Empirical likelihood confidence bands for distribution functions with missing responses

Wang, Qihua; Qin, Yongsong

doi:10.1016/j.jspi.2010.03.044

Cited by 19 publications

(10 citation statements)

References 18 publications

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“…This is a rather standard assumption in the literature on missing data; see, for example, Wang and Qin (2010) and Cheng and Chu (1996). The following result is an immediate consequence of the main theorem of Devroye and Krzyzak (1989).…”

Section: The Proposed Kernel Regression Estimatormentioning

confidence: 99%

Kernel regression estimation for incomplete data with applications

Mojirsheibani

Reese

2015

Stat Papers

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Methods are proposed to construct kernel estimators of a regression function in the presence of incomplete data. Furthermore, exponential upper bounds are derived on the performance of the L p norms of the proposed estimators, which can then be used to establish various strong convergence results. The presence of incomplete data points are handled by a Horvitz-Thompson-type inverse weighting approach, where the unknown selection probabilities are estimated by both kernel regression and least-squares methods. As an immediate application of these results, the problem of nonparametric classification with partially observed data will be studied.

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Section: The Proposed Kernel Regression Estimatormentioning

confidence: 99%

Kernel regression estimation for incomplete data with applications

Mojirsheibani

Reese

2015

Stat Papers

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“…Let

p_{1}, p_{2}, \dots, p_{n}

be nonnegative numbers with total sum being unity. Hence, the empirical log‐likelihood function for

F false(y false)

at any fixed point y , evaluated at

θ

, is defined by

ℓ_{n} false(θ, y; γ false) = - 2 max \{\sum_{i = 1}^{n} log ({italicnp}_{i}) | p_{i} \geq 0, \sum_{i = 1}^{n} p_{i} = 1, \sum_{i = 1}^{n} p_{i} {g_{i} (y; γ) - θ} = 0\} .

As pointed out in Wang & Qin (), the empirical log‐likelihood function

ℓ_{n} false(θ, y; γ false)

cannot be immediately applied to inference for

θ

ℓ_{n} false(θ, y; γ false)

contains unknown terms such as

γ

and

F_{0} false(y false| X_{i} false)

. In other words,

θ

is not identifiable.…”

Section: Empirical Likelihood Estimationmentioning

confidence: 99%

“…For example, Kaplan & Meier () presented the well‐known Kaplan‐Meier estimator of distribution function for censored data; Groeneboom & Wellner () proposed a nonparametric maximum likelihood estimator of distribution function for interval censored data; and Cheng & Chu () presented a distribution‐free imputation estimator using nonparametric kernel regression. Recently, Wang & Qin () discussed estimation of the distribution function of a response variable with missing data based on an augmented inverse probability weighted empirical log‐likelihood function. They showed that the augmented inverse probability weighted estimator converges weakly to a zero‐mean Gaussian process and the empirical log‐likelihood ratio function converges weakly to the square of a Gaussian process with mean zero and variance one.…”

Section: Introductionmentioning

confidence: 99%

Robust estimation of distribution functions and quantiles with non‐ignorable missing data

Zhao

Tang

2013

Can J Statistics

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This paper considers several robust estimators for distribution functions and quantiles of a response variable when some responses may not be observed under the non‐ignorable missing data mechanism. Based on a particular semiparametric regression model for non‐ignorable missing response, we propose a nonparametric/semiparametric estimation method and an augmented inverse probability weighted imputation method to estimate the distribution function and quantiles of a response variable. Under some regularity conditions, we derive asymptotic properties of the proposed distribution function and quantile estimators. Two empirical log‐likelihood functions are also defined to construct confidence intervals for distribution function of a response variable. Simulation studies show that our proposed methods are robust. In particular, the semiparametric estimator is more efficient than the nonparametric estimator, and the inverse probability weighted imputation estimator is bias‐corrected. The Canadian Journal of Statistics 41: 575–595; 2013 © 2013 Statistical Society of Canada

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“…This inspires the development of some approaches, including the imputation, inverse probability weighting and doubly robust methods. See, for example, Rosenbaum and Rubin (1983), Hahn (1998), Hirano et al (2003), Cao et al (2009), Rotnitzky et al (2012), Firpo (2007), Wang and Qin (2010), Hu et al (2011) , Zhang et al (2011), and Markus and Blaise (2013). Many early literature established asymptotic theory on estimation and inference problem with missing responses in the classical setting where the dimension of covariable vector is a constant.…”

Section: Introductionmentioning

confidence: 99%

A Convex Programming Solution Based Debiased Estimator for Quantile with Missing Response and High-dimensional Covariables

Wang

2020

Preprint

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This paper is concerned with the estimating problem of response quantile with high dimensional covariates when response is missing at random. Some existing methods define root-n consistent estimators for the response quantile. But these methods require correct specifications of both the conditional distribution of response given covariates and the selection probability function. In this paper, a debiased method is proposed by solving a convex programming. The estimator obtained by the proposed method is asymptotically normal given a correctly specified parametric model for the condition distribution function, without the requirement to specify and estimate the selection probability function. Moreover, the proposed estimator is asymptotically more efficient than the existing estimators. The proposed method is evaluated by a simulation study and is illustrated by a real data example.

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Empirical likelihood confidence bands for distribution functions with missing responses

Cited by 19 publications

References 18 publications

Kernel regression estimation for incomplete data with applications

Kernel regression estimation for incomplete data with applications

Robust estimation of distribution functions and quantiles with non‐ignorable missing data

A Convex Programming Solution Based Debiased Estimator for Quantile with Missing Response and High-dimensional Covariables

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