A. Lopatatzidis & Peter J. Green

In many statistical applications it is required to estimate the whole distribution of a response as it varies with covariates, and not just its expectation. A way to deal with this is to estimate quantiles of the response. One application of such a method would be to construct reference curves for biometrical or econometric measurements. A partially parametric approach based on penalized likelihood is described and then further developed. It is based on the LMS method by replacing the normality assumption imposed there with the gamma distribution. It is shown that gamma distribution provides a realistic alternative to the normal. The LMS parameters for the two distributions can be related, while both produce very similar centiles. Moreover, the gamma version of the LMS avoids a range of problems introduced by the parametrization of the normal version. It also provides a natural setting for generalizing the available methodology in order to include the whole spectrum of the continuous exponential family distributions, since the proposed method is essentially building on the generalized gamma distribution. Confidence intervals for quantiles are obtained using a bootstrap method and numerical algorithms are discussed. Also numerical examples are provided to illustrate the use of the proposed method.

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