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Quantile regression

Conventional linear regression assumes homogeneity of variance across the range of x-values. Often this assumption is clearly not met and the data points assume a triangular shape, with variance increasing as the value of X increases. This is sometimes best dealt with by a logarithmic transformation of the response variable. An alternative model is that there is an upper limiting factor, but that many observations lie below this level because of the impact of other unmeasured factors. This is the model adopted by quantile regression. captured by different quantiles

Whilst conventional linear regression gives estimates that approximate the mean of the response variable, quantile regression gives estimates that approximate its median and other quantiles. The estimates are semiparametric in the sense that no parametric distributional form (for example normal or Poisson), is assumed for the random error part of the mode1. Quantile regression functions are estimated by minimizing an asymmetrically-weighted sum of absolute residual errors. The upper quantile is a more appropriate representation of the limiting factor than the central estimate of a conventional regression.