Quantile regression is a type of regression analysis used in statistics and econometrics. Whereas the method of least squares estimates the conditional mean of the response variable across values of the predictor variables, quantile regression estimates the conditional median (or other quantiles) of the response variable. There is also a method for predicting the conditional geometric mean of the response variable,. Quantile regression is an extension of linear regression used when the conditions of linear regression are not met. It was introduced by Roger Koenker in 1978. As a complementary and extended approach to the least squares method, quantile regression addresses the limitations of least squares method in the presence of heteroscedasticity and ensures the robustness of quantile regression through its robustness to outliers, which compensates for the weakness of least squares method in dealing with outlier data.
Advantages and applications One advantage of quantile regression relative to ordinary least squares regression is that the quantile regression estimates are more robust against outliers in the response measurements. However, the main attraction of quantile regression goes beyond this and is advantageous when conditional quantile functions are of interest. Different measures of central tendency and statistical dispersion can be used to more comprehensively analyze the relationship between variables. In ecology, quantile regression has been proposed and used as a way to discover more useful predictive relationships between variables in cases where there is no relationship or only a weak relationship between the means of such variables. The need for and success of quantile regression in ecology has been attributed to the complexity of interactions between different factors leading to data with unequal variation of one variable for different ranges of another variable. Another application of quantile regression is in the areas of growth charts, where percentile curves are commonly used to screen for abnormal growth.
History The idea of estimating a median regression slope, a major theorem about minimizing sum of the absolute deviances and a geometrical algorithm for constructing median regression was proposed in 1760 by Ruđer Josip Bošković, a Jesuit Catholic priest from Dubrovnik. He was interested in the ellipticity of the earth, building on Isaac Newton's suggestion that its rotation could cause it to bulge at the equator with a corresponding flattening at the poles. He finally produced the first geometric procedure for determining the equator of a rotating planet from three observations of a surface feature. More importantly for quantile regression, he was able to develop the first evidence of the least absolute criterion and preceded the least squares introduced by Legendre in 1805 by fifty years. Other thinkers began building upon Bošković's idea such as Pierre-Simon Laplace, who developed the so-called "methode de situation." This led to Francis Edgeworth's plural median - a geometric approach to median regression - and is recognized as the precursor of the simplex method. The works of Bošković, Laplace, and Edgeworth were recognized as a prelude to Roger Koenker's contributions to quantile regression. Median regression computations for larger data sets are quite tedious compared to the least squares method, for which reason it has historically generated a lack of popularity among statisticians, until the widespread adoption of computers in the latter part of the 20th century.
Background: quantiles Quantile regression expresses the conditional quantiles of a dependent variable as a linear function of the explanatory variables. Crucial to the practicality of quantile regression is that the quantiles can be expressed as the solution of a minimization problem, as we will show in this section before discussing conditional quantiles in the next section.
Quantile of a random variable
Let Y {\displaystyle Y} be a real-valued random variable with cumulative distribution function F Y ( y ) = P ( Y ≤ y ) {\displaystyle F_{Y}(y)=P(Y\leq y)} . The τ {\displaystyle \tau } th quantile of Y is defined as
q Y ( τ ) := F Y − 1 ( τ ) := inf { y : F Y ( y ) ≥ τ } , {\displaystyle q_{Y}(\tau ):=F_{Y}^{-1}(\tau ):=\inf \left\{y:F_{Y}(y)\geq \tau \right\},}
where 0 < τ < 1 {\displaystyle 0<\tau <1} . Define the loss function as
… excerpt ends here. Continue reading the full article.
