Score estimation of monotone partially linear index model

Abstract

This paper studies semiparametric estimation of a partially linear single index model with a monotone link function. Our estimator is an extension of the score-type estimator developed by Balabdaoui et al. (2019) for the monotone single index model, which profiles out the unknown link function by isotonic regression. An attractive feature of the proposed estimator is that it is free from tuning parameters for nonparametric smoothing. We show that our estimator for the finite-dimensional components is $\sqrt{n}$-consistent and asymptotically normal. By introducing an additional smoothing to obtain the efficient score, we propose an asymptotically efficient estimator for the finite-dimensional components. Furthermore, we establish the asymptotic validity of a bootstrap inference method based the score-type estimator, which is also free from tuning parameters. A simulation study illustrates the usefulness of the proposed method.

Acknowledgments

Otsu acknowledges financial support from the ERC Consolidator grant (SNP 615882). London School of Economics

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 Examples of such parametrisation are the spherical coordinate system $\mathbb{S}:[0,\pi {]}^{d-2}\times [0,2\pi ]\to {\mathcal{S}}_{d-1}$ with $\begin{array}{rl}\mathbb{S}\left(\gamma \right)& =(\cos ({\gamma }_1),\sin ({\gamma }_1)\cos ({\gamma }_2),\sin ({\gamma }_1)\sin ({\gamma }_2)\cos ({\gamma }_3),\dots ,\sin ({\gamma }_1)\cdots \sin ({\gamma }_{d-2})\cos ({\gamma }_{d-1}),\\ & \sin \left({\gamma }_1\right)\cdots \sin \left({\gamma }_{d-2}\right)\sin \left({\gamma }_{1-2}\right){)}^{\mathrm{\prime }},\end{array}$ and the half sphere $\mathbb{S}:\{\gamma \in [0,1{]}^{d-1}:||\gamma ||\le 1\}\to {\mathcal{S}}_{d-1}$ with $\mathbb{S}\left(\gamma \right)=({\gamma }_1,\dots ,{\gamma }_{d-1},\sqrt{1-{\gamma }_1^2-\cdots -{\gamma }_{d-1}^2}{)}^{\mathrm{\prime }}.$

2 We say that ${\theta }^{\ast }$ is a zero-crossing of a real-valued function $\zeta :\mathrm{\Theta }\to \mathbb{R}$ if each open neighbourhood of ${\theta }^{\ast }$ contains points ${\theta }_1,{\theta }_2\in \mathrm{\Theta }$ such that $\zeta \left({\theta }_1\right)\zeta \left({\theta }_2\right)\le 0$. This definition can be extended to a vector of functions, where a zero-crossing vector has each of its component to be a zero-crossing in the corresponding dimension.

3 Similar to other estimators by BGH or Groeneboom and Hendrickx (2018), our zero-crossing estimator $\hat{\theta }$ may not be unique. Indeed there are many flat parts in ${\phi }_n\left(\theta \right)$, and the intersection of ${\phi }_n\left(\theta \right)$ and zero could be an interval. In this case, any point on this interval will satisfy the results in Theorems 2.1 and 2.3.

4 We note that even for single index models, the convergence rate and asymptotic distribution of the least square estimator, $\arg \underset{\gamma }{min}\left\{\underset{\psi \in \mathcal{M}}{min}\frac{1}{n}\sum _{i=1}^n\right\{Y_i-\psi \left(Z_i^{\mathrm{\prime }}\mathbb{S}\right(\gamma \left)\right){\}}^2\}$, is an open problem.

5 Let W be an integrable random variable with the density $w\mapsto h(w,{\vartheta }_2)\exp \left\{{\vartheta }_2^{-1}w\ell \right({\vartheta }_1)-{\vartheta }_2^{-1}B(\ell \left({\vartheta }_1\right)\left)\right\}$, where ${\vartheta }_1$ is the mean, ${\vartheta }_2$ is a dispersion parameter, ℓ is a real-valued function with a strictly positive first derivative on an open interval, B is a real-valued function, and h is a normalising function. Balabdaoui et al. (2019, Proposition 9.2) showed that there exist c>0 and M>0 such that $E\left[|W{|}^m\right]\le m!M^{m-2}c$ for all integers $m\ge 2$. This proposition can be adapted to provide primitive conditions for A5 on the conditional distribution $Y-X^{\mathrm{\prime }}\text{β}|Z=z$, where the parameters ${\vartheta }_1$ and ${\vartheta }_2$ may vary with z.

6 Due to discontinuity in ${\hat{\psi }}_{n\theta }$, we can only guarantee the existence of $\hat{\theta }$ with probability approaching one. Similar to other zero-crossing estimators using isotonic regression, its existence for a given sample size is an open question.

7 For example, the conditional expectation $\mu \left(z\right)=E\left[X|z^{\mathrm{\prime }}\mathbb{S}\right({\gamma }_0\left)\right]$ in $V_{x,z}$ and $V_{x,z,{\psi }^{\mathrm{\prime }}}$ can be estimated by $\hat{\mu }\left(z\right)=\frac{\sum _{i=1}^{n}K\left(\frac{Z_i^{\mathrm{\prime }}\mathbb{S}\left(\hat{\gamma }\right)-z^{\mathrm{\prime }}\mathbb{S}\left(\hat{\gamma }\right)}{b}\right)X_i}{\sum _{i=1}^{n}K\left(\frac{Z_i^{\mathrm{\prime }}\mathbb{S}\left(\hat{\gamma }\right)-z^{\mathrm{\prime }}\mathbb{S}\left(\hat{\gamma }\right)}{b}\right)},$ where K is a kernel function (e.g. Gaussian and Epanechnikov) and b is a bandwidth.

8 Similar to $\hat{\theta }$, the zero-crossing estimator $\tilde{\theta }$ may not be unique. If the intersection of ${\xi }_{nh}\left(\theta \right)$ and zero is an interval, any point on this interval satisfies the result in Theorem 2.2.

9 More precisely, the estimator SSE_L is obtained by a zero-crossing of ${\phi }_n^L\left(\theta \right)=\left[\begin{array}{c}\frac{1}{n}\sum _{i=1}^n{X}_i\{Y_i-X_i^{\mathrm{\prime }}\text{β}-{\hat{\psi }}_{n\alpha }(Z_i^{\mathrm{\prime }}\alpha \left)\right\}\\ \frac{1}{n}(1-{\alpha }^{\mathrm{\prime }}\alpha )\sum _{i=1}^n{X}_i\{Y_i-X_i^{\mathrm{\prime }}\text{β}-{\hat{\psi }}_{n\alpha }(Z_i^{\mathrm{\prime }}\alpha \left)\right\}\end{array}\right],$ and ESE_L is defined analogously.