Extract various types of residuals from unit quantile regression models.
Arguments
- object
fitted model object of class
unitquantreg.- type
character indicating type of residuals. The options are
"quantile","cox-snell","working"and"partial".- ...
currently not used.
Value
Numeric vector of residuals extract from an object of class
unitquantreg.
Details
The residuals method can compute quantile
and Cox-Snell residuals. These residuals are defined, respectively, by
$$r_{Q} = \Phi^{-1}\left[ F(y_i \mid \widehat{\mu}_i, \widehat{\theta}_i)\right]$$
and
$$r_{CS} = -\log\left[1- F(y_i \mid \widehat{\mu}_i, \widehat{\theta}_i)\right]$$ where \(\widehat{\mu}_i\) and \(\widehat{\theta}_i\) are the fitted values of parameters \(\mu\) and \(\theta\), \(F(\cdot \mid \cdot, \cdot)\) is the cumulative distribution function (c.d.f.) and \(\Phi(\cdot)\) is the c.d.f. of standard Normal distribution.
Apart from the variability due the estimates of parameters,if the fitted regression model is correctly specified then the quantile residuals, \(r_Q\), follow a standard Normal distribution and the Cox-Snell residuals, \(r_{CS}\), follow a standard exponential distribution.