, Volume 9, Issue 6, pp 419440
Computation of Bounds on Population Parameters When the Data Are Incomplete
 Joel L. HorowitzAffiliated withDepartment of Economics, Northwestern University
 , Charles F. ManskiAffiliated withDepartment of Economics, Northwestern University
 , Maria PonomarevaAffiliated withDepartment of Economics, Northwestern University
 , Jörg StoyeAffiliated withDepartment of Economics, Northwestern University
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This paper continues our research on the identification and estimation of statistical functionals when the sampling process produces incomplete data due to missing observations or interval measurement of variables. Incomplete data usually cause population parameters of interest in applications to be unidentified except under untestable and often controversial assumptions. However, it is often possible to identify sharp bounds on these parameters. The bounds are functionals of the population distribution of the available data and do not rely on untestable assumptions about the process through which data become incomplete. They contain all logically possible values of the population parameters. Moreover, every parameter value within the bounds is consistent with some model of the process that generates incomplete data. The bounds can be estimated consistently by replacing the population distribution of the data with the empirical distribution in the functionals that give the bounds. In practice, this is straightforward in some circumstances but computationally burdensome in others; in general, the bounds are the solutions to nonconvex mathematical programming problems that can be difficult to solve. Horowitz and Manski (Censoring of Outcomes and Regressors Due to Survey Nonresponse: Identification and Estimation Using Weights and Imputations, Journal of Econometrics 84 (1998), pp. 37–58; Nonparametric Analysis of Randomized Experiments with Missing Covariate and Outcome Data, Journal of the American Statistical Association 95 (2000), pp. 77–84) studied nonparametric mean regression with missing data. In this paper, we first describe the general problem. We then present new findings on the computation of bounds on best linear predictors under square loss. We describe a genetic algorithm to compute sharp bounds and a minimax approach yielding simple but nonsharp outer bounds. We use actual data to demonstrate the computations.
 Title
 Computation of Bounds on Population Parameters When the Data Are Incomplete
 Journal

Reliable Computing
Volume 9, Issue 6 , pp 419440
 Cover Date
 200312
 DOI
 10.1023/A:1025865520086
 Print ISSN
 13853139
 Online ISSN
 15731340
 Publisher
 Kluwer Academic Publishers
 Additional Links
 Topics
 Authors

 Joel L. Horowitz ^{(1)}
 Charles F. Manski ^{(1)}
 Maria Ponomareva ^{(1)}
 Jörg Stoye ^{(1)}
 Author Affiliations

 1. Department of Economics, Northwestern University, 2001 Sheridan Road, Evanston, IL, 602082600, USA