Optimal shrinkage estimations in partially linear single-index models for binary longitudinal data

Abstract

This paper focuses on the optimal estimation strategies of partially linear single-index models (PLSIM) for binary longitudinal data. Fitting model between the response and covariates may cause complexity and the linear terms may not be adequate to represent the relationship. In this situation, the PLSIM containing both linear and nonlinear terms is preferable. The objective of this paper is to develop optimal estimation strategies such as, pretest and shrinkage methods, for the analysis of binary longitudinal data under the PLSIM where some regression parameters are subject to restrictions. We estimate the nonparametric component using kernel estimating equations, and then use profile estimating equations to estimate the unrestricted and restricted estimators. To apply the pretest and shrinkage methods, we fit two models: one includes all covariates and the other restricts the regression parameters based on the auxiliary information. The unrestricted and restricted estimators are then combined optimally to get the pretest and shrinkage estimators. We also derive the asymptotic properties of the estimators in terms of biases and risks. Monte Carlo simulations are also conducted to examine the relative performance of the proposed estimators to the unrestricted estimator. An empirical application is also be used to illustrate the usefulness of our methodology.

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Acknowledgements

The authors gratefully acknowledge an associate editor and two anonymous referees, whose comments greatly improved the form and content of this paper.The research of Dr. Shakhawat Hossain was supported by the Natural Sciences and Engineering Research Council of Canada.

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Appendix

Appendix

Proof of Theorem 4.2

Let \(\hat{\varvec{\xi }}=(\hat{\varvec{\xi }}_1^{\top }, \hat{\varvec{\xi }}_2^{\top })^{\top }\) be the solution of \(\sum _{i=1}^n S_i(\varvec{\xi }) = 0\). For parameter \(\varvec{\xi }_2\), we can show under \(H_0\) that \(\sqrt{n}(\hat{\varvec{\xi }}_2-\varvec{0})\) is multivariate normal with mean \(\varvec{0}\) and variance \({\varvec{{\mathcal {I}}}}_{\xi _{22}}\) as \(n\rightarrow \infty \). Hence \(n\hat{\varvec{\xi }}_2^{\top } {\varvec{{\mathcal {I}}}}_{\xi _{22}}^{-1} \hat{\varvec{\xi }}_2\) converges to \(\chi _{p_2}^2\) distribution.

Using the idea of the proof of Result 1 of Lin and Carroll (2001), it can be shown by Taylor expansion that \((\hat{\varvec{\xi }}-\varvec{\xi }) =n^{-1}\varvec{\varOmega }_{\varvec{\xi }} \varvec{S}(\varvec{\xi }_1, \varvec{\xi }_2) +o_p(1)\) and if \(H_0\) is true, we can write

$$\begin{aligned} \hat{\varvec{\xi }}_2 -\varvec{0} = \frac{1}{n} \left( \varvec{\varOmega }^{(1)} \varvec{S}_{\varvec{\xi }_1} (\varvec{\xi }_1,\varvec{0}) + \varvec{\varOmega }_{\varvec{\xi }_{22}} \varvec{S}_{\varvec{\xi }_2} (\varvec{\xi }_1,\varvec{0})\right) +o_p(1) , \end{aligned}$$

where \(\varvec{\varOmega }^{(1)}\) is the lower left of \(p_2\times (q+p_1)\) submatrix of \(\varvec{\varOmega }_{\varvec{\xi }}\). Plug-in the value of \(\hat{\varvec{\xi }}_2\) in the following expression

$$\begin{aligned}&n\hat{\varvec{\xi }}_{2}^{\top } {\varvec{{\mathcal {I}}}}_{{\varvec{\xi }}_{22}}^{-1}\hat{\varvec{\xi }}_{2} \nonumber \\&\quad = n^{-1} {\varvec{S}}_{ {\varvec{\xi }}_1}^{\top }(\varvec{\xi }_1, \varvec{0}) \left( [{\varvec{\varOmega }}^{(1)}]^{\top } {\varvec{{\mathcal {I}}}}_{\varvec{\xi }_{22}}^{-1} {\varvec{\varOmega }}^{(1)}\right) {\varvec{S}}_{\varvec{\xi }_1}( \varvec{\xi }_1, \varvec{0}) \nonumber \\&\qquad + n^{-1} {\varvec{S}}_{\varvec{\xi }_2}^{\top }(\varvec{\xi }_1, \varvec{0}) {\varvec{\varOmega }}_{\varvec{\xi }_{22}}^{\top } {\varvec{{\mathcal {I}}}}_{\varvec{\xi }_{22}}^{-1} {\varvec{\varOmega }}^{(1)} {\varvec{S}}_{\varvec{\xi }_1}(\varvec{\xi }_1, \varvec{0}) \nonumber \\&\qquad +n^{-1} {\varvec{S}}_{\varvec{\xi }_1}^{\top }(\varvec{\xi }_1, \varvec{0}) \left( [{\varvec{\varOmega }}^{(1)}]^{\top } {\varvec{{\mathcal {I}}}}_{\varvec{\xi }_{22}}^{-1} {\varvec{\varOmega }}_{\varvec{\xi }_{22}}\right) {\varvec{S}}_{\varvec{\xi }_2}(\varvec{\xi }_1, \varvec{0}) \nonumber \\&\qquad + n^{-1} {\varvec{S}}_{\varvec{\xi }_2}^{\top }(\varvec{\xi }_1, \varvec{0})\left( {\varvec{\varOmega }}_{\varvec{\xi }_{22}}^{\top } {\varvec{{\mathcal {I}}}}_{\varvec{\xi }_{22}}^{-1} {\varvec{\varOmega }}_{\varvec{\xi }_{22}}\right) {\varvec{S}}_{\varvec{\xi }_2}( \varvec{\xi }_1, \varvec{0}). \end{aligned}$$
(9)

which has an asymptotic \(\chi _{p_2}^2\) distribution with \(p_2\) degrees of freedom. To show this, assume that \({\hat{\varXi }}_L^*\) be the new version of relation (9) when \( \varvec{\xi }_1\) is replaced by \(\tilde{\varvec{\xi }}_1\). Now the relation (9) differs from \({\hat{\varXi }}_L^*\) by an \(o_p(1)\) term as \(n\rightarrow \infty \). Thus \({\hat{\varXi }}_L^*\) converges in distribution to \(\chi _{p_2}^2\). We know that \({\varvec{S}}_{\varvec{\xi }_1}(\tilde{\varvec{\xi }}_1, \varvec{0})=\varvec{0}\) and this \({\varvec{S}}_{\varvec{\xi }_1}(\tilde{\varvec{\xi }}_1, \varvec{0})\) involves in the first three terms of (9) when \( \varvec{\xi }_1\) is replaced by \(\tilde{\varvec{\xi }}_1\). Thus \({\hat{\varXi }}_L^*= {\hat{\varXi }}_L\) and hence we complete the proof of the theorem.

Under local alternative (6), the following theorem and Lemma facilitates the theoretical derivation of ADB and ADR of the RPSIM, PTSIM, SESIM, and PSESIM estimators in Theorems 4.1.2 and 4.1.3. \(\square \)

Proof of Theorem 4.1.1

$$\begin{aligned} \text{ E }(\varvec{\psi }_1)= & {} \text{ E }\left( \sqrt{n}(\hat{\varvec{\xi }} - \varvec{\xi })\right) = \varvec{0}.\\ \text{ E }(\varvec{\psi }_2)= & {} \text{ E }\left( \sqrt{n}(\tilde{\varvec{\xi }} - \varvec{\xi })\right) \\= & {} \text{ E }\left( \sqrt{n}(\hat{\varvec{\xi }}_1- \varvec{\xi }_1+ {\varvec{{\mathcal {I}}}}_{\varvec{\xi }_{11}}^{-1} {\varvec{{\mathcal {I}}}}_{\varvec{\xi }_{12}} \hat{\varvec{\xi }}_2)\right) ,~~\text{ see } Lawless and Singhal (1978)\\= & {} \varvec{0} + {\varvec{{\mathcal {I}}}}_{\varvec{\xi }_{11}}^{-1} {\varvec{{\mathcal {I}}}}_{\varvec{\xi }_{12}} \sqrt{n}\left( \frac{\varvec{\delta }}{\sqrt{n}}\right) ={\varvec{{\mathcal {I}}}}_{\varvec{\xi }_{11}}^{-1} {\varvec{{\mathcal {I}}}}_{\varvec{\xi }_{12}}\varvec{\delta } =\varvec{\gamma }. \end{aligned}$$
$$\begin{aligned} \text{ E }(\varvec{\psi }_3)= & {} \text{ E }\left( \sqrt{n}(\hat{\varvec{\xi }} - \tilde{\varvec{\xi }})\right) = \text{ E }\left( \sqrt{n}(\hat{\varvec{\xi }} - \varvec{\xi })-\sqrt{n}(\tilde{\varvec{\xi }} - \varvec{\xi })\right) = - \varvec{\gamma }.\\ \text{ Var }(\varvec{\psi }_1)= & {} \text{ Var }\left( \sqrt{n}(\hat{\varvec{\xi }} - \varvec{\xi })\right) = {\varvec{{\mathcal {I}}}}_{\varvec{\xi }_{11.2}}^{-1} ={\mathcal {A}}_{11}.\\ \text{ Var }(\varvec{\psi }_2)= & {} \text{ Var }\left( \sqrt{n}(\tilde{\varvec{\xi }} - \varvec{\xi })\right) =\text{ Var }\left( \sqrt{n}(\hat{\varvec{\xi }}_1- \varvec{\xi }_1+ {\varvec{{\mathcal {I}}}}_{\varvec{\xi }_{11}}^{-1} {\varvec{{\mathcal {I}}}}_{\varvec{\xi }_{12}} \hat{\varvec{\xi }}_2)\right) \\= & {} \text{ Var }\left( \sqrt{n}\left( \hat{\varvec{\xi }}_1- \varvec{\xi }_1\right) \right) + {\varvec{{\mathcal {I}}}}_{\varvec{\xi }_{11}}^{-1} {\varvec{{\mathcal {I}}}}_{\varvec{\xi }_{12}} \text{ Var }\left( \sqrt{n} \hat{\varvec{\xi }}_2\right) {\varvec{{\mathcal {I}}}}_{\varvec{\xi }_{21}} {\varvec{{\mathcal {I}}}}_{\varvec{\xi }_{11}}^{-1} \\&+ 2 \text{ Cov }\left( \sqrt{n}\left( \hat{\varvec{\xi }}_1- \varvec{\xi }_1\right) , {\varvec{{\mathcal {I}}}}_{\varvec{\xi }_{11}}^{-1} {\varvec{{\mathcal {I}}}}_{\varvec{\xi }_{12}} \sqrt{n} \hat{\varvec{\xi }}_2\right) \\= & {} {\varvec{{\mathcal {I}}}}_{\varvec{\xi }_{11.2}}^{-1} - {\varvec{{\mathcal {I}}}}_{\varvec{\xi }_{11}}^{-1} {\varvec{{\mathcal {I}}}}_{\varvec{\xi }_{12}} {\varvec{{\mathcal {I}}}}_{\varvec{\xi }_{22.1}}^{-1} {\varvec{{\mathcal {I}}}}_{\varvec{\xi }_{21}} {\varvec{{\mathcal {I}}}}_{\varvec{\xi }_{11}}^{-1} = {\mathcal {A}}_{22}. \end{aligned}$$
$$\begin{aligned} \text{ Var }(\varvec{\psi }_3)= & {} \text{ Var }\left( \sqrt{n}(\hat{\varvec{\xi }} - \tilde{\varvec{\xi }})\right) = \text{ Var }\left( \sqrt{n}(\hat{\varvec{\xi }} - \varvec{\xi })-\sqrt{n}(\tilde{\varvec{\xi }} - \varvec{\xi })\right) \\= & {} {\varvec{{\mathcal {I}}}}_{\varvec{\xi }_{11}}^{-1} {\varvec{{\mathcal {I}}}}_{\varvec{\xi }_{12}} {\varvec{{\mathcal {I}}}}_{\varvec{\xi }_{22.1}}^{-1} {\varvec{{\mathcal {I}}}}_{\varvec{\xi }_{21}} {\varvec{{\mathcal {I}}}}_{\varvec{\xi }_{11}}^{-1} = {\mathcal {A}}_{33}.\\ \text{ Cov }(\varvec{\psi }_1, \varvec{\psi }_2)= & {} \text{ Cov } \left( \sqrt{n}(\tilde{\varvec{\xi }} - \varvec{\xi }), \sqrt{n}(\tilde{\varvec{\xi }} - \varvec{\xi })\right) \\= & {} {\varvec{{\mathcal {I}}}}_{\varvec{\xi }_{11.2}}^{-1} - {\varvec{{\mathcal {I}}}}_{\varvec{\xi }_{11}}^{-1} {\varvec{{\mathcal {I}}}}_{\varvec{\xi }_{12}} {\varvec{{\mathcal {I}}}}_{\varvec{\xi }_{22.1}}^{-1} {\varvec{{\mathcal {I}}}}_{\varvec{\xi }_{21}} {\varvec{{\mathcal {I}}}}_{\varvec{\xi }_{11}}^{-1} = {\mathcal {A}}_{12} = {\mathcal {A}}_{21}^{\top }.\\ \text{ Cov }(\varvec{\psi }_1, \varvec{\psi }_3)= & {} \text{ Cov } \left( \sqrt{n}(\tilde{\varvec{\xi }} - \varvec{\xi }), \sqrt{n}(\hat{\varvec{\xi }} - \tilde{\varvec{\xi }})\right) \\= & {} \text{ Cov } \left( \sqrt{n}(\tilde{\varvec{\xi }} - \varvec{\xi }), \sqrt{n}(\tilde{\varvec{\xi }} - \varvec{\xi })-\sqrt{n} (\tilde{\varvec{\xi }} - \varvec{\xi })\right) \\= & {} {\varvec{{\mathcal {I}}}}_{\varvec{\xi }_{11}}^{-1} {\varvec{{\mathcal {I}}}}_{\varvec{\xi }_{12}} {\varvec{{\mathcal {I}}}}_{\varvec{\xi }_{22.1}}^{-1} {\varvec{{\mathcal {I}}}}_{\varvec{\xi }_{21}} {\varvec{{\mathcal {I}}}}_{\varvec{\xi }_{11}}^{-1}= {\mathcal {A}}_{13} = {\mathcal {A}}_{31}^{\top } \end{aligned}$$
$$\begin{aligned} \text{ Cov }(\varvec{\psi }_2, \varvec{\psi }_3)= & {} \text{ Cov } \left( \sqrt{n}(\tilde{\varvec{\xi }} - \varvec{\xi }), \sqrt{n}(\hat{\varvec{\xi }} - \tilde{\varvec{\xi }})\right) \\= & {} \text{ Cov } \left( \sqrt{n}(\tilde{\varvec{\xi }} - \varvec{\xi }), \sqrt{n}(\hat{\varvec{\xi }}-\varvec{\xi }) - \sqrt{n}(\tilde{\varvec{\xi }}-\varvec{\xi })\right) \\= & {} \text{ Cov } \left( \sqrt{n}(\tilde{\varvec{\xi }} - \varvec{\xi }), \sqrt{n}(\hat{\varvec{\xi }}-\varvec{\xi })\right) - \text{ Cov } \left( \sqrt{n}(\tilde{\varvec{\xi }} - \varvec{\xi }), \sqrt{n}(\tilde{\varvec{\xi }}-\varvec{\xi })\right) \\= & {} {\varvec{{\mathcal {I}}}}_{\varvec{\xi }_{11.2}}^{-1} - {\varvec{{\mathcal {I}}}}_{\varvec{\xi }_{11}}^{-1} {\varvec{{\mathcal {I}}}}_{\varvec{\xi }_{12}} {\varvec{{\mathcal {I}}}}_{\varvec{\xi }_{22.1}}^{-1} {\varvec{{\mathcal {I}}}}_{\varvec{\xi }_{21}} {\varvec{{\mathcal {I}}}}_{\varvec{\xi }_{11}}^{-1} -{\varvec{{\mathcal {I}}}}_{\varvec{\xi }_{11.2}}^{-1} \\&\qquad + {\varvec{{\mathcal {I}}}}_{\varvec{\xi }_{11}}^{-1} {\varvec{{\mathcal {I}}}}_{\varvec{\xi }_{12}} {\varvec{{\mathcal {I}}}}_{\varvec{\xi }_{22.1}}^{-1} {\varvec{{\mathcal {I}}}}_{\varvec{\xi }_{21}} {\varvec{{\mathcal {I}}}}_{\varvec{\xi }_{11}}^{-1}=\varvec{0}. \end{aligned}$$

\(\square \)

Lemma

Let \(\varvec{X}\sim \text{ n } (\varvec{\mu }, \varvec{{\mathcal {I}}})\), where \(\varvec{{\mathcal {I}}}\) is a nonnegative definite matrix. Also let \(\varvec{Q}\) be symmetric and positive definite matrix such that \(\varvec{Q}^{1/2} \varvec{{\mathcal {I}}} \varvec{Q}^{1/2}\) is an idempotent matrix, and \(\varvec{Q} \varvec{{\mathcal {I}}} \varvec{Q}\varvec{\mu }=\varvec{Q}\varvec{\mu }\). Then for all \(\varphi \), Borel measurable and real-valued integrable function

$$\begin{aligned}&1 \text{ E }\left( \varphi \left( \varvec{X}^{\top } \varvec{Q} \varvec{{\mathcal {I}}} \varvec{Q}\varvec{X}\right) \varvec{X}\right) = \varvec{\theta }\text{ E }\left( \varphi \left( \chi _{p+2}^2(\varvec{\mu }^{\top } \varvec{Q} \varvec{\mu }) \right) \right) \\&2 \text{ E }\left( \varphi \left( \varvec{X}^{\top } \varvec{Q} \varvec{{\mathcal {I}}} \varvec{Q}\varvec{X}\right) \varvec{X}^{\top }\varvec{M}\varvec{X}\right) = \text{ E }\left( \varphi \left( \chi _{p+2}^2(\varvec{\mu }^{\top } \varvec{Q} \varvec{\mu }) \right) \right) \text{ tr }(\varvec{A\varvec{{\mathcal {I}}}})\\&\qquad +\text{ E }\left( \varphi \left( \chi _{p+4}^2(\varvec{\mu }^{\top } \varvec{Q} \varvec{\mu }) \right) \right) \varvec{\mu }^{\top } \varvec{M} \varvec{\mu }, \end{aligned}$$

where \(\varvec{M}\) is nonnegative definite matrix.

The outline of the proof of this lemma is given in Nkurunziza et al. (2013).

Proof of Theorem 4.1.2

In this proof, we derive the bias expressions of the proposed estimators. It is obvious that \(\text{ ADB }(\hat{\varvec{\xi }})=\varvec{0} \). The ADB of RPSIM (\(\tilde{\varvec{\xi }}\)), PTSIM (\(\hat{\varvec{\xi }}_p\)), SESIM(\(\hat{\varvec{\xi }}_S\)), and PSESIM (\(\hat{\varvec{\xi }}_{S+}\)) estimators are as follows:

$$\begin{aligned} \text{ ADB }(\tilde{\varvec{\xi }})&= \text{ E } \left( \lim _{n\rightarrow \infty } \sqrt{n} (\tilde{\varvec{\xi }} - \varvec{\xi }) \right) = \text{ E }(\varvec{\psi }_2) = -{\varvec{{\mathcal {I}}}}_{\varvec{\xi }_{11}}^{-1} {\varvec{{\mathcal {I}}}}_{\varvec{\xi }_{12}} \varvec{\delta } = \varvec{\gamma }\\ \text{ ADB }(\hat{\varvec{\xi }}_{P})&= \text{ E } \left( \lim _{n\rightarrow \infty } \sqrt{n} (\hat{\varvec{\xi }}_{P} - \varvec{\xi }) \right) = \varvec{0}- \text{ E } \left( \lim _{n\rightarrow \infty } \sqrt{n} I\left( {\hat{\varXi }}_L \le \chi ^2_{\kappa +2, \alpha }\right) \right) \left( \hat{\varvec{\xi }} - \tilde{\varvec{\xi }}\right) \\&= - \text{ E } \left( \lim _{n\rightarrow \infty } I\left( {\hat{\varXi }}_L \le \chi ^2_{p_2, \alpha }, \varDelta \right) \varvec{\psi }_3 \right) = \varvec{\gamma } H_{\kappa +4} \left( \chi ^2_{\kappa +2, \alpha }, \varDelta \right) \\ \text{ ADB }(\hat{\varvec{\xi }}_{S})&= \text{ E } \left( \lim _{n\rightarrow \infty } \sqrt{n} (\hat{\varvec{\xi }}_{S} - \varvec{\xi }) \right) = - \text{ E } \left( \lim _{n\rightarrow \infty } \sqrt{n} (\kappa {\hat{\varXi }}_L^{-1}(\hat{\varvec{\xi }} - \tilde{\varvec{\xi }})) \right) \\&= -\kappa \text{ E } \left( \lim _{n\rightarrow \infty } \varvec{\psi }_3 {\hat{\varXi }}_L^{-1}\right) =\kappa \varvec{\gamma } \text{ E }(Z_1^{-1})\\ \text{ ADB }(\hat{\varvec{\xi }}_{S+})&= \text{ E } \left( \lim _{n\rightarrow \infty } \sqrt{n} (\hat{\varvec{\xi }}_{S+} - \varvec{\xi }) \right) \\&= \text{ E } \left( \lim _{n\rightarrow \infty } \sqrt{n}(\hat{\varvec{\xi }}_{S} - \varvec{\xi }) - \sqrt{n} \left( 1-\kappa {\hat{\varXi }}_L^{-1}\right) I\left( {\hat{\varXi }}_L< \kappa \right) (\hat{\varvec{\xi }}-\tilde{\varvec{\xi }}) \right) \\&= \text{ ADB }(\hat{\varvec{\xi }}_{S}) - \text{ E } \left( \lim _{n\rightarrow \infty } \varvec{\psi }_3 \left( 1-\kappa {\hat{\varXi }}_L^{-1}\right) I\left( {\hat{\varXi }}_L< \kappa \right) \right) \\&= \text{ ADB }(\hat{\varvec{\xi }}_{S})+ \varvec{\gamma } \text{ E } \left( I\left( {\hat{\varXi }}_L< \kappa \right) \right) - \varvec{\gamma }\kappa \text{ E }\left( \varvec{\psi }_3 {\hat{\varXi }}_L^{-1}I\left( {\hat{\varXi }}_L< \kappa \right) \right) \\&= \text{ ADB }(\hat{\varvec{\xi }}_{S})+ \varvec{\gamma } H_{\kappa +4}(\kappa ,\varDelta ) - \varvec{\gamma }\kappa \text{ E }\left( Z_1^{-1} I\left( Z_1< \kappa \right) \right) \end{aligned}$$

Proof of Theorem 4.1.3

Based on the definition of ADR function, it is necessary to derive the asymptotic covariance matrices for the four estimators. The covariance matrix of any estimator \(\hat{\varvec{\xi }}^*\) is defined as:

$$\begin{aligned} \text{ Cov } (\hat{\varvec{\xi }}^*) = \text{ E }\left( \lim _{n\rightarrow \infty } n (\hat{\varvec{\xi }}^*-\varvec{\xi }) (\hat{\varvec{\xi }}^*-\varvec{\xi })^{\top } \right) . \end{aligned}$$

First, we will start deriving the covariance matrices of the UPSIM and RPSIM:

$$\begin{aligned} \text{ Cov } (\hat{\varvec{\xi }})= & {} \text{ E }\left( \lim _{n\rightarrow \infty } \sqrt{n} (\hat{\varvec{\xi }} -\varvec{\xi }) \sqrt{n}( \hat{\varvec{\xi }} -\varvec{\xi })^{\top } \right) = \text{ E }(\varvec{\psi } \varvec{\psi } ^{\top }) = {\varvec{{\mathcal {A}}}}_{12} ={\varvec{{\mathcal {I}}}}^{-1}_{11.2}.\\ \text{ Cov }( \tilde{\varvec{\xi }})= & {} \text{ E }\left( \lim _{n\rightarrow \infty } \sqrt{n}(\tilde{\varvec{\xi }} -\varvec{\xi }) \sqrt{n}(\tilde{\varvec{\xi }} -\varvec{\xi })^{\top }\right) = \text{ E }(\varvec{\psi }_2 \varvec{\psi }_2^{\top })= {\varvec{{\mathcal {A}}}}_{12} +\varvec{\gamma } \varvec{\gamma }^{\top }. \end{aligned}$$

Second, we derive the covariance matrix of the pretest estimator:

$$\begin{aligned} \text{ Cov }( \hat{\varvec{\xi }}_{P})= & {} \text{ E }\left( \lim _{n\rightarrow \infty } \sqrt{n}(\hat{\varvec{\xi }}_{P} - \varvec{\xi }) \sqrt{n} (\hat{\varvec{\xi }}_{P} - \varvec{\xi })^{\top } \right) \\= & {} \text{ E } \left( \varvec{\psi }_1 \varvec{\psi }_1^{\top } + \varvec{\psi }_3 \varvec{\psi }_3^{\top } \lim _{n\rightarrow \infty } I \left( {\hat{\varXi }}_L \le \chi ^2_{p_2, \alpha }\right) -2 \varvec{\psi }_3 \varvec{\psi }_1^{\top } \lim _{n\rightarrow \infty } I \left( {\hat{\varXi }}_L \le \chi ^2_{p_2, \alpha }\right) \right) \\= & {} {\varvec{{\mathcal {A}}}_{11}} + {\varvec{{\mathcal {A}}}_{13}} H_{\kappa + 4}\left( \chi ^2_{\kappa + 2,\alpha }, \varDelta \right) + \varvec{\gamma } \varvec{\gamma }^{\top } H_{\kappa + 6}\left( \chi ^2_{\kappa + 2,\alpha }, \varDelta \right) \\&- 2 \text{ E } \left( \varvec{\psi }_3 \varvec{\psi }_1^{\top } \lim _{n\rightarrow \infty } I\left( {\hat{\varXi }}_L \le \chi ^2_{p_2, \alpha }\right) \right) . \end{aligned}$$

Consider the fourth term:

$$\begin{aligned}&\text{ E } \left( \varvec{\psi }_3 \varvec{\psi }_1^{\top } \lim _{n\rightarrow \infty } I \left( {\hat{\varXi }}_L \le \chi ^2_{p_2, \alpha }\right) \right) =\text{ E } \left( \text{ E } \left( \varvec{\psi }_3 \varvec{\psi }_1^{\top } \lim _{n\rightarrow \infty } I \left( {\hat{\varXi }}_L \le \chi ^2_{p_2, \alpha }\right) | \varvec{\psi }_3\right) \right) \\&\quad = \text{ E } \left( \varvec{\psi }_3 \left( \varvec{\psi }_3^{\top } - \text{ E }\left( \varvec{\psi }_3\right) ^{\top }\right) \lim _{n\rightarrow \infty } I \left( {\hat{\varXi }}_L \le \chi ^2_{p_2, \alpha }\right) \right) \\&\quad = \text{ E } \left( \varvec{\psi }_3 \varvec{\psi }_3^{\top } \lim _{n\rightarrow \infty } I \left( {\hat{\varXi }}_L \le \chi ^2_{p_2, \alpha }\right) \right) -\text{ E } \left( \varvec{\psi }_3 \lim _{n\rightarrow \infty } I \left( {\hat{\varXi }}_L \le \chi ^2_{p_2, \alpha }\right) \right) \text{ E }\left( \varvec{\psi }_3\right) ^{\top } \\&\quad = {\varvec{{\mathcal {A}}}_{13}} H_{\kappa +4}\left( \chi ^2_{\kappa + 2, \alpha }, \varDelta \right) + \varvec{\gamma } \varvec{\gamma }^{\top } H_{\kappa +6}\left( \chi ^2_{\kappa + 2, \alpha }, \varDelta \right) \\&\qquad - \varvec{\gamma } \varvec{\gamma }^{\top } H_{\kappa +4}\left( \chi ^2_{\kappa + 2, \alpha }, \varDelta \right) \\&\quad = {\varvec{{\mathcal {A}}}_{13}} H_{\kappa + 4}\left( \chi ^2_{\kappa + 2, \alpha }, \varDelta \right) + \varvec{\gamma } \varvec{\gamma }^{\top } H_{\kappa + 6}\left( \chi ^2_{\kappa + 2, \alpha }, \varDelta \right) - \varvec{\gamma } \varvec{\gamma }^{\top } H_{\kappa + 4}\left( \chi ^2_{\kappa + 2, \alpha }, \varDelta \right) ,\\ \text{ Cov }( \hat{\varvec{\xi }}_{P})= & {} {\varvec{{\mathcal {I}}}}^{-1}_{11.2}+ \lim _{n\rightarrow \infty } \text{ E } \left( \varvec{\psi }_3 \varvec{\psi }_3^{\top } I\left( {\hat{\varXi }}_L \le \chi ^2_{p_2, \alpha }\right) \right) -2 \lim _{n\rightarrow \infty } \text{ E } \left( \varvec{\psi }_3 \varvec{\psi } ^{\top } I\left( {\hat{\varXi }}_L \le \chi ^2_{p_2, \alpha }\right) \right) \\&\quad = {\varvec{{\mathcal {I}}}}^{-1}_{11.2} + {\varvec{{\mathcal {A}}}_{13}} H_{\kappa + 4}\left( \chi ^2_{\kappa + 2, \alpha }, \varDelta \right) + \varvec{\gamma } \varvec{\gamma }^{\top } H_{\kappa + 6}\left( \chi ^2_{\kappa + 2, \alpha }, \varDelta \right) \\&\qquad -2 \left( {\varvec{{\mathcal {A}}}_{13}} H_{\kappa + 4}\left( \chi ^2_{\kappa + 2, \alpha }, \varDelta \right) + \varvec{\gamma } \varvec{\gamma }^{\top } H_{\kappa + 6}\left( \chi ^2_{\kappa + 2, \alpha }, \varDelta \right) \right. \\&\left. -\varvec{\gamma } \varvec{\gamma }^{\top } H_{\kappa + 4}\left( \chi ^2_{\kappa + 2, \alpha }, \varDelta \right) \right) \\&\quad = {\varvec{{\mathcal {I}}}}^{-1}_{11.2} - {\varvec{{\mathcal {A}}}_{13}} H_{\kappa + 4}\left( \chi ^2_{\kappa + 2, \alpha }, \varDelta \right) - \varvec{\gamma } \varvec{\gamma }^{\top } H_{\kappa + 6}\left( \chi ^2_{\kappa + 2, \alpha }, \varDelta \right) \\&\qquad + 2 \varvec{\gamma } \varvec{\gamma }^{\top } H_{\kappa + 4}\left( \chi ^2_{\kappa + 2, \alpha }, \varDelta \right) . \end{aligned}$$

Third, we derive the covariance matrices of the shrinkage and positive shrinkage estimators:

$$\begin{aligned} \text{ Cov } ( \hat{\varvec{\xi }}_{S})= & {} \text{ E } \left( \lim _{n\rightarrow \infty } \sqrt{n}(\hat{\varvec{\xi }}_{S} - \varvec{\xi }) \sqrt{n} (\hat{\varvec{\xi }}_{S} - \varvec{\xi })^{\top } \right) \\= & {} \text{ E } \Bigg (\lim _{n\rightarrow \infty } \sqrt{n} \left( \hat{\varvec{\xi }} - \varvec{\xi } - \kappa {\hat{\varXi }}_L^{-1}(\hat{\varvec{\xi }} - \tilde{\varvec{\xi }})\right) \sqrt{n} \left( \hat{\varvec{\xi }} - \varvec{\xi } - \kappa {\hat{\varXi }}_L^{-1}(\hat{\varvec{\xi }} - \tilde{\varvec{\xi }})\right) ^{\top } \Bigg ) \\= & {} \text{ E } (\varvec{\psi }_1 \varvec{\psi }_1^{\top }) + \kappa ^2 \text{ E } \left( \varvec{\psi }_3 \varvec{\psi }_3^{\top } \lim _{n\rightarrow \infty } {\hat{\varXi }}_L^{-2}\right) -2 \kappa \text{ E } \left( \varvec{\psi }_3 \varvec{\psi }_1^{\top } \lim _{n\rightarrow \infty } {\hat{\varXi }}_L^{-1}\right) \\= & {} {\varvec{{\mathcal {I}}}}^{-1}_{11.2} + {\varvec{{\mathcal {A}}}_{13}} \text{ E }\left( Z_1^{-2}\right) + \varvec{\gamma } \varvec{\gamma }^{\top } \text{ E }\left( Z_2^{-2}\right) -2 \kappa \text{ E } \left( \varvec{\psi }_3 \varvec{\psi }_1^{\top } \lim _{n\rightarrow \infty } {\hat{\varXi }}_L^{-1}\right) . \end{aligned}$$

Consider the last term:

$$\begin{aligned}&\text{ E } (\varvec{\psi }_3 \varvec{\psi }_1^{\top } \lim _{n\rightarrow \infty } {\hat{\varXi }}_L^{-1})=\text{ E } \left( \text{ E } (\varvec{\psi }_3 \varvec{\psi }_1^{\top } \lim _{n\rightarrow \infty } {\hat{\varXi }}_L^{-1} | \varvec{\psi }_3)\right) \\&\quad = \text{ E } \left( \varvec{\psi }_3 \text{ E } (\varvec{\psi }_1^{\top } | \varvec{\psi }_3) \lim _{n\rightarrow \infty } {\hat{\varXi }}_L^{-1}\right) + \text{ E } \left( \varvec{\psi }_3 \left( \varvec{\psi }_3^{\top } - \text{ E }(\varvec{\psi }_3\right) ^{\top }) \lim _{n\rightarrow \infty } {\hat{\varXi }}_L^{-1}\right) \\&\quad = \text{ E } \left( \varvec{\psi }_3 \varvec{\psi }_3^{\top } \lim _{n\rightarrow \infty } {\hat{\varXi }}_L^{-1}\right) - \text{ E } \left( \varvec{\psi }_3 \lim _{n\rightarrow \infty } {\hat{\varXi }}_L^{-1}\right) \text{ E }\left( \varvec{\psi }_3\right) ^{\top } \\&\quad = {\varvec{{\mathcal {A}}}_{13}} \text{ E } \left( \chi ^{-2}_{p_2 + 2}(\varDelta )\right) + \varvec{\gamma } \varvec{\gamma }^{\top } \text{ E } \left( \chi ^{-2}_{p_2 + 4}(\varDelta )\right) - \varvec{\gamma } \varvec{\gamma }^{\top } \text{ E } \left( \chi ^{-2}_{p_2 + 2}(\varDelta )\right) \\&\quad = {\varvec{{\mathcal {A}}}_{13}} \text{ E } \left( Z_1^{-1}\right) + \varvec{\gamma } \varvec{\gamma }^{\top } \text{ E } \left( Z_2^{-1}\right) - \varvec{\gamma } \varvec{\gamma }^{\top } \text{ E } \left( Z_1^{-1}\right) . \end{aligned}$$

Hence

$$\begin{aligned} \text{ Cov } ( \hat{\varvec{\xi }}_{S})= & {} {\varvec{{\mathcal {I}}}}^{-1}_{11.2} + \kappa ^2 \left( {\varvec{{\mathcal {A}}}_{13}} \text{ E } (Z_1^{-2}) + \varvec{\gamma } \varvec{\gamma }^{\top } \text{ E }(Z_2^{-2}) \right) \\&-2 \kappa \left( {\varvec{{\mathcal {A}}}_{13}} \text{ E }(Z_1^{-1}) + \varvec{\gamma } \varvec{\gamma }^{\top } \text{ E } (Z_2^{-1}) - \varvec{\gamma } \varvec{\gamma }^{\top } \text{ E } (Z_1^{-1})\right) \\= & {} {\varvec{{\mathcal {I}}}}^{-1}_{11.2} + \left( \kappa ^2 \text{ E } \left( Z_1^{-2}\right) -2 \kappa \text{ E } (Z_1^{-1}) \right) {\varvec{{\mathcal {A}}}_{13}}\\&+ \left( \kappa ^2 \text{ E } (Z_2^{-2}) +2 \kappa \text{ E } (Z_1^{-1}) -2 \kappa \text{ E } (Z_2^{-1}) \right) \varvec{\gamma } \varvec{\gamma }^{\top }. \end{aligned}$$

Let \(F_m(\varDelta )=\left( 1-\kappa {\hat{\varXi }}_L^{-1}\right) ^m I \left( {\hat{\varXi }}_L < \kappa \right) \), where \(m=1,2\)

$$\begin{aligned} \text{ Cov } (\hat{\varvec{\xi }}_{S+})= & {} \text{ E } \left( \lim _{n\rightarrow \infty } \sqrt{n} (\hat{\varvec{\xi }}_{S+} - \varvec{\xi }) \sqrt{n} (\hat{\varvec{\xi }}_{S+} - \varvec{\xi })^{\top }\right) , \\= & {} \text{ E } \left( \lim _{n\rightarrow \infty } \sqrt{n} (\hat{\varvec{\xi }}_{S} - \varvec{\xi }) \sqrt{n} (\hat{\varvec{\xi }}_{S} - \varvec{\xi })^{\top }\right) \\&+ \text{ E } \left( \lim _{n\rightarrow \infty } F_2(\varDelta ) \sqrt{n} (\hat{\varvec{\xi }} - \tilde{\varvec{\xi }}) \sqrt{n} (\hat{\varvec{\xi }} - \tilde{\varvec{\xi }})^{\top }\right) \\&-2 \text{ E } \left( \lim _{n\rightarrow \infty } F_1(\varDelta ) \sqrt{n} (\hat{\varvec{\xi }} - \tilde{\varvec{\xi }}) \sqrt{n}(\hat{\varvec{\xi }}_{S} - \varvec{\xi })^{\top }\right) \\= & {} \text{ Cov } ( \hat{\varvec{\xi }}_{S}) + \text{ E } \left( \lim _{n\rightarrow \infty } F_2(\varDelta ) \varvec{\psi }_3 \varvec{\psi }_2^{\top }\right) \\- & {} 2 \text{ E } \left( \lim _{n\rightarrow \infty } F_1(\varDelta ) \varvec{\psi }_3 \left( \varvec{\psi }_2^{\top } + \left( 1-\kappa {\hat{\varXi }}_L^{-1}\right) \varvec{\psi }_3^{\top }\right) \right) , \\= & {} \text{ Cov } ( \hat{\varvec{\xi }}_{S}) - \text{ E } \left( \lim _{n\rightarrow \infty }F_2(\varDelta )\varvec{\psi }_3 \varvec{\psi }_3^{\top }\right) -2 \text{ E } \left( \lim _{n\rightarrow \infty } F_1(\varDelta ) \varvec{\psi }_3 \varvec{\psi }_2^{\top }\right) . \end{aligned}$$

\(\square \)

Consider the second term:

$$\begin{aligned}&- \text{ E } \left( \lim _{n\rightarrow \infty } F_2(\varDelta )\varvec{\psi }_3 \varvec{\psi }_3^{\top }\right) = - \text{ E } \left( \lim _{n\rightarrow \infty } \left( 1-\kappa {\hat{\varXi }}_L^{-1}\right) ^2 I \left( {\hat{\varXi }}_L< \kappa \right) \varvec{\psi }_3 \varvec{\psi }_3^{\top }\right) \\&\quad = - {\varvec{{\mathcal {A}}}_{13}} \text{ E } \left( I \left( Z_1< \kappa \right) \left( 1 - \kappa Z_1^{-1} \right) ^2\right) - \varvec{\gamma } \varvec{\gamma }^{\top } \text{ E } \left( I \left( Z_2 < \kappa \right) \left( 1 - \kappa Z_2^{-1} \right) ^2\right) . \end{aligned}$$

Consider the third term:

$$\begin{aligned}&-2 \text{ E } \left( \lim _{n\rightarrow \infty } F_1(\varDelta ) \varvec{\psi }_3 \varvec{\psi }_2^{\top }\right) = -2 \text{ E } \left( \lim _{n\rightarrow \infty } \varvec{\psi }_3 \text{ E } \left( F_1(\varDelta ) \varvec{\psi }_2^{\top } | \varvec{\psi }_3 \right) \right) \\&\quad = -2 \text{ E } \left( \lim _{n\rightarrow \infty } \varvec{\psi }_3 \left( \text{ E } \left( \varvec{\psi }_2^{\top }\right) + \text{ cov } \left( \varvec{\psi }_3, \varvec{\psi }_2\right) \left( \varvec{\psi }_3 - \text{ E }\left( \varvec{\psi }_3\right) \right) \right) F_1(\varDelta )\right) \\&\quad = -2 \text{ E } \left( \lim _{n\rightarrow \infty } \varvec{\psi }_3 \text{ E } \left( \varvec{\psi }_2^{\top }\right) F_1(\varDelta ) + \varvec{0}\right) \\&\quad = -2 \text{ E } \left( \lim _{n \rightarrow \infty } \varvec{\psi }_3 I \left( {\hat{\varXi }}_L< \kappa \right) - \kappa {\hat{\varXi }}_L^{-1} \varvec{\psi }_3 I \left( {\hat{\varXi }}_L< \kappa \right) \right) \text{ E } \left( \varvec{\psi }_2^{\top }\right) \\&\quad = 2 H_{\kappa + 4} (\kappa , \varDelta ) \varvec{\gamma } \varvec{\gamma }^{\top } - 2 \kappa \text{ E } \left( Z_1^{-1} I \left( Z_1< \kappa \right) \right) \varvec{\gamma } \varvec{\gamma }^{\top } \\&\quad = \left( 2 H_{\kappa + 4} (\kappa , \varDelta ) - 2 \kappa \text{ E } \left( Z_1^{-1} I \left( Z_1< \kappa \right) \right) \right) \varvec{\gamma } \varvec{\gamma }^{\top }. \end{aligned}$$

Finally,

$$\begin{aligned} \text{ Cov } \left( \hat{\varvec{\xi }}_{S+}\right)= & {} \text{ Cov } \left( \hat{\varvec{\xi }}_{S}\right) - \text{ E } \left( \left( 1 - \kappa Z_1^{-1}\right) ^2 I \left( Z_1< \kappa \right) \right) {\varvec{{\mathcal {A}}}_{13}} \\&+ \left( 2 H_{\kappa + 4} (\kappa , \varDelta ) - 2 \kappa \text{ E } \left( Z_1^{-1} I \left( Z_1< \kappa \right) \right) \right. \\&- \left. \text{ E } \left( \left( 1 - \kappa Z_2^{-1} \right) ^2 I \left( Z_2 < \kappa \right) \right) \right) \varvec{\gamma } \varvec{\gamma }^{\top }. \end{aligned}$$

The ADR expressions in Theorem 4.4 now follow from (8) which completes the proof.

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Hossain, S., Lac, L.A. Optimal shrinkage estimations in partially linear single-index models for binary longitudinal data. TEST (2021). https://doi.org/10.1007/s11749-021-00753-3

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Keywords

  • Asymptotic distributional bias and risk
  • Generalized estimating equations
  • Monte Carlo simulation
  • Partially linear single-index models
  • Pretest and shrinkage estimators

Mathematics Subject Classification

  • 62G08
  • 62G10
  • 62G20