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Abstract : |
In observational studies subjects may self select, thereby creating a biased sample. Such problems arise frequently, for example, in astronomical, biomedical, animal, and oil studies, survey sampling and econometrics. For a typical subject, let Y denote the value of interest and suppose that Y has an unknown density function f. Further, let w(y) denote the probability that the subject includes itself in the study given Y = y. Then the conditional density of Y given that it is observed is f (y) = w(y)f(y)=, where is a normalizing constant. The problem of estimating w and f from a biased sample X 1; : : : ; X n independently from f is considered when f is known to belong to a parametric family, say f = f ` , where ` is a vector of unknown parameters, and w is assumed to be non-decreasing. An algorithm for computing the maximum likelihood estimator of (w; `) is developed, and consistency is established. Simulations are used to show that our method is feasible with moderate sample size, and applications to animal and oil data are given., |
