\name{Lehmann}
\alias{Lehmann}
\title{
  Proportional Reverse Time Hazards Linear Regression
}
\description{
  Non-normal linear regression for Lehmann-alternatives
}
\usage{
Lehmann(formula, data, subset, weights, offset, cluster, na.action = na.omit, ...)
}
\arguments{
  \item{formula}{an object of class \code{"formula"}: a symbolic description 
          of the model structure to be
          fitted.  The details of model specification are given under
          \code{\link{tram}} and in the package vignette.
  }
  \item{data}{an optional data frame, list or environment (or object
          coercible by \code{as.data.frame} to a data frame) containing the
          variables in the model.  If not found in \code{data}, the
          variables are taken from \code{environment(formula)}.
  }
  \item{subset}{an optional vector specifying a subset of observations to be
          used in the fitting process.
  }
  \item{weights}{an optional vector of weights to be used in the fitting
          process.  Should be \code{NULL} or a numeric vector. If present,
          the weighted log-likelihood is maximised.
  }
  \item{offset}{this can be used to specify an _a priori_ known component to
          be included in the linear predictor during fitting.  This
          should be \code{NULL} or a numeric vector of length equal to the
          number of cases.
  }
  \item{cluster}{optional factor with a cluster ID employed for computing
          clustered covariances.
  }
  \item{na.action}{a function which indicates what should happen when the data
          contain \code{NA}s.  The default is set to \code{na.omit}.
  }
  \item{\dots}{additional arguments to \code{\link{tram}}.}
}
\details{
This transformation model uses the cumulative distribution function for the
standard Gumbel maximum extreme value distribution to map the shifted
transformation function into probabilities. The exponential of the shift
paramater can be interpreted as a Lehmann-alternative or 
reverse time hazard ratio.
}
\value{
An object of class \code{Lehmann}, with corresponding \code{coef},
\code{vcov}, \code{logLik}, \code{estfun}, \code{summary}, 
\code{print}, \code{plot} and \code{predict} methods.
}
\references{

  Erich L. Lehmann (1953), The Power of Rank Tests,
  \emph{The Annals of Mathematical Statistics}, \bold{24}(1),
  23-43.

  Torsten Hothorn, Lisa Moest, Peter Buehlmann (2018), Most Likely
  Transformations, \emph{Scandinavian Journal of Statistics}, \bold{45}(1),
  110--134, \doi{10.1111/sjos.12291}.

}
\examples{

  data("BostonHousing2", package = "mlbench")

  lm(cmedv ~ crim + zn + indus + chas + nox + rm + age + dis + 
             rad + tax + ptratio + b + lstat, data = BostonHousing2)

  Lehmann(cmedv ~ chas + crim + zn + indus + nox + 
                  rm + age + dis + rad + tax + ptratio + b + lstat, 
                  data = BostonHousing2)
}
\keyword{models}
\keyword{regression}
\keyword{smooth}
