Abstract : A procedure for estimating the parameters of a quantile regression function is investigated. The procedure is based on the work of Parzen (1979a) in the theory of quantile functions and is applicable to a wide range of distributional families. The procedure assumes the quantile functions of k populations to be location-scale shifts of a common quantile function. First, a goodness-of-fit procedure for determining the common distributional shape of the k populations generalizes the one-population data modeling techniques of Parzen (1979a). An estimator of the shape parameter of a distribution is also investigated. The methods of Ogawa (1951) and Eubank (1979) are then used for estimating the location and scale parameters of the k populations. A regression model for the location and scale parameters is specified, and the resulting estimators of the regression parameters are used to determine a regression function for any quantiles of the observed data. Finally it is shown that inferences about the quantile relationships can be based on the asymptotic normality of the estimated parameters. The procedures are applied to some published data sets.