Computation of Bounds on Population Parameters When the Data Are Incomplete
Rent the article at a discountRent now
* Final gross prices may vary according to local VAT.Get Access
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 non-convex 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 min-imax approach yielding simple but non-sharp outer bounds. We use actual data to demonstrate the computations.
- Andrews, D. W. K.: Inconsistency of the Bootstrap When a Parameter Is on the Boundary of the Parameter Space, Econometrica 68 (2000), pp. 399–405.
- Bickel, P. J. and Freedman, D.A.: Some Asymptotic Theory for the Bootstrap,Annals of Statistics 9 (1981), pp. 1196–1217.
- Fischhoff, B. and Bruine deBruin, W. Fifty/Fifty = 50%?, Journal of BehavioralDecision Making 12 (1999), pp. 149–163.
- Heckman, J.: The Common Structure of Statistical Models of Truncation, Sample Selection, and Limited Dependent Variables and a Simple Estimator for Such Models, Annals of Economic and Social Measurement 5 (1976), pp. 479–492.
- Horowitz, J. L. and Manski, C. F.: Censoring of Outcomes and Regressors Due to Survey Nonresponse: Identification and Estimation Using Weights and Imputations, Journal of Econometrics 84 (1998), pp. 37–58.
- Horowitz, J. L. and Manski, C. F.: Imprecise Identification fromIncomplete Data, in: Proceedings of the Second International Symposium on Imprecise Probabilities and Their Applications, 2001, http://ippserv.rug.ac.be/″isipta01/proceedings/index.html.
- Horowitz, J. L. and Manski, C. F.: Nonparametric Analysis of Randomized Experiments with Missing Covariate and Outcome Data, Journal of the American Statistical Association 95 (2000), pp. 77–84.
- Imbens, G. and Manski, C.: Confidence Intervals for Partially Identified Parameters, Department of Economics, University of California at Berkeley, 2003, in process.
- Judd, K. L.: Numerical Methods in Economics, The MIT Press, Cambridge, 1998.
- Manski, C. F.: Identification Problems in the Social Sciences, Harvard University Press, Cambridge, 1995.
- Manski, C. F. and Straub, J.: Worker Perceptions of Job Insecurity in the Mid-1990s: Evidence fromthe Survey of Economic Expectations, Journal ofHuman Resources 35 (2000), pp. 447–479.
- Manski, C. F. and Tamer, E.: Inference on Regressions with Interval Data on a Regressor or Outcome, Econometrica 70 (2002), pp. 519–546.
- Myrveit, I., Stensrud, E., and Olsson, U.: Analysing Data Sets with Missing Data: An Empirical Evaluation of Imputation Methods and Likelihood-Based Methods, IEEE Transactions on Software Engineering 27 (2001), pp. 999–1013.
- Politis, D. N. and Romano, J. P.: Large Sample Confidence Regions Based on Subsamples under Minimal Assumptions, Annals of Statistics 22 (1994), pp. 2031–2050.
- Rosenbaum, P.: Observational Studies, Springer-Verlag, New York, 1995.
- Rubin, D.: Inference and Missing Data, Biometrika 63 (1976), pp. 581–590.
- Scharfstein, D., Rotnitzky, A., and Robins, J.: Adjusting for Nonignorable Drop-Out Using Semiparametric NonresponseModels, Journal of the American Statistical Association 94 (1999), pp. 1096–1120.
- Vansteelandt, S. and Goetghebeur, E.: Analyzing the Sensitivity of Generalized Linear Models to Incomplete Outcomes Via the IDEAlgorithm, Journal of Computational and Graphical Statistics 10 (2001), pp. 656–672.
- Zaffalon, M.: Exact Credal Treatment of Missing Data, Journal of Statistical Planning and Inference 105 (2002), pp. 105–122.
- Computation of Bounds on Population Parameters When the Data Are Incomplete
Volume 9, Issue 6 , pp 419-440
- Cover Date
- Print ISSN
- Online ISSN
- Kluwer Academic Publishers
- Additional Links