}({\beta}|\cdot),I_{K(C-1)}), \end{array} $$
(b.4)

justifying that z1,…,zi,…,zK, are uncorrelated (C − 1)-dimensional vectors. Further, re-express (b.3) as

$$ \begin{array}{@{}rcl@{}} [\hat{{\beta}}_{GQL}-{\beta}] &\approx & -{V}_{K}({\beta},\tau_{\sigma},\sigma_{\gamma},\phi)\left( \begin{array}{ccccc}{M}_{1}(\cdot) &{\ldots} & {M}_{i}(\cdot) & {\ldots} & {M}_{K}(\cdot) \end{array}\right) \\ &\times & \left( \begin{array}{ccccc}({z}_{1}-\tilde{{\pi}}_{1}({\beta}|\cdot))\\ \vdots \\ ({z}_{i}-\tilde{{\pi}}_{i}({\beta}|\cdot)) \\ {\vdots} \\ ({z}_{K}-\tilde{{\pi}}_{K}({\beta}|\cdot)) \end{array}\right) \\ &=&-K{V}_{K}({\beta},\tau_{\gamma},\sigma_{\gamma},\phi)\frac{1}{K}{\sum}^{K}_{i=1}{M}_{i} ({z}_{i}-\tilde{{\pi}}_{i}({\beta}|\cdot)) \\ &=&-K{V}_{K}({\beta},\tau_{\gamma},\sigma_{\gamma},\phi)\frac{1}{K}{\sum}^{K}_{i=1}{h}_{i}({\beta}|\cdot) \\ &=&-K{V}_{K}({\beta},\tau_{\gamma},\sigma_{\gamma},\phi)\bar{{h}}({\beta}|\cdot). \end{array} $$
(b.5)

Notice that

$$ \begin{array}{@{}rcl@{}} &&E[{h}_{i}({\beta}|\cdot)]=0, \text{var}[{h}_{i}({\beta}|\cdot)]={M}_{i}{M}'_{i}, \text{and} \\ &&\text{cov}[{h}_{i}({\beta}|\cdot),{h}'_{j}({\beta}|\cdot)]=0, \end{array} $$
(b.6)

because cov[zi,zj′] = 0 by Eq. b.4. Thus, \(\bar {{h}}({\beta }|\cdot )\) in Eq. b.5 has its mean and covariance as

$$ \begin{array}{@{}rcl@{}} &&{}E[\bar{{h}}({\beta}|\cdot)]=0, \\ &&{}\text{and} \text{cov}[\bar{{h}}({\beta}|\cdot)] =\frac{1}{K^{2}}{\sum}^{K}_{i=1}{M}_{i}{M}'_{i}=\frac{1}{K^{2}}{M}{M}' =\frac{1}{K^{2}}{V}^{-1}_{K}({\beta},\tau_{\gamma},\sigma_{\gamma},\phi). \end{array} $$
(b.7)

We now assume that the following regularity condition holds.

ASSUMPTION 3

Suppose that hi(β|⋅) with moment properties as in Eq. b.7 satisfy the Lindeberg regularity condition that

$$ {\lim}_{K \rightarrow \infty}V_{K}(\cdot){\sum}^{K}_{i=1}{\sum}_{\{{h}'_{i}V_{K}{h}_{i}\}>\epsilon}{h}_{i}{h}'_{i}g({h}_{i})=0, $$
(b.8)

for all 𝜖 > 0, g(⋅) being the p-dimensional probability distribution of hi(⋅).

One may then exploit the Lindeberg-Feller central limit theorem (Amemiya 1985, Theorem 3.3.6) and obtain the limiting distribution of

$$ \left[\text{cov}(\bar{{h}}({\beta}|\cdot) )\right]^{-\frac{1}{2}} \bar{{h}}({\beta}|\cdot)=w_{K} (\text{say}) $$
(b.9)

as

$$ {\lim}_{K \rightarrow \infty} w_{K} \rightarrow_{d} N(0,I_{(p+1)(C-1)}). $$
(b.10)

Now because by Eq. b.9 it follows from Eq. b.5 that

$$ \begin{array}{@{}rcl@{}} &&[\hat{{\beta}}_{GQL}-{\beta}] \approx -{V}^{\frac{1}{2}}_{K}({\beta},\tau_{\gamma},\sigma_{\gamma},\phi) \left[{V}^{-1}_{K}(\cdot)/K^{2}\right]^{-\frac{1}{2}}\bar{{h}}({\beta}|\cdot) \\ &=&-{V}^{\frac{1}{2}}_{K}({\beta},\tau_{\gamma},\sigma_{\gamma},\phi)w_{K}. \end{array} $$
(b.11)

Hence by Eq. b.10,

$$ {\lim}_{K \rightarrow \infty}[\hat{\beta}_{GQL}-{\beta}] \rightarrow_{d} N(0,{V}^{\frac{1}{2}}_{K}({\beta},\tau_{\gamma},\sigma_{\gamma},\phi)I_{(p+1)(C-1)} {V}^{\frac{1}{2}}_{K}({\beta},\tau_{\gamma},\sigma_{\gamma},\phi)).$$

That is

$$ {\lim}_{K \rightarrow \infty}[\hat{\beta}_{GQL}-{\beta}] \rightarrow_{d} N(0,{V}_{K}({\beta},\tau_{\gamma},\sigma_{\gamma},\phi))$$

justifying the limiting distributional result in Eq. 4.3 under the Theorem 4.1.

Appendix C: Formulas for the Derivatives in Estimating Equations

Computation of the Derivatives \(\frac {\partial {\pi _{i,c}}({\beta }|\cdot )}{\partial {\beta }}\) in Eq. 4.6:

Recall from Eq. 2.2 that β = (β1′,…,βc′,…,βC− 1′) is a (C − 1)(p + 1)-dimensional vector of regression parameters,βc being the (p + 1)-dimensional vector corresponding to the c-th category. Because this β parameter vector is involved in πi,c (marginal probability for i-th spatial location (si) response to be in c-th category) in the way as shown by Eq. 2.8, we can compute the derivatives in Eq. 4.6 as follows.

$$ \begin{array}{@{}rcl@{}} &&\frac{\partial {\pi_{i,c}}({\beta}|\cdot)}{\partial {\beta}} \end{array} $$
(c.1)
$$ \begin{array}{@{}rcl@{}} &\!\!\!\!=&\!\!\!\!\left\{\!\!\!\begin{array}{ll} {\int}^{\infty}_{-\infty}\frac{\partial }{\partial \beta}[\exp(x^{\prime}_{i}\beta_{c}+\tau_{\gamma,c}a'_{i}\gamma_{i})/ [1+{\sum}^{C-1}_{u=1}\exp(x^{\prime}_{i}\beta_{u} +\tau_{\gamma,u}a'_{i}\gamma_{i})]] \\ g^{*}_{N}(\gamma_{i}) {\Pi}^{n_{i}}_{u_{i}=1}d\tilde{\gamma}_{s_{iu_{i}}} & \text{for} c = 1,\ldots,C-1\!\!\!\! \end{array} \right. \\ &\!\!\!\!=&\!\!\!\!{\int}^{\infty}_{-\infty}\left[\left\{{\pi}^{*}_{i,c}(\beta,\tau_{\gamma},\gamma_{i}) (\delta_{c}-{\pi}^{*}_{i}(\beta,\tau_{\gamma},\gamma_{i})) \right\} \otimes x_{i}\right]g^{*}_{N}(\gamma_{i}) {\Pi}^{n_{i}}_{u_{i}=1}d\tilde{\gamma}_{s_{iu_{i}}} \end{array} $$
(c.2)

where δc = (0 ⊗ 1c− 1′,1,0 ⊗ 1C− 1−c′), \({\pi }^{*}_{i,c}(\cdot ) \equiv {\pi }^{*}_{i,c}(\beta ,\tau _{\gamma },\gamma _{i})\) is given by Eq. 2.2, and \({\pi }^{*}_{i}(\cdot )=({\pi }^{*}_{i,1}(\cdot ),\ldots ,{\pi }^{*}_{i,c}(\cdot ), \ldots ,{\pi }^{*}_{i,C-1}(\cdot ))'.\)

Next, by using the non-singular transformation from γi to gi as in Eq. a.2 and the standardized form of the binomial variable \(v_{iu_{i}}\) as in Eq. a.3, this integration in Eq. c.2 may be approximated as

$$ \begin{array}{@{}rcl@{}} &&\frac{\partial {\pi_{i,c}}({\beta}|\cdot)}{\partial {\beta}}\\ &=&{\sum}^{V}_{v_{i1}=0}{\ldots} {\sum}^{V}_{v_{iu_{i}}=0}\ldots {\sum}^{V}_{v_{in_{i}}=0} \\&&\left[\left\{\tilde{\pi}^{*}_{i,c}({a^{*}}_{i1},\ldots, {a^{*}}_{iu_{i}},\ldots,{a^{*}}_{in_{i}}; v_{i1},\ldots,v_{iu_{i}},\ldots,v_{in_{i}})\right. \right. \\ &\times & \left. \left. (\delta_{c}-\tilde{\pi}^{*}_{i}({a^{*}}_{i1},\ldots, {a^{*}}_{iu_{i}},\ldots,{a^{*}}_{in_{i}}; v_{i1},\ldots,v_{iu_{i}},\ldots,v_{in_{i}}))\right\} \otimes x_{i}\right] \\ &\times & {\Pi}^{n_{i}}_{u_{i}=1}\left\{\left( \begin{array}{ccccc}V \\ v_{iu_{i}} \end{array}\right)(1/2)^{v_{iu_{i}}}(1/2)^{V-v_{iu_{i}}}\right\}. \end{array} $$
(c.3)

Computation of the Derivatives \(\frac {\partial {\pi _{i,c}}({\tau _{\gamma }}|\cdot )}{\partial {\tau _{\gamma }}}\) in Eq. 4.16:

The formula for the probability function πi,c(τγ|⋅) for c = 1,…,C − 1 is defined in Eq. 2.8, where τγ,1 = 1. Thus, the estimating equation given by Eq. 4.15 is written for τγ = (τγ,2,…,τγ,c,…,τγ,C− 1). Note that as τγ,1 = 1, the pattern for the derivatives \(\frac {\partial {\pi _{i,c}}({\tau _{\gamma }}|\cdot )}{\partial {\tau _{\gamma }}}\) for c = 2,…,C − 1, will be similar and they will be different than that of \(\frac {\partial {\pi _{i,1}}({\tau _{\gamma }}|\cdot )}{\partial {\tau _{\gamma }}}.\) Following Eqs. c.1c.3, these derivatives may be obtained as

$$ \begin{array}{@{}rcl@{}} &&{}\quad \frac{\partial {\pi_{i,1}}({\tau_{\gamma}}|\cdot)}{\partial {\tau_{\gamma}}} \\ &&{}=-{\int}^{\infty}_{-\infty}\left[ {\pi}^{**}_{i}(\beta,\tau_{\gamma},\gamma_{i}) \otimes \{{\pi}^{*}_{i,1}(\beta,\tau_{\gamma},\gamma_{i})a'_{i}\gamma_{i}\}\right]g^{*}_{N}(\gamma_{i}) {\Pi}^{n_{i}}_{u_{i}=1}d\tilde{\gamma}_{s_{iu_{i}}}, \end{array} $$
(c.4)

and

$$ \begin{array}{@{}rcl@{}} &&{}\quad\frac{\partial {\pi_{i,c}}({\tau_{\gamma}}|\cdot)}{\partial {\tau_{\gamma}}}: c=2,\ldots,C-1\\ &&{}={\int}^{\infty}_{-\infty}\left[\left\{{\pi}^{*}_{i,c}(\beta,\tau_{\gamma},\gamma_{i}) (\delta^{*}_{c}-{\pi}^{**}_{i}(\beta,\tau_{\gamma},\gamma_{i})) \right\} \otimes a'_{i}\gamma_{i}\right]g^{*}_{N}(\gamma_{i}) {\Pi}^{n_{i}}_{u_{i}=1}d\tilde{\gamma}_{s_{iu_{i}}} \end{array} $$
(c.5)

where \(\delta ^{*}_{c}=(0\otimes 1'_{c-2},1,0 \otimes 1'_{C-2-(c-1)})',\)\({\pi }^{*}_{i,c}(\cdot ) \equiv {\pi }^{*}_{i,c}(\beta ,\tau _{\gamma },\gamma _{i})\) is given by Eq. 2.2, and \({\pi }^{**}_{i}(\cdot )=({\pi }^{*}_{i,2}(\cdot ),\ldots ,{\pi }^{*}_{i,c}(\cdot ), \ldots ,{\pi }^{*}_{i,C-1}(\cdot ))'.\)

Next by similar algebras as in Eqs. c.2 and c.3, the above two integrals in Eqs. c.4 and c.5 may be computed as follows by using the Binomial approximation:

$$ \begin{array}{@{}rcl@{}} && \frac{\partial {\pi_{i,1}}({\tau_{\gamma}}|\cdot)}{\partial {\tau_{\gamma}}} \\ &\!\!\!\!\!\!\!\!=&\!\!\!\!\!\!\!{\sum}^{V}_{v_{i1}=0}{\ldots} {\sum}^{V}_{v_{iu_{i}}=0}\ldots {\sum}^{V}_{v_{in_{i}}=0} \left[{\pi}^{**}_{i}({a^{*}}_{i1},\ldots, {a^{*}}_{iu_{i}},\ldots,{a^{*}}_{in_{i}}; v_{i1},\ldots,v_{iu_{i}},\ldots,v_{in_{i}}) \right. \!\! \\ & \otimes & \left. \{{\pi}^{*}_{i,1}({a^{*}}_{i1},\ldots, {a^{*}}_{iu_{i}},\ldots,{a^{*}}_{in_{i}}; v_{i1},\ldots,v_{iu_{i}},\ldots,v_{in_{i}}){a^{*}}'_{i}g_{i}\}\right] \\ &\times & {\Pi}^{n_{i}}_{u_{i}=1}\left\{\left( \begin{array}{ccccc}V \\ v_{iu_{i}} \end{array}\right)(1/2)^{v_{iu_{i}}}(1/2)^{V-v_{iu_{i}}}\right\}, \end{array} $$
(c.6)

where \(g_{i}=(g_{i1},\ldots ,g_{iu_{i}},\ldots ,g_{in_{i}})'\) with \(g_{iu_{i}}(v_{iu_{i}})\) as given in Eq. a.3.

$$ \begin{array}{@{}rcl@{}} &&\frac{\partial {\pi_{i,c}}({\tau_{\gamma}}|\cdot)}{\partial {\tau_{\gamma}}}: c=2,\ldots,C-1 \end{array} $$
(c.7)
$$ \begin{array}{@{}rcl@{}} &=&{\sum}^{V}_{v_{i1}=0}{\ldots} {\sum}^{V}_{v_{iu_{i}}=0}\ldots {\sum}^{V}_{v_{in_{i}}=0}\\&& \left[\left\{\pi^{*}_{i,c}({a^{*}}_{i1},\ldots, {a^{*}}_{iu_{i}},\ldots,{a^{*}}_{in_{i}};v_{i1},\ldots,v_{iu_{i}},\ldots,v_{in_{i}})\right. \right.\\ &\times & \left. \left. (\delta^{*}_{c}-\pi^{**}_{i}({a^{*}}_{i1},\ldots, {a^{*}}_{iu_{i}},\ldots,{a^{*}}_{in_{i}}; v_{i1},\ldots,v_{iu_{i}},\ldots,v_{in_{i}}))\right\} \otimes {a^{*}}'_{i}g_{i}\right]\\ &\times & {\Pi}^{n_{i}}_{u_{i}=1}\left\{\left( \begin{array}{ccccc}V \\ v_{iu_{i}} \end{array}\right)(1/2)^{v_{iu_{i}}}(1/2)^{V-v_{iu_{i}}}\right\}. \end{array} $$
(c.8)

Computation of the Derivatives \(\frac {\partial \pi _{i,c}(\cdot )} {\partial \sigma ^{2}_{\gamma }}\) and \(\frac {\partial \lambda _{ij,cr}(\cdot )} {\partial \sigma ^{2}_{\gamma }}\) in Eq. 4.27:

Similar to the derivatives in Eq. c.1 with respect to β, we write the derivative of the same function \(\pi _{i,c}({\sigma ^{2}_{\gamma }}|\cdot )\) with respect to \(\sigma ^{2}_{\gamma },\) as

$$ \begin{array}{@{}rcl@{}} &&\frac{\partial {\pi_{i,c}}({\sigma^{2}_{\gamma}}|\cdot)}{\partial {\sigma^{2}_{\gamma}}} \\ &\!\!\!\!\!=&\!\!\!\!\!\left\{\!\!\!\begin{array}{ll} {\int}^{\infty}_{-\infty}\frac{\partial}{\partial \sigma^{2}_{\gamma}}[\exp(x^{\prime}_{i}\beta_{c} + \tau_{\gamma,c}a'_{i}\gamma_{i})/ [1 + {\sum}^{C-1}_{u=1}\exp(x^{\prime}_{i}\beta_{u} +\tau_{\gamma,u}a'_{i}\gamma_{i})]] \\ g^{*}_{N}(\gamma_{i}) {\Pi}^{n_{i}}_{u_{i}=1}d\tilde{\gamma}_{s_{iu_{i}}} & \text{for} c=1,\ldots,C - 1 \end{array} \right.\\ &\!\!\!\!\!=&\!\!\!\!\!{\int}^{\infty}_{-\infty}\frac{\partial }{\partial \sigma^{2}_{\gamma}}\left[\exp(x^{\prime}_{i}\beta_{c}+\sigma_{\gamma}\tau_{\gamma,c}a'_{i}{C}^{\frac{1}{2}}_{ii} {g}_{i})/ [1+{\sum}^{C-1}_{u=1}\exp(x^{\prime}_{i}\beta_{u} +\sigma_{\gamma}\tau_{\gamma,u}{a}'_{i}{C}^{\frac{1}{2}}_{ii}{g}_{i})]\right] \\ &\!\!\!\!\!\times &\!\!\!\!\! g^{*}_{N}(g_{i}) {\Pi}^{n_{i}}_{u_{i}=1}dg_{iu_{i}}, \end{array} $$
(c.9)

using \({\Phi }_{i}(\sigma ^{2}_{\gamma },\phi )=\sigma ^{2}_{\gamma } {C}_{ii}(\phi )\) from Eqs. a.1 into a.2.

Because σγ is unplugged in Eq. c.9, following the notation from Eq. a.6, the derivative and integration gives the formula as

$$ \begin{array}{@{}rcl@{}} &&\frac{\partial {\pi_{i,c}}({\sigma^{2}_{\gamma}}|\cdot)}{\partial {\sigma^{2}_{\gamma}}} \\ &=&\frac{1}{2\sigma_{\gamma}} {\sum}^{V}_{v_{i1}=0}\ldots {\sum}^{V}_{v_{iu_{i}}=0}\ldots {\sum}^{V}_{v_{in_{i}}=0}\{\tau_{\gamma,c}a'_{i}{C}^{\frac{1}{2}}_{ii} {g}_{i}\}\\&& \left[ \tilde{\pi}^{*}_{i,c}(\beta,\tau_{\gamma},\sigma_{\gamma},\phi; {a^{*}}_{i1},\ldots, {a^{*}}_{iu_{i}},\ldots,{a^{*}}_{in_{i}};v_{i1},\ldots,v_{iu_{i}},\ldots,v_{in_{i}}) \right. \\ &\!\!\!\times &\!\!\!\!\!\! \left. \left\{\tilde{\pi}^{*}_{(i,c)D}(\beta,\tau_{\gamma},\sigma_{\gamma},\phi; {a^{*}}_{i1},\ldots, {a^{*}}_{iu_{i}},\ldots,{a^{*}}_{in_{i}};v_{i1},\ldots,v_{iu_{i}}, \ldots,v_{in_{i}})\right\}^{-1}\right] \\ &\!\!\!\times & \!\!\!{\Pi}^{n_{i}}_{u_{i}=1}\left\{\left( \begin{array}{ccccc}V \\ v_{iu_{i}} \end{array}\right)(1/2)^{v_{iu_{i}}}(1/2)^{V-v_{iu_{i}}}\right\}, \end{array} $$
(c.10)

where gi in terms of vi is defined in Eq. a.3.

Next to compute \(\frac {\partial \lambda _{ij,cr}(\cdot )} {\partial \sigma ^{2}_{\gamma }}\) for c,r = 1,…,C − 1, we follow the formula for λij,cr(⋅) from (a.15), and compute the desired derivative as

$$ \begin{array}{@{}rcl@{}} &&\frac{\partial \lambda_{ij,cr}(\sigma^{2}_{\gamma}|\cdot)}{\partial \sigma^{2}_{\gamma}} \\ &=&{\int}^{\infty}_{-\infty}\frac{\partial }{\partial \sigma^{2}_{\gamma}} \left[\frac{\tilde{\pi}^{*}_{(i,c)N}(\cdot)}{\tilde{\pi}^{*}_{(i,c)D}(\cdot )}\frac{\tilde{\pi}^{*}_{(j,r)N}(\cdot )} {\tilde{\pi}^{*}_{(j,r)D}(\cdot)}\right] {\Pi}^{n_{i}+n^{*}_{j}}_{k=1}g^{*}_{N}(g^{\dagger}_{ij,k}) dg^{\dagger}_{ij,k}. \end{array} $$
(c.11)

Further because \(\tilde {\Phi }_{ij}(\sigma ^{2}_{\gamma },\phi )=\sigma ^{2}_{\gamma } {{C}}^{\dag }_{ij}(\phi )\) as in Eq. a.9, we can unplug \(\sigma ^{2}_{\gamma }\) and re-express (a.14) as

$$ \gamma_{i}=\sigma_{\gamma} \tilde{A}(\phi){g}^{\dagger}_{ij}, \text{and} \gamma_{j}=\sigma_{\gamma} {\tilde{B}}^{*}(\phi){g}^{\dagger}_{ij}. $$
(c.12)

It then follows that

$$ \begin{array}{@{}rcl@{}} &&\frac{\partial \lambda_{ij,cr}(\sigma^{2}_{\gamma}|\cdot)}{\partial \sigma^{2}_{\gamma}} \\ &=&{\int}^{\infty}_{-\infty}\frac{1}{2\sigma_{\gamma}}\left[\{\tau_{\gamma,c}a'_{i}\tilde{A}(\phi) g^{\dagger}_{ij}\}[\{\tilde{\pi}^{*}_{i,c}(\cdot)(\tilde{\pi}^{*}_{(i,c)D}(\cdot))^{-1}\} \tilde{\pi}^{*}_{j,r}(\cdot)] \right. \\ &+&\left. \{\tau_{\gamma,r}a'_{j}{\tilde{B}}^{*}(\phi) g^{\dagger}_{ij}\}[\{\tilde{\pi}^{*}_{j,r}(\cdot)(\tilde{\pi}^{*}_{(j,r)D}(\cdot))^{-1}\} \tilde{\pi}^{*}_{i,c}(\cdot)]\right]{\Pi}^{n_{i}+n^{*}_{j}}_{k=1}g^{*}_{N}(g^{\dagger}_{ij,k}) dg^{\dagger}_{ij,k} \\ &=&\frac{1}{2\sigma_{\gamma}} {\sum}^{V}_{v_{ij,1}=0}\ldots {\sum}^{V}_{v_{ij,k}=0}\ldots {\sum}^{V}_{v_{ij,n_{i}+n^{*}_{j}}=0}\\&&\left[\{\tau_{\gamma,c}a'_{i}\tilde{A}(\phi) g^{\dagger}_{ij}\}[\{\tilde{\pi}^{*}_{i,c}(\cdot)(\tilde{\pi}^{*}_{(i,c)D}(\cdot))^{-1}\} \tilde{\pi}^{*}_{j,r}(\cdot)] \right. \\ &+&\left. \{\tau_{\gamma,r}a'_{j}{\tilde{B}}^{*}(\phi) g^{\dagger}_{ij}\}[\{\tilde{\pi}^{*}_{j,r}(\cdot)(\tilde{\pi}^{*}_{(j,r)D}(\cdot))^{-1}\} \tilde{\pi}^{*}_{i,c}(\cdot)]\right] \\ &\times & {\Pi}^{n_{i}+n^{*}_{j}}_{k=1}\left\{\left( \begin{array}{ccccc}V \\ v_{ij,k} \end{array}\right)(1/2)^{v_{ij,k}}(1/2)^{V-v_{ij,k}}\right\}, \end{array} $$
(c.13)

where \(g^{\dagger }_{ij,k}\) in terms of vij,k is defined in Eq. a.16.

Computation of the Derivative \(\frac {\partial \lambda _{i(i-1),cr}(\cdot )} {\partial \phi }\) in Eq. 4.36:

Notice that the structure of the scale matrices \(\tilde {A}(\phi )\) and \({\tilde {B}}^{*}(\phi )\) as a function of ϕ is known from the relationships (a.9), and (a.11)–(a.13). More specifically, one writes from Eqs. a.9 and a.11 that

$$ \begin{array}{@{}rcl@{}} &&{C^{\dagger}}^{\frac{1}{2}}_{ij}(\phi) = \left( \begin{array}{c} \tilde{A}(\phi) \\ \tilde{B}(\phi) \end{array} \right) = \left( \begin{array}{c} \tilde{A}_{1}(\phi):n^{*}_{i} \times (n_{i}+n^{*}_{j}) \\ \tilde{A}_{2}(\phi):n_{ij} \times (n_{i}+n^{*}_{j}) \\ \tilde{B}(\phi):n^{*}_{j} \times (n_{i}+n^{*}_{j}) \end{array} \right), \end{array} $$
(c.14)

and from Eq. a.13

$$ \begin{array}{@{}rcl@{}} && \left( \begin{array}{c} \tilde{B}(\phi) \\ \tilde{A}_{2}(\phi) \end{array} \right) ={\tilde{B}}^{*}(\phi). \end{array} $$
(c.15)

Hence by similar calculations as in Eq. c.13, we compute the desired derive as

$$ \begin{array}{@{}rcl@{}} &&\frac{\partial \lambda_{i(i-1),cr}(\phi|\cdot)} {\partial \phi} \\ &=&\sigma_{\gamma} {\sum}^{V}_{v_{i(i-1),1}=0}\ldots {\sum}^{V}_{v_{i(i-1),k}=0}\ldots {\sum}^{V}_{v_{i(i-1),n_{i}+n^{*}_{i-1}}=0}\\&&\left[\{\tau_{\gamma,c}a'_{i}\frac{\partial \tilde{A}(\phi)}{\partial \phi} g^{\dagger}_{i(i-1)}\}[\{\tilde{\pi}^{*}_{i,c}(\cdot)(\tilde{\pi}^{*}_{(i,c)D}(\cdot))^{-1}\} \tilde{\pi}^{*}_{i-1,r}(\cdot)] \right. \\ &+&\left. \{\tau_{\gamma,r}a'_{i-1}\frac{\partial {\tilde{B}}^{*}(\phi)}{\partial \phi} g^{\dagger}_{i(i-1)}\}[\{\tilde{\pi}^{*}_{i-1,r}(\cdot)(\tilde{\pi}^{*}_{(i-1,r)D}(\cdot))^{-1}\} \tilde{\pi}^{*}_{i,c}(\cdot)]\right] \\ &\times & {\Pi}^{n_{i}+n^{*}_{i-1}}_{k=1}\left\{\left( \begin{array}{ccccc}V \\ v_{i(i-1),k} \end{array}\right)(1/2)^{v_{i(i-1),k}}(1/2)^{V-v_{i(i-1),k}}\right\}. \end{array} $$
(c.16)

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Sutradhar, B.C., Rao, R.P. Asymptotic Inferences in a Multinomial Logit Mixed Model for Spatial Categorical Data. Sankhya A (2022). https://doi.org/10.1007/s13171-022-00282-7

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Keywords

  • Categorical/multinomial responses in a spatial setup
  • Moving correlations
  • Multinomial mixed logits
  • Normality and consistency of the estimators
  • Spatial correlations
  • Spatial statistics

AMS (2000) subject classification

  • Primary 62F10
  • 62F12