Abstract
We study efficient nonparametric estimation of distribution functions of several scientifically meaningful sub-populations from data consisting of mixed samples where the sub-population identifiers are missing. Only probabilities of each observation belonging to a sub-population are available. The problem arises from several biomedical studies such as quantitative trait locus (QTL) analysis and genetic studies with ungenotyped relatives where the scientific interest lies in estimating the cumulative distribution function of a trait given a specific genotype. However, in these studies subjects' genotypes may not be directly observed. The distribution of the trait outcome is therefore a mixture of several genotype-specific distributions. We characterize the complete class of consistent estimators which includes members such as one type of nonparametric maximum likelihood estimator (NPMLE) and least squares or weighted least squares estimators. We identify the efficient estimator in the class that reaches the semiparametric efficiency bound, and we implement it using a simple procedure that remains consistent even if several components of the estimator are mis-specified. In addition, our close inspections on two commonly used NPMLEs in these problems show the surprising results that the NPMLE in one form is highly inefficient, while in the other form is inconsistent. We provide simulation procedures to illustrate the theoretical results and demonstrate the proposed methods through two real data examples.
Highlights
AMS 2000 subject classifications: Primary 62G05, 62G20; secondary 62G99
We provide nonparametric estimation in the sense that we do not make any distributional assumption on the conditional distributions
Comparing φeff with φOW LS obtained in Appendix A.2, we find that the optimal WLS (OWLS) is optimal among the WLS family, it does not reach the semiparametric efficiency bound
Summary
The traditional approach to estimating F (t) is maximum likelihood estimator for a parametric model or NPMLE for a nonparametric model, a very simple weighted estimator can be used if we formulate the same problem from a different angle. Viewing the qi’s as covariates and I(Si ≤ t) as response variables, the covariates and the responses are linked by F (t) via a familiar linear regression model. Denote by M an arbitrary n × n diagonal matrix. En)T ∈ Rn. we obtain the general WLS estimator. The simplest estimator is the OLS where we set M = In, derived in Fine et al (2004) using a different formulation, while the most efficient WLS estimator is obtained when we assign M to be a diagonal matrix with the ith diagonal entry equals vi−1. Standard iteratively re-weighted estimation procedure can be used to obtain this optimal WLS (OWLS) estimator. The presence of the matrix M allows the flexibility to derive other WLS estimators to achieve desired properties such as robustness
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