Partial Identification of Probability Distributions
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From the reviews:
"Charles Manski has produced a nice and compact text written with extreme care, providing technical detail, and mathematical proofs where needed." Biometrics, March 2005
"This book is an excellent and rigorous presentation of the state of research in the area of partial identification of populations and credible inference, in which the author has made many important contributions. . The overall quality of the book is very good. . The main part of each chapter is written in a textbook style. . Clearly, both methodology and the applications presented are intended to provide statisticians with a good foundation for further study in the subject . ." (Evdokia Xekalaki, Zentralblatt MATH, Vol. 1047 (22), 2004)
"I found the material very pertinent, departing, as it does, from the usual parametric approach in which the conclusions depend rather critically on the probability model adopted. Given a chance, it will make the traditionalist, like me, stop and think and perhaps, try to mend their ways a little. The main part of each chapter is written in textbook style, but fairly formally and rigorously . . At the end of each chapter appear 'Complements', giving examples and extensions, and 'Endnotes'. ." (M. J. Crowder, Short Book Reviews, Vol. 23 (3), 2003)
"This book, containing ten chapters, is the first comprehensive presentation of the theory of partial identification of probability distributions. It gives an overview of the research into this topic." (M. Riedel, Mathematical Reviews, Issue 2006 g)
"The book is carefully and thoughtfully written. Some chapters start with a cogent section on the "anatomy of the problem," and all end with complements addressing specific contexts." (Alan F. Kaar, Journal of the American Statistical Association, Vol. 102, No. 477, 2007)
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Charles F. Manski is Board of Trustees Professor at Northwestern University. He is author of Identification Problems in the Social Sciences and Analog Estimation Methods in Econometrics. He is a Fellow of the American Academy of Arts and Sciences, the American Association for the Advancement of Science, and the Econometric Society.
Content
Statistical inference uses sample data to draw conclusions about a population of interest. However, data alone do not suffice. Inference always requires assumptions about the population and the sampling process. Statistical theory illuminates the logic of inference by showing how data and assumptions combine to yield conclusions.
Empirical researchers should be concerned with both the logic and the credibility of their inferences. Credibility is a subjective matter, yet I take there to be wide agreement on a principle that I shall call:
The Law of Decreasing Credibility: The credibility of inference decreases with the strength of the assumptions maintained.
This principle implies that empirical researchers face a dilemma as they decide what assumptions to maintain: Stronger assumptions yield inferences that may be more powerful but less credible. Statistical theory cannot resolve the dilemma but can clarify its nature.
It is useful to distinguish combinations of data and assumptions that point-identify a population parameter of interest from ones that place the parameter within a set-valued identification region. Point identification is the fundamental necessary condition for consistent point estimation of a parameter. Strengthening an assumption that achieves point identification may increase the attainable precision of estimates of the parameter. Statistical theory has had much to say about this matter. The classical theory of local asymptotic efficiency characterizes, through the Fisher information matrix, how attainable precision increases as more is assumed known about a population distribution. Nonparametric regression analysis shows how the attainable rate of convergence of estimates increases as more is assumed about the shape of the regression. These and other achievements provide important guidance to empirical researchers as they weigh the credibility and precision of alternative point estimates.
Statistical theory has had much less to say about inference on population parameters that are not point-identified (see the historical note at the end of this Introduction). It has been commonplace to think of identification as a binary event - a parameter is either identified or it is not - and to view point identification as a precondition for meaningful inference. Yet there is enormous scope for fruitful inference using data and assumptions that partially identify population parameters. This book explains why and shows how.
Origin and Organization of the Book
The book has its roots in my research on nonparametric regression analysis with missing outcome data, initiated in the late 1980s. Empirical researchers estimating regressions commonly assume that missingness is random, in the sense that the observability of an outcome is statistically independent of its value. Yet this and other point-identifying assumptions have regularly been criticized as implausible. So I set out to determine what random sampling with partial observability of outcomes reveals about mean and quantile regressions if nothing is known about the missingness process or if assumptions weak enough to be widely credible are imposed. The findings were sharp bounds whose forms vary with the regression of interest and with the maintained assumptions. These bounds can readily be estimated using standard methods of nonparametric regression analysis.
Study of regression with missing outcome data stimulated investigation of more general incomplete data problems. Some sample realizations may have unobserved outcomes, some may have unobserved covariates, and others may be entirely missing. Sometimes interval data on outcomes or covariates are available, rather than point measurements. Random sampling with incomplete observation of outcomes and covariates generically yields partial identification of regressions. The challenge is to describe and estimate the identification regions produced by incomplete-data processes when alternative assumptions are maintained. researchers estimating regressions commonly assume that missingness is random, in the sense that the observability of an outcome is statistically independent of its value. Yet this and other point-identifying assumptions have regularly been criticized as implausible. So I set out to determine what random sampling with partial observability of outcomes reveals about mean and quantile regressions if nothing is known about the missingness process or if assumptions weak enough to be widely credible are imposed. The findings were sharp bounds whose forms vary with the regression of interest and with the maintained assumptions. These bounds can readily be estimated using standard methods of nonparametric regression analysis.
