

   FFiitt aa GGeenneerraalliizzeedd NNoonnlliinneeaarr RReeggrreessssiioonn MMooddeell ffoorr TThhrreeee
   PPaarraammeetteerr DDiissttrriibbuuttiioonnss

        gnlr3(y, dist="normal", mu=NULL, shape=NULL, family=NULL,
             linear=NULL, pmu=NULL, pshape=NULL, pfamily=NULL, censor=F,
             exact=F, wt=1, delta=1, common=F, envir=sys.frame(sys.parent()),
             print.level=0, typsiz=abs(p), ndigit=10, gradtol=0.00001,
             stepmax=10*sqrt(p%*%p), steptol=0.00001, iterlim=100, fscale=1)

   AArrgguummeennttss::

          y: The response vector for uncensored data, two
             columns for censored data, with the second being
             the censoring indicator (1: uncensored, 0: right
             censored, -1: left censored.), or an object of
             class, response (created by `restovec') or
             repeated (created by `rmna').

   distribution: Either a character string containing the name
             of the distribution or a function giving the -log
             likelihood and calling the location and shape
             functions.

         mu: A user-specified function of `pmu', and possibly
             `linear', giving the regression equation for the
             location. This may contain a linear part as the
             second argument to the function.  It may also be a
             formula beginning with ~, specifying either a lin-
             ear regression function for the location parameter
             in the Wilkinson and Rogers notation or a general
             function with named unknown parameters. If none is
             supplied, the location is taken to be constant
             unless the linear argument is given.

      shape: A user-specified function of `pshape', and possi-
             bly `linear', giving the regression equation for
             the dispersion or shape parameter. This may con-
             tain a linear part as the second argument to the
             function. It may also be a formula beginning with
             ~, specifying either a linear regression function
             for the shape parameter in the Wilkinson and
             Rogers notation or a general function with named
             unknown parameters. If none is supplied, this
             parameter is taken to be constant unless the lin-
             ear argument is given. This parameter is the loga-
             rithm of the usual one.

     family: A user-specified function of `pfamily', and possi-
             bly `linear', for the regression equation of the
             third (family) parameter of the distribution. This
             may contain a linear part that is the second argu-
             ment to the function. It may also be a formula
             beginning with ~, specifying either a linear
             regression function for the family parameter in
             the Wilkinson and Rogers notation or a general
             function with named unknown parameters. If neither
             is supplied, this parameter is taken to be con-
             stant unless the linear argument is given. In most
             cases, this parameter is the logarithm of the
             usual one.

     linear: A formula beginning with ~, specifying the linear
             part of the regression function for the location
             parameters or list of three such expressions for
             the location, shape, and/or family parameters.

        pmu: Vector of initial estimates for the location
             parameters.  If `mu' is a formula with unknown
             parameters, their estimates must be supplied
             either in their order of appearance in the expres-
             sion or in a named list.

     pshape: Vector of initial estimates for the shape parame-
             ters.  If `shape' is a formula with unknown param-
             eters, their estimates must be supplied either in
             their order of appearance in the expression or in
             a named list.

    pfamily: Vector of initial estimates for the family parame-
             ters.  If `family' is a formula with unknown
             parameters, their estimates must be supplied
             either in their order of appearance in the expres-
             sion or in a named list.

      exact: If TRUE, fits the exact likelihood function for
             continuous data by integration over intervals of
             observation, i.e. interval censoring.

         wt: Weight vector.

      delta: Scalar or vector giving the unit of measurement
             (always one for discrete data) for each response
             value, set to unity by default - for example, if a
             response is measured to two decimals, delta=0.01.
             If the response is transformed, this must be mul-
             tiplied by the Jacobian. The transformation cannot
             contain unknown parameters. For example, with a
             log transformation, `delta=1/y'. (The delta values
             for the censored response are ignored.)

     common: If TRUE, at least two of `mu', `shape', and `fam-
             ily' must both be functions with, as argument, a
             vector of parameters having some or all elements
             in common between them so that indexing is in com-
             mon between them; all parameter estimates must be
             supplied in `pmu'. If FALSE, parameters are dis-
             tinct between the two functions and indexing
             starts at one in each function.

      envir: Environment in which model formulae are to be
             interpreted or a data object of class, repeated,
             tccov, or tvcov.  If `y' has class `repeated', it
             is used as the environment.

     others: Arguments controlling `nlm'.

   DDeessccrriippttiioonn::

        `gnlr3' fits user specified nonlinear regression equa-
        tions to one, two, or all three parameters of three
        parameter distributions (Box-Cox transformed normal,
        generalized inverse Gauss, generalized logistic,
        Hjorth, generalized gamma, Burr, generalized Weibull,
        power exponential, Student t, and generalized extreme
        value).

        Nonlinear regression models can be supplied as formulae
        where parameters are unknowns. Factor variables cannot
        be used and parameters must be scalars. (See `fin-
        terp'.)

   VVaalluuee::

        A list of class gnlr is returned.  The printed output
        includes the -log likelihood (not the deviance), the
        corresponding AIC, the maximum likelihood estimates,
        standard errors, and correlations. A list is returned
        that contains all of the relevant information calcu-
        lated, including error codes.

   AAuutthhoorr((ss))::

        J.K. Lindsey

   SSeeee AAllssoo::

        `fmr', `finterp', `glm', `gnlr', `lm'.

   EExxaammpplleess::

        y <- rgamma(20,2,1)
        sex <- c(rep(0,10),rep(1,10))
        sexf <- gl(2,10)
        age <- rpois(20,10)
        # linear regression with the generalized gamma distribution
        mu <- function(p) p[1]+p[2]*sex+p[3]*age
        gnlr3(y, dist="gamma", mu=mu, pmu=rep(1,3), pshape=0, pfamily=0)
        # or equivalently
        gnlr3(y, dist="gamma", mu=~sexf+age, pmu=rep(1,3),
             pshape=0, pfamily=0)
        # or
        gnlr3(y, dist="gamma", linear=~sex+age, pmu=rep(1,3),
             pshape=0, pfamily=0)
        # or
        gnlr3(y, dist="gamma", mu=~b0+b1*sex+b2*age,
             pmu=list(b0=1,b1=1,b2=1), pshape=0, pfamily=0)
        #
        # nonlinear regression with generalized gamma distribution
        mu <- function(p, linear) p[1]+exp(linear)
        gnlr3(y, dist="gamma", mu=mu, linear=~sex+age, pmu=rep(1,4),
             pshape=0, pfamily=0)
        # or equivalently
        gnlr3(y, dist="gamma", mu=~b4+exp(b0+b1*sex+b2*age),
             pmu=list(b0=1,b1=1,b2=1,b4=1), pshape=0, pfamily=0)
        #
        # include regression for the shape parameter with same mu function
        shape <- function(p) p[1]+p[2]*sex+p[3]*age
        gnlr3(y, dist="gamma", mu=mu, linear=~sexf+age, shape=shape,
             pmu=rep(1,4), pshape=rep(0,3), pfamily=0)
        # or equivalently
        gnlr3(y, dist="gamma", mu=mu, linear=list(~sexf+age,~sex+age,NULL),
             pmu=rep(1,4), pshape=rep(0,3), pfamily=0)
        # or
        gnlr3(y, dist="gamma", mu=mu, linear=~sexf+age,
             shape=~c0+c1*sex+c2*age, pmu=rep(1,4),
             pshape=list(c0=0,c1=0,c2=0), pfamily=0)
        # include regression for the family parameter with same mu
        # and shape functions
        family <- function(p) p[1]+p[2]*sex+p[3]*age
        gnlr3(y, dist="gamma", mu=mu, linear=~sexf+age, shape=shape,
             family=shape, pmu=rep(1,4), pshape=rep(0,3), pfamily=rep(0,3))
        # or equivalently
        gnlr3(y, dist="gamma", mu=mu,
             linear=list(~sex+age,~sex+age,~sex+age), pmu=rep(1,4),
             pshape=rep(0,3), pfamily=rep(0,3))
        # or
        gnlr3(y, dist="gamma", mu=~b4+exp(b0+b1*sex+b2*age),
             shape=~c0+c1*sex+c2*age, family=~d0+d1*sex+d2*age,
             pmu=list(b0=1,b1=1,b2=1,b4=1), pshape=list(c0=0,c1=0,c2=0),
             pfamily=list(d0=0,d1=0,d2=0))

