Rank-Based Empirical Likelihood for Regression Models with Responses Missing at Random

Abstract

In this paper, a general regression model with responses missing at random is considered. From an imputed rank-based objective function, a rank-based estimator is derived and its asymptotic distribution is established under mild conditions. Inference based on the normal approximation approach results in under coverage or over coverage issues. In order to address these issues, we propose an empirical likelihood approach based on the rank-based objective function, from which its asymptotic distribution is established. Extensive Monte Carlo simulation experiments under different settings of error distributions with different response probabilities are considered. The simulation results show that the proposed approach has better performance for the regression parameters compared to the normal approximation approach and its least-squares counterpart. Finally, a data example is provided to illustrate our method.

Appendix

This Appendix contains assumptions used in the development of theoretical results as well as the proof of the main results.

1.1 Assumptions

(I ₁ ):

φ is a nondecreasing, bounded, and twice continuously differentiable score function with bounded derivatives, defined on (0, 1), and, satisfying:

$$\displaystyle \begin{aligned}\quad \int_{0}^{1}\varphi(u)du=0\quad and\quad \int_{0}^{1}\varphi^{2}(u)du=1.\end{aligned}$$

(I ₂ ):

g(⋅) being a function of two variables x and β, it is required that g has continuous derivatives with respect to β that are bounded up to order 3 by p-integrable functions of x, independent of β, p ≥ 1.

(I ₃ ):

K(⋅) is a regular kernel of order r > 2, with window b _n satisfying $\displaystyle {nb_{n}^{4r}\to 0}$, $C(\log {n}/n)^{\gamma }<b_{n}<h_{n}$, for any C > 0, γ = 1 − 2∕p, p > 2 and h _n is a bandwidth such that $C(\log {n}/n)^{\gamma }<h_{n}<1$ with h _n → 0 as n →∞.

(I ₄ ):

sup_x E[|Y |^p|x = x] < ∞, for p ≥ 1 and inf_x Δ(x) > 0.

(I ₅ ):

For fixed n, ${\boldsymbol \beta }_{0,n}\in Int(\mathcal {B})$ is the unique minimizer of E[D _n(β)] such that lim_n→∞ β _0,n = β ₀

(I ₆ ):

The model error has a distribution with a finite Fisher information.

(I ₆ ):

$Var(\sqrt {n}S_{n}^{j}({\boldsymbol \beta }_0))\to \varSigma _{{\boldsymbol \beta }_{0}}^{j}$, where $\varSigma _{{\boldsymbol \beta }_{0}}^{j}$ is positive definite.

(I ₇ ):

Set H = B(B ^τ B)⁻¹ B ^τ, where B = ∇_β g(x, β ₀). H is the projection matrix onto the column space of B, which in this case represents the tangent space generated by B and let h _iin, i = 1, ⋯ , n, be the leverage values that stand for the diagonal entries of H. We assume that lim_n→∞max_1≤i≤n h _iin = 0

Assumptions (I ₂) − (I ₄) are necessary and sufficient to ensure the strong consistency of $\widehat {\pi }({\mathbf x})$ used in the imputation process. On the other hand, assumptions (I ₁), (I ₅) − (I ₇) together with the previous assumptions are necessary to establish the asymptotic properties (consistency and asymptotic normality distribution) of the rank-based estimators of β ₀. An elaborate discussion about these assumptions can be found in Hettmansperger and McKean (2011), Bindele and Abebe (2012), and Bindele (2017).

By definition of $S_{n}^{j}({\boldsymbol \beta })$, $\widehat {{\boldsymbol \beta }}_{n}^{j}$ is solution to the equation $S_{n}^{j}({\boldsymbol \beta })=\mathbf {0}$. As in Brunner and Denker (1994), assume without loss of generality that ∥λ _i∥ = 1 and define

$$\displaystyle \begin{aligned} J_{jn}(s) &=\frac{1}{n}\sum_{i=1}^{n}F_{ij}(s),\ \ \hat{J}_{jn}(s)=\frac{1}{n}\sum_{i=1}^{n}I(\nu_{ij}({\boldsymbol \beta}_{0})\leq s),\ \ F_{jn}(s)=\frac{1}{n}\sum_{i=1}^{n}{\boldsymbol \lambda}_{i}F_{ij}(s)\\ \hat{F}_{jn}(s)&=\frac{1}{n}\sum_{i=1}^{n}{\boldsymbol \lambda}_{i}I(\nu_{ij}({\boldsymbol \beta}_{0})\leq s),\quad T_{n}^{j}({\boldsymbol \beta}_{0})=S_{n}^{j}({\boldsymbol \beta}_{0})-E\big[S_{n}^{j}({\boldsymbol \beta}_{0}) \big]. \end{aligned} $$

The following lemma due to Brunner and Denker (1994) is a key for establishing asymptotic normality of the rank gradient function for dependent data.

Lemma 3.1

Let ς _jn be the minimum eigenvalue of W _jn = V ar(U _jn) with U _jn given by

$$\displaystyle \begin{aligned} U_{jn}=\int\varphi(J_{jn}(s))(\hat{F}_{jn}-F_{jn})(ds)+\int\varphi'(J_{jn}(s))(\hat{J}_{jn}(s)-J_{jn}(s))F_{jn}(ds)\;. \end{aligned}$$

Suppose that ς _jn ≥ Cn ^a for some constants $C, a \in \mathbb {R}$ and m(n) is such that M ₀ n ^γ ≤ m(n) ≤ M ₁ n ^γ for some constants 0 < M ₀ ≤ M ₁ < ∞ and 0 < γ < (a + 1)∕2. Then $\displaystyle {m(n){\mathbf W}_{jn}^{-1}T_{n}^{j}({\boldsymbol \beta }_{0})}$ is asymptotically standard multivariate normal, provided φ is twice continuously differentiable with bounded second derivative.

We provide a sketch of the proof of this lemma. A detailed proof can be found in Brunner and Denker (1994).

Proof

Set

$$\displaystyle \begin{aligned}B_{jn}=-\int(\hat{F}_{jn}-F_{jn})d\varphi(J_{jn})+\int(\hat{J}_{jn}-J_{jn})\frac{dF_{jn}}{dJ_{jn}}d\varphi(J_{jn}).\end{aligned}$$

Brunner and Denker (1994) showed that W _jn = n ² V ar(B _jn), as U _n = nB _n. From its definition, $S_{n}^{j}({\boldsymbol \beta })$ can be rewritten as

$$\displaystyle \begin{aligned}S_{n}^{j}({\boldsymbol \beta}_{0})=\frac{1}{n}\sum_{i=1}^{n}{\boldsymbol \lambda}_{i}\varphi\Big(\frac{R(\nu_{ij}(\beta_{0}))}{n+1}\Big)=\int\varphi\Big(\frac{n}{n+1}\hat{J}_{jn}\Big)dF_{jn}.\end{aligned}$$

By (I ₅), since ${\boldsymbol \beta }_{0}=\displaystyle \lim _{n\to \infty } \operatorname *{\mathrm {Argmin}}_{{\boldsymbol \beta }\in \mathcal {B}}E\{D_{n}^{j}({\boldsymbol \beta })\}$, we have $E\{S_{n}^{j}({\boldsymbol \beta }_{0})\}\to \mathbf {0}$ as n →∞. From the fact that Var(ε|x) > 0, there exists a positive constant C such that ς _jn ≥ Cn ² which satisfies the assumptions of Lemma 3.1, as φ is twice continuously differentiable with bounded derivatives, γ < (a + 1)∕2 with a = 2, M ₀ = M ₁ = 1, γ = 1, and m(n) = n. Thus, for n large enough, $\displaystyle {n{\mathbf W}_{jn}^{-1}T_{n}^{j}({\boldsymbol \beta }_{0})\approx n{\mathbf W}_{jn}^{-1}S_{n}^{j}({\boldsymbol \beta }_{0})}$, which converges to a multivariate standard normal, by Lemma 3.1. A direct application of Slutsky’s Lemma and putting Σ _jn = n ^−1∕2 W _jn give $\displaystyle \sqrt {n}S_{n}^{j}({\boldsymbol \beta }_{0}) \ \xrightarrow {\mathcal {D}} \ N_{p}(\mathbf {0}, \varSigma _{{\boldsymbol \beta }_{0}}^{j})$, j = 1, 2, where $\varSigma _{{\boldsymbol \beta }_{0}}^{j}=\displaystyle {\lim _{n\to \infty }\varSigma _{jn}\varSigma _{jn}^{\tau }}$.

Proof of Theorem 3.2. Let C be an arbitrary positive constant. Recall from Eq. (3.5) that the log likelihood ratio of β ₀ is given by

$$\displaystyle \begin{aligned}-2\log R_{n}^{j}({\boldsymbol \beta}_{0})=-2\log\prod_{i=1}^{n}\big(1+{\boldsymbol \xi}^{\tau}\eta_{ij}({\boldsymbol \beta}_{0})\big)^{-1}=2\sum_{i=1}^{n}\log\big(1+{\boldsymbol \xi}^{\tau}\eta_{ij}({\boldsymbol \beta}_{0})\big).\end{aligned} $$

Under (I ₁) and (I ₂), there exist a positive constant M and a function h ∈ L ^p, p ≥ 1 such that |φ(t)|≤ M for all t ∈ (0, 1), and ∥∇_β g(x _i, β ₀)∥≤ h(x _i), where ∥⋅∥ stands for the L ²- norm. From this, max_1≤i≤n∥∇_β g(x _i, β ₀)∥ = o _p(n ^1∕2) since E(|h(x _i)|^P) < ∞, p ≥ 1. Also, since $\varSigma _{jn}\varSigma _{jn}^{\tau }\to \varSigma _{{\boldsymbol \beta }_{0}}^{j}\;\; a.s.$, Σ _jn is almost surely bounded. Thus, ∥η _ij(β ₀)∥≤ M ×max_1≤i≤n h(x _i), which implies that

$$\displaystyle \begin{aligned} \max_{1\leq i\leq n}\|\eta_{ij}({\boldsymbol \beta}_{0})\|=o_{p}(n^{1/2})\quad and \quad \frac{1}{n}\sum_{i=1}^{n}\|\eta_{ij}({\boldsymbol \beta}_{0})\|{}^{3}=o_{p}(n^{1/2}).\end{aligned} $$

(3.6)

Moreover, ${\boldsymbol \varLambda }_{nj}=Var(\sqrt {n}S_{n}^{j}({\boldsymbol \beta }_{0}))=n^{-1}\sum _{i=1}^{n}\eta _{ij}({\boldsymbol \beta }_{0})\eta _{ij}^{\tau }({\boldsymbol \beta }_{0})=\varSigma _{{\boldsymbol \beta }_{0}}^{j}+o_{p}(1)$ by assumption (I ₆), from which $\varSigma _{{\boldsymbol \beta }_{0}}^{j}$ is assumed to be positive definite. Hence, following the proof of Lemma 3.1, we have $\varSigma _{jn}\varSigma _{jn}^{\tau }-{\boldsymbol \varLambda }_{nj}\to {0}\; a.s.$ Since $\sqrt {n}S_{n}^{j}({\boldsymbol \beta }_{0})\xrightarrow {\mathcal {D}}N(0,\varSigma _{{\boldsymbol \beta }_{0}}^{j})$, we have $\|S_{n}^{j}({\boldsymbol \beta }_{0})\|=O_{p}(n^{-1/2})$. Now from Eq. (3.4), using similar arguments as those in Owen (1990), it is obtained that ∥ξ∥ = O _p(n ^−1∕2). On the other hand, performing a Taylor expansion to the right-hand side of Eq. (3.5) results in

$$\displaystyle \begin{aligned}-2\log R_{n}^{j}({\boldsymbol \beta}_{0})=2\sum_{i=1}^{n}\Big[{\boldsymbol \xi}^{\tau}\eta_{ij}({\boldsymbol \beta}_{0})-\frac{1}{2}\big({\boldsymbol \xi}^{\tau}\eta_{ij}({\boldsymbol \beta}_{0})\big)^2\Big]+{\boldsymbol \gamma}_{n},\end{aligned} $$

where $\displaystyle {{\boldsymbol \gamma }_{n}=O_{P}(1)\sum _{i=1}^{n}|{\boldsymbol \xi }^{\tau }\eta _{ij}({\boldsymbol \beta }_{0})|{ }^{3}}$. Now, using similar arguments as in Owen (2001), we have

$$\displaystyle \begin{aligned} \begin{array}{rcl} -2\log R_{n}^{j}({\boldsymbol \beta}_{0})&\displaystyle =&\displaystyle \sum_{i=1}^{n}{\boldsymbol \xi}^{\tau}\eta_{ij}({\boldsymbol \beta}_{0})+o_{P}(1)\\ &\displaystyle =&\displaystyle \Big(\frac{1}{n}\sum_{i=1}^{n}\eta_{ij}({\boldsymbol \beta}_{0})\Big)^{\tau}(n{\boldsymbol \varLambda}_{nj})^{-1}\Big(\frac{1}{n}\sum_{i=1}^{n}\eta_{ij}({\boldsymbol \beta}_{0})\Big)+o_{p}(1)\\ &\displaystyle =&\displaystyle \Big(\sqrt{n}{\boldsymbol \varLambda}^{-1/2}_{nj}S_{n}^{j}({\boldsymbol \beta}_{0})\Big)^{\tau}\Big(\sqrt{n}{\boldsymbol \varLambda}^{-1/2}_{nj}S_{n}^{j}({\boldsymbol \beta}_{0})\Big)+ o_{p}(1). \end{array} \end{aligned} $$

Using Slutsky’s lemma, we have $\displaystyle {\sqrt {n}{\boldsymbol \varLambda }_{nj}^{-1/2}S_{n}^{j}({\boldsymbol \beta }_{0})\xrightarrow {\mathcal {D}} N_{p}(0, I_{p})}$ as n →∞, and therefore,