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Biometrika 2005 92(3):543-557; doi:10.1093/biomet/92.3.543
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© 2005 Biometrika Trust

Smooth quantile ratio estimation

Francesca Dominici1, Leslie Cope2, Daniel Q. Naiman3 and Scott L. Zeger4

1 Department Biostatistics, Bloomberg School of Public Health, Johns Hopkins University, 615 N. Wolfe Street, Baltimore, Maryland 21205-2179, U.S.A. fdominic{at}jhsph.edu, 2 Department of Oncology, The Sidney Kimmel Comprehensive Cancer Center, Johns Hopkins University, 550 North Broadway, Baltimore, Maryland 21205-2011, U.S.A. cope{at}jhu.edu, 3 Department of Applied Mathematics and Statistics, The Whiting School of Engineering, Johns Hopkins University, 3400 North Charles Street, Baltimore, Maryland 21218-2682, U.S.A. daniel.naiman{at}jhu.edu, 4 Department Biostatistics, Bloomberg School of Public Health, Johns Hopkins University, 615 N. Wolfe Street, Baltimore, Maryland 21205-2179, U.S.A. szeger{at}jhsph.edu

We propose a novel approach to estimating the mean difference between two highly skewed distributions. The method, which we call smooth quantile ratio estimation, smooths, over percentiles, the ratio of the quantiles of the two distributions. The method defines a large class of estimators, including the sample mean difference, the maximum likelihood estimator under log-normal samples and the L-estimator. We derive asymptotic properties such as consistency and asymptotic normality, and also provide a closed-form expression for the asymptotic variance. In a simulation study, we show that smooth quantile ratio estimation has lower mean squared error than several competitors, including the sample mean difference and the log-normal parametric estimator in several realistic situations. We apply the method to the 1987 National Medicare Expenditure Survey to estimate the difference in medical expenditures between persons suffering from the smoking attributable diseases, lung cancer and chronic obstructive pulmonary disease, and persons without these diseases.

Key Words: Comparing means; Health expenditure; Log-normal; Order statistic; Q–Q plot; Regression spline; Smoking

Received November 2003. Revised January 2005.


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