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Minimum phi-divergence estimators for multinomial logistic regression with complex sample design

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Abstract

This article develops the theoretical framework needed to study the multinomial regression model for complex sample design with pseudo-minimum phi-divergence estimators. The numerical example and the simulation study propose new estimators for the parameter of the logistic regression with overdispersed multinomial distributions for the response variables, the pseudo-minimum Cressie–Read divergence estimators, as well as new estimators for the intra-cluster correlation coefficient. The simulation study shows that the Binder’s method for the intra-cluster correlation coefficient exhibits an excellent performance when the pseudo-minimum Cressie–Read divergence estimator, with \(\lambda =\frac{2}{3}\), is plugged.

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Acknowledgements

We would like to thank the referees for their helpful comments and suggestions. Their comments have improved the paper. This research is partially supported by Grants MTM2012-33740, MTM2015-67057-P and ECO2015-66593-P from Ministerio de Economia y Competitividad (Spain).

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Authors and Affiliations

  1. Department of Statistics and Operations Research, Complutense University of Madrid, Madrid, Spain

    Elena Castilla, Nirian Martín & Leandro Pardo

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  1. Elena Castilla
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  2. Nirian Martín
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  3. Leandro Pardo
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Correspondence to Leandro Pardo.

A Proof of results

A Proof of results

1.1 A.1 Proof of Theorem 1

Proof

The pseudo-minimum phi-divergence estimator of \(\varvec{\beta }\), \( \widehat{\varvec{\beta }}_{\phi ,P}\), is obtained by solving the system of equations \(\frac{\partial }{\partial \varvec{\beta }}d_{\phi }\left( \widehat{\varvec{p}},\varvec{\pi }\left( \varvec{\beta }\right) \right) =\varvec{0}_{dk}\), and then, it is also obtained from \( \varvec{u}_{\phi }\left( \varvec{\beta }\right) =\varvec{0}_{dk}\), where

$$\begin{aligned} \varvec{u}_{\phi }\left( \varvec{\beta }\right) =-\frac{\tau }{ \phi ^{\prime \prime }(1)}\frac{\partial }{\partial \varvec{\beta }}d_{\phi }\left( \widehat{\varvec{p}},\varvec{\pi }\left( \varvec{\beta } \right) \right) =\sum \limits _{h=1}^{H}\sum \limits _{i=1}^{n_{h}}\varvec{u} _{\phi ,hi}\left( \varvec{\beta }\right) , \end{aligned}$$

with

$$\begin{aligned} \varvec{u}_{\phi ,hi}\left( \varvec{\beta }\right)&=-\frac{ w_{hi}m_{hi}}{\phi ^{\prime \prime }(1)}\frac{\partial }{\partial \varvec{ \beta }}d_{\phi }\left( \tfrac{\widehat{\varvec{y}}_{hi}}{m_{hi}}, \varvec{\pi }_{hi}(\varvec{\beta })\right) =\frac{w_{hi}m_{hi}}{ \phi ^{\prime \prime }(1)}\sum \limits _{s=1}^{d+1}\frac{\partial \pi _{his}( \varvec{\beta })}{\partial \varvec{\beta }}f_{\phi ,his}(\tfrac{\widehat{ y}_{his}}{m_{hi}},\varvec{\beta }) \\&=\frac{w_{hi}m_{hi}}{\phi ^{\prime \prime }(1)}\frac{\partial \varvec{\pi } _{hi}^{T}(\varvec{\beta })}{\partial \varvec{\beta }}\varvec{f} _{\phi ,hi}(\tfrac{\widehat{\varvec{y}}_{hi}}{m_{hi}},\varvec{\beta }), \end{aligned}$$

and

$$\begin{aligned} \varvec{f}_{\phi ,hi}\left( \tfrac{\widehat{\varvec{y}}_{hi}}{m_{hi}}, \varvec{\beta }\right) =\left( f_{\phi ,hi1}\left( \tfrac{\widehat{y}_{hi1}}{m_{hi}}, \varvec{\beta }\right) ,\ldots ,f_{\phi ,hi,d+1}\left( \tfrac{\widehat{y}_{hi,d+1}}{m_{hi}}, \varvec{\beta }\right) \right) ^{T}. \end{aligned}$$

Since

$$\begin{aligned} \frac{\partial \varvec{\pi }_{hi}^{T}(\varvec{\beta })}{\partial \varvec{\beta }}=\left( \varvec{I}_{d\times d},\varvec{0} _{d\times 1}\right) \varvec{\Delta }(\varvec{\pi }_{hi}\left( \varvec{\beta }\right) )\otimes \varvec{x}_{hi}, \end{aligned}$$

the expression of \(\varvec{u}_{\phi ,hi}\left( \varvec{\beta }\right) \) is rewritten as (16)–(17). \(\square \)

1.2 A.2 Proof of Theorem 2

Proof

From Theorem 1 and by following the same steps of the linearization method of Binder (1983),

$$\begin{aligned} \mathbf {G}\left( \varvec{\beta }\right) =\lim _{n\rightarrow \infty } \varvec{V}\left[ \tfrac{1}{\sqrt{n}}\varvec{U}_{\phi }\left( \varvec{ \beta }\right) \right] \quad \text {and}\quad \mathbf {H}\left( \varvec{\beta } \right) =-\lim _{n\rightarrow \infty }\frac{1}{n}\frac{\partial \varvec{U} _{\phi }^{T}\left( \varvec{\beta }\right) }{\partial \varvec{\beta }}, \end{aligned}$$

where \(\varvec{U}_{\phi }\left( \varvec{\beta }\right) \) is the random variable generator of \(\varvec{u}_{\phi }\left( \varvec{\beta }\right) \) is given by (15). Taking into account that \(f_{\phi ,his}(\pi _{his}(\varvec{\beta }),\varvec{\beta })=0\) and \(f_{\phi ,his}^{\prime }(\pi _{his}(\varvec{\beta }),\varvec{\beta })=\frac{1}{\pi _{his}( \varvec{\beta })}\phi ^{\prime \prime }\left( 1\right) \), a first Taylor expansion of \(f_{\phi ,his}(\tfrac{\widehat{Y}_{his}}{m_{hi}},\varvec{ \beta })\) given in (18) is

$$\begin{aligned} f_{\phi ,his}\left( \tfrac{\widehat{Y}_{his}}{m_{hi}},\varvec{\beta }\right)&=f_{\phi ,his}\left( \pi _{his}\left( \varvec{\beta }\right) ,\varvec{\beta }\right) +f_{\phi ,his}^{\prime }\left( \pi _{his}\left( \varvec{\beta }\right) ,\varvec{\beta }\right) \left( \tfrac{ \widehat{Y}_{his}}{m_{hi}}-\pi _{his}\left( \varvec{\beta }\right) \right) \nonumber \\&\quad +o\left( \tfrac{\widehat{ Y}_{his}}{m_{hi}}-\pi _{his}\left( \varvec{\beta }\right) \right) =\frac{\phi ^{\prime \prime }\left( 1\right) }{\pi _{his}\left( \varvec{\beta }\right) } \left( \tfrac{Y_{his}}{m_{hi}}-\pi _{his}\left( \varvec{\beta }\right) \right) \nonumber \\&\quad +o\left( \tfrac{\widehat{Y} _{his}}{m_{hi}}-\pi _{his}\left( \varvec{\beta }\right) \right) , \end{aligned}$$
(28)

i.e.,

$$\begin{aligned} \varvec{f}_{\phi ,hi}\left( \tfrac{\widehat{\varvec{Y}}_{hi}}{m_{hi}}, \varvec{\beta }\right) =\phi ^{\prime \prime }\left( 1\right) \mathrm {diag}^{-1}\left( \varvec{\pi }_{hi}\left( \varvec{\beta }\right) \right) \left( \tfrac{\widehat{\varvec{Y}} _{hi}}{m_{hi}}-\varvec{\pi }_{hi}\left( \varvec{\beta }\right) \right) +o\left( \tfrac{ \widehat{\varvec{Y}}_{hi}}{m_{hi}}-\varvec{\pi }_{hi}\left( \varvec{ \beta }\right) \right) , \end{aligned}$$

and hence,

$$\begin{aligned} \frac{1}{\sqrt{n}}\varvec{U}_{\phi }\left( \varvec{\beta }\right)= & {} \frac{1}{\sqrt{n}}\sum \limits _{h=1}^{H}\sum \limits _{i=1}^{n_{h}}w_{hi}m_{hi} \frac{\partial \varvec{\pi }_{hi}^{T}\left( \varvec{\beta }\right) }{\partial \varvec{\beta }}\mathrm {diag}^{-1}\left( \varvec{\pi }_{hi}\left( \varvec{\beta }\right) \right) \left( \tfrac{\widehat{\varvec{Y}}_{hi}}{m_{hi}}-\varvec{\pi }_{hi}\left( \varvec{\beta }\right) \right) \\&\quad +\sum \limits _{h=1}^{H}\sqrt{\eta _{h}^{*}}o\left( \frac{1}{\sqrt{n_{h}}}\left( \sum _{i=1}^{n_{h}}\widehat{\varvec{Y}} _{hi}-\sum _{i=1}^{n_{h}}m_{hi}\varvec{\pi }_{hi}\left( \varvec{\beta } \right) \right) \right) . \end{aligned}$$

From the Central Limit Theorem

$$\begin{aligned} \frac{1}{\sqrt{n_{h}}}\left( \sum _{i=1}^{n_{h}}\widehat{\varvec{Y}} _{hi}-\sum _{i=1}^{n_{h}}m_{hi}\varvec{\pi }_{hi}\left( \varvec{\beta } \right) \right) \underset{n_{h}\rightarrow \infty }{\overset{\mathcal {L}}{ \longrightarrow }}\mathcal {N}\left( \varvec{0}_{d+1},\lim _{n_{h}\rightarrow \infty }\tfrac{1}{n_{h}}\sum _{i=1}^{n_{h}}\varvec{V}\left[ \widehat{ \varvec{Y}}_{hi}\right] \right) , \end{aligned}$$

then

$$\begin{aligned} o\left( \frac{1}{\sqrt{n_{h}}}\left( \sum _{i=1}^{n_{h}}\widehat{ \varvec{Y}}_{hi}-\sum _{i=1}^{n_{h}}m_{hi}\varvec{\pi }_{hi}( \varvec{\beta })\right) \right) =o\left( o_{p}(\varvec{1} _{d+1})\right) =o_{p}(\varvec{1}_{d+1}), \end{aligned}$$

and thus,

$$\begin{aligned} \frac{1}{\sqrt{n}}\varvec{U}_{\phi }\left( \varvec{\beta }\right) = \frac{1}{\sqrt{n}}\sum \limits _{h=1}^{H}\sum \limits _{i=1}^{n_{h}}w_{hi}\frac{ \partial \log \varvec{\pi }_{hi}^{T}(\varvec{\beta })}{\partial \varvec{\beta }}(\widehat{\varvec{y}}_{hi}-m_{hi}\varvec{\pi } _{hi}(\varvec{\beta }))+o_{p}(\varvec{1}_{dk}). \end{aligned}$$

Since

$$\begin{aligned} \frac{\partial \log \varvec{\pi }_{hi}^{T}(\varvec{\beta })}{\partial \varvec{\beta }}\varvec{\pi }_{hi}(\varvec{\beta })&=\frac{ \partial \varvec{\pi }_{hi}^{T}(\varvec{\beta })}{\partial \varvec{ \beta }}\mathrm {diag}^{-1}(\varvec{\pi }_{hi}(\varvec{\beta })) \varvec{\pi }_{hi}(\varvec{\beta }) \\&=\frac{\partial \varvec{\pi }_{hi}^{T}(\varvec{\beta })}{\partial \varvec{\beta }}\varvec{1}_{d+1}=\frac{\partial \left( \varvec{\pi } _{hi}^{T}(\varvec{\beta })\varvec{1}_{d+1}\right) }{\partial \varvec{\beta }}=\varvec{0}_{dk},\\ \frac{\partial \log \varvec{\pi }_{hi}^{T}(\varvec{\beta })}{\partial \varvec{\beta }}\widehat{\varvec{y}}_{hi}&=\frac{\partial \varvec{\pi }_{hi}^{T}(\varvec{\beta })}{\partial \varvec{\beta }} \mathrm {diag}^{-1}(\varvec{\pi }_{hi}(\varvec{\beta }))\widehat{ \varvec{Y}}_{hi} \\&=\left( \left( \varvec{I}_{d\times d},\varvec{0}_{d\times 1}\right) \varvec{\Delta }(\varvec{\pi }_{hi}\left( \varvec{\beta } \right) )\otimes \varvec{x}_{hi}\right) \mathrm {diag}^{-1}(\varvec{\pi }_{hi}(\varvec{\beta }))\widehat{\varvec{Y}}_{hi} \\&=\left( \varvec{I}_{d\times d},\varvec{0}_{d\times 1}\right) \varvec{\Delta }(\varvec{\pi }_{hi}\left( \varvec{\beta }\right) ) \mathrm {diag}^{-1}(\varvec{\pi }_{hi}(\varvec{\beta }))\widehat{ \varvec{Y}}_{hi}\otimes \varvec{x}_{hi} \\&=\left( \varvec{I}_{d\times d},\varvec{0}_{d\times 1}\right) \left( \widehat{\varvec{Y}}_{hi}-\varvec{\pi }_{hi}\left( \varvec{\beta } \right) \varvec{\pi }_{hi}\left( \varvec{\beta }\right) ^{T}\mathrm { diag}^{-1}\left( \pi _{hi}\left( \varvec{\beta }\right) \right) \widehat{ \varvec{Y}}_{hi}\right) \otimes \varvec{x}_{hi} \\&=\left( \varvec{I}_{d\times d},\varvec{0}_{d\times 1}\right) \left( \widehat{\varvec{Y}}_{hi}-m_{hi}\varvec{\pi }_{hi}\left( \varvec{ \beta }\right) \right) \otimes \varvec{x}_{hi} \\&=\left( \widehat{\varvec{y}}_{hi}^{*}-m_{hi}\varvec{\pi } _{hi}^{*}\left( \varvec{\beta }\right) \right) \otimes \varvec{x} _{hi}, \end{aligned}$$

it follows that

$$\begin{aligned} \frac{1}{\sqrt{n}}\varvec{U}_{\phi }\left( \varvec{\beta }\right) = \frac{1}{\sqrt{n}}\sum \limits _{h=1}^{H}\sum \limits _{i=1}^{n_{h}}w_{hi}\left( \widehat{\varvec{y}}_{hi}^{*}-m_{hi}\varvec{\pi }_{hi}^{*}( \varvec{\beta })\right) \otimes \varvec{x}_{hi}+o_{p}(\varvec{1} _{dk}), \end{aligned}$$
(29)

Then, \(\mathbf {H}\left( \varvec{\beta }_{0}\right) \) is the limit of

$$\begin{aligned} -\frac{1}{n}\frac{\partial }{\partial \varvec{\beta }}\varvec{U}_{\phi }^{T}\left( \varvec{\beta }\right)&=\frac{1}{n}\sum \limits _{h=1}^{H} \sum \limits _{i=1}^{n_{h}}w_{hi}m_{hi}\frac{\partial }{\partial \varvec{ \beta }}\varvec{\pi }_{hi}^{*}(\varvec{\beta })\otimes \varvec{x} _{hi}+o_{p}(\varvec{1}_{dk\times dk}) \\&=\frac{1}{n}\sum \limits _{h=1}^{H}\sum \limits _{i=1}^{n_{h}}w_{hi}m_{hi} \varvec{\Delta }\left( \varvec{\pi }_{hi}^{*}\left( \varvec{\beta } \right) \right) \otimes \varvec{x}_{hi}+o_{p}(\varvec{1}_{dk\times dk}), \end{aligned}$$

as n increases, and hence, \(\mathbf {H}\left( \varvec{\beta }\right) =\lim _{n\rightarrow \infty }\mathbf {H}_{n}\left( \varvec{\beta }\right) \). On the other hand, from (29) it follows that

$$\begin{aligned} \frac{1}{\sqrt{n}}\varvec{U}_{\phi }\left( \varvec{\beta }\right) = \frac{1}{\sqrt{n}}\varvec{U}\left( \varvec{\beta }\right) +o_{p}( \varvec{1}_{dk}), \end{aligned}$$

and this justifies that \(\mathbf {G}\left( \varvec{\beta }\right) =\lim _{n\rightarrow \infty }\mathbf {G}_{n}\left( \varvec{\beta }\right) \). \(\square \)

1.3 A.3 Proof of Theorem 3

Proof

If \(\varvec{V}[\widehat{\varvec{Y}}_{hi}]=\nu _{m_{h}}m_{h} \varvec{\Delta }(\varvec{\pi }_{hi}\left( \varvec{\beta } _{0}\right) )\), then from the expression of \(\mathbf {G}_{n_{h}}^{(h)}\left( \varvec{\beta }_{0}\right) \) given in Theorem 2,

$$\begin{aligned} \mathbf {G}_{n_{h}}^{(h)}\left( \varvec{\beta }_{0}\right)&=\frac{1}{ n_{h}}\sum \limits _{i=1}^{n_{h}}w_{h}^{2}\varvec{V}\left[ \widehat{ \varvec{Y}}_{hi}^{*}\right] \otimes \varvec{x}_{hi}\varvec{x} _{hi}^{T}=\nu _{m_{h}}w_{h}\frac{1}{n_{h}}\sum \limits _{i=1}^{n_{h}}w_{h}m_{h}\varvec{\Delta }\left( \varvec{\pi } _{hi}^{*}( \varvec{\beta }_{0}) \right) \otimes \varvec{x}_{hi} \varvec{x}_{hi}^{T} \\&=\nu _{m_{h}}w_{h}\mathbf {H}_{n_{h}}^{(h)}\left( \varvec{\beta } _{0}\right) . \end{aligned}$$

Hence, from

$$\begin{aligned} \mathrm {trace}\left( \mathbf {H}_{n_{h}}^{(h)}\left( \varvec{\beta } _{0}\right) ^{-1}\mathbf {G}_{n_{h}}^{(h)}\left( \varvec{\beta } _{0}\right) \right) =\nu _{m_{h}}w_{h}dk, \end{aligned}$$

and consistency of \(\mathbf {H}_{n_{h}}^{(h)}(\widehat{\varvec{\beta }} _{\phi ,P})\) and \(\widehat{\mathbf {G}}_{n_{h}}^{(h)}(\widehat{\varvec{ \beta }}_{\phi ,P})\),

$$\begin{aligned} \widehat{\nu }_{m_{h}}(\widehat{\varvec{\beta }}_{\phi ,P})=\frac{1}{dk} \mathrm {trace}\left( \frac{1}{w_{h}}\mathbf {H}_{n_{h}}^{(h)}(\widehat{ \varvec{\beta }}_{\phi ,P})^{-1}\widehat{\mathbf {G}}_{n_{h}}^{(h)}( \widehat{\varvec{\beta }}_{\phi ,P})\right) , \end{aligned}$$

is proved with

$$\begin{aligned} \frac{1}{w_{h}}\mathbf {H}_{n_{h}}^{(h)}\left( \widehat{\varvec{\beta }}_{\phi ,P}\right) ^{-1}\widehat{\mathbf {G}}_{n_{h}}^{(h)}\left( \widehat{\varvec{\beta }} _{\phi ,P}\right)&=\left( \sum \limits _{i=1}^{n_{h}}m_{h}\varvec{\Delta }\left( \varvec{\pi }_{hi}^{*}\left( \widehat{\varvec{\beta }}_{\phi ,P}\right) \right) \otimes \varvec{x}_{hi}\varvec{x}_{hi}^{T}\right) ^{-1} \\&\quad \times \sum \limits _{i=1}^{n_{h}}\left( \varvec{v}_{hi}\left( \widehat{ \varvec{\beta }}_{\phi ,P}\right) -\varvec{\bar{v}}_{h}\left( \widehat{\varvec{ \beta }}_{\phi ,P}\right) \right) \left( \varvec{v}_{hi}\left( \widehat{\varvec{ \beta }}_{\phi ,P}\right) -\varvec{\bar{v}}_{h}\left( \widehat{\varvec{\beta }} _{\phi ,P}\right) \right) ^{T}, \\ \varvec{v}_{hi}\left( \widehat{\varvec{\beta }}_{\phi ,P}\right)&=\frac{1}{w_{h}} \varvec{u}_{hi}\left( \varvec{\beta }\right) , \end{aligned}$$

which is equivalent to (25). \(\square \)

1.4 A.4 Proof of Theorem 4

Proof

The mean vector and variance-covariance matrix of

$$\begin{aligned} \varvec{Z}_{hi}^{*}\left( \varvec{\beta }_{0}\right) =\sqrt{m_{h}}\varvec{ \Delta }^{-\frac{1}{2}}\left( \varvec{\pi }_{hi}^{*}\left( \varvec{\beta } _{0}\right) \right) \left( \tfrac{\widehat{\varvec{Y}}_{hi}^{*}}{m_{h}}- \varvec{\pi }_{hi}^{*}\left( \varvec{\beta }_{0}\right) \right) , \end{aligned}$$

are, respectively,

$$\begin{aligned} \varvec{E}[\varvec{Z}_{hi}^{*}(\varvec{\beta }_{0})]&= \varvec{0}_{d}, \\ \varvec{V}[\varvec{Z}_{hi}^{*}(\varvec{\beta }_{0})]&=\nu _{m_{h}}\varvec{I}_{d}, \end{aligned}$$

for \(h=1,\ldots ,H\). An unbiased estimator of \(\varvec{V}[\varvec{Z} _{hi}^{*}(\varvec{\beta }_{0})]\) is

$$\begin{aligned} \widehat{\varvec{V}}[\varvec{Z}_{hi}^{*}(\varvec{\beta }_{0})]= \frac{1}{n_{h}}\sum \limits _{i=1}^{n_{h}}\varvec{Z}_{hi}^{*}( \varvec{\beta }_{0})\varvec{Z}_{hi}^{*T}(\varvec{\beta }_{0}), \end{aligned}$$

from which is derived

$$\begin{aligned} E\left[ \mathrm {trace}\widehat{\varvec{V}}[\varvec{Z}_{hi}^{*}( \varvec{\beta }_{0})]\right]&=\mathrm {trace}\varvec{V}[\varvec{Z }_{hi}^{*}(\varvec{\beta }_{0})], \\ E\left[ \frac{1}{n_{h}}\sum \limits _{i=1}^{n_{h}}\mathrm {trace}\left( \varvec{Z}_{hi}^{*}(\varvec{\beta }_{0})\varvec{Z}_{hi}^{*T}(\varvec{\beta }_{0})\right) \right]&=\mathrm {trace}\left( \nu _{m_{h}} \varvec{I}_{d}\right) , \\ E\left[ \frac{1}{n_{h}}\sum \limits _{i=1}^{n_{h}}\varvec{Z}_{hi}^{*T}( \varvec{\beta }_{0})\varvec{Z}_{hi}^{*}(\varvec{\beta }_{0}) \right]&=\nu _{m_{h}}d, \\ E\left[ \frac{1}{n_{h}d}\sum \limits _{i=1}^{n_{h}}\varvec{Z}_{hi}^{*T}(\varvec{\beta }_{0})\varvec{Z}_{hi}^{*}(\varvec{\beta }_{0}) \right]&=\nu _{m_{h}}. \end{aligned}$$

This expression suggests using

$$\begin{aligned} \widetilde{\nu }_{m_{h}}\left( \widehat{\varvec{\beta }}_{\phi ,P}\right)&=\frac{1}{ n_{h}d}\sum \limits _{i=1}^{n_{h}}\widehat{\varvec{z}}_{hi,\phi ,P}^{*T}\left( \widehat{\varvec{\beta }}_{\phi ,P}\right) \widehat{\varvec{z}}_{hi,\phi ,P}^{*}\left( \widehat{\varvec{\beta }}_{\phi ,P}\right) \\&=\frac{1}{n_{h}d}m_{h}\left( \tfrac{\widehat{\varvec{y}}_{hi}^{*}}{ m_{h}}-\varvec{\pi }_{hi}^{*}\left( \widehat{\varvec{\beta }}_{\phi ,P}\right) \right) ^{T}\varvec{\Delta }^{-1}\left( \varvec{\pi }_{hi}^{*}\left( \widehat{\varvec{\beta }}_{\phi ,P}\right) \right) \left( \tfrac{\widehat{\varvec{y} }_{hi}^{*}}{m_{h}}-\varvec{\pi }_{hi}^{*}\left( \widehat{\varvec{ \beta }}_{\phi ,P}\right) \right) \\&=\frac{1}{n_{h}d}m_{h}\left( \tfrac{\widehat{\varvec{y}}_{hi}}{m_{h}}- \varvec{\pi }_{hi}\left( \widehat{\varvec{\beta }}_{\phi ,P}\right) \right) ^{T} \varvec{\Delta }^{-}\left( \varvec{\pi }_{hi}\left( \widehat{\varvec{\beta }} _{\phi ,P}\right) \right) \left( \tfrac{\widehat{\varvec{y}}_{hi}^{*}}{m_{h}}- \varvec{\pi }_{hi}^{*}\left( \widehat{\varvec{\beta }}_{\phi ,P}\right) \right) ,\\ \widehat{\varvec{z}}_{hi,\phi ,P}^{*}&=\sqrt{m_{h}}\varvec{\Delta }^{-\frac{1}{2}}\left( \varvec{\pi }_{hi}^{*}\left( \widehat{\varvec{\beta }} _{\phi ,P}\right) \right) \left( \tfrac{\widehat{\varvec{y}}_{hi}^{*}}{m_{h}}- \varvec{\pi }_{hi}^{*}\left( \widehat{\varvec{\beta }}_{\phi ,P}\right) \right) . \end{aligned}$$

Finally, since \(\varvec{\Delta }^{-}(\varvec{\pi }_{hi}(\widehat{ \varvec{\beta }}_{\phi ,P}))=\mathrm {diag}^{-1}(\varvec{\pi }_{hi}( \widehat{\varvec{\beta }}_{\phi ,P}))\) is a possible expression for the generalized inverse, the desired result for \(\widetilde{\nu }_{m_{h}}( \widehat{\varvec{\beta }}_{\phi ,P})\) is obtained. \(\square \)

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Castilla, E., Martín, N. & Pardo, L. Minimum phi-divergence estimators for multinomial logistic regression with complex sample design. AStA Adv Stat Anal 102, 381–411 (2018). https://doi.org/10.1007/s10182-017-0311-6

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