

   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

        gnlr(y, dist="normal", pmu=NULL, pshape=NULL, mu=NULL, shape=NULL,
        linear=NULL, exact=F, wt=1, delta=1, shfn=F,
        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: A response vector for uncensored data or a two
             column matrix for binomial data. For censored
             data, two columns with the second being the cen-
             soring indicator (1: uncensored, 0: right cen-
             sored, -1: left censored.) It may also be an
             object of class, response.

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

        pmu: Vector of initial estimates for the location
             parameters.

     pshape: Vector of initial estimates for the shape parame-
             ters.

         mu: 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
             language expression beginning with ~, specifying a
             linear regression function for the location param-
             eter. If neither is supplied, the location is
             taken to be constant unless the linear argument is
             given.

      shape: User-specified function of `pshape', and possibly
             `linear' and/or `mu', giving the regression equa-
             tion for the dispersion or shape parameter. This
             may contain a linear part as the second argument
             to the function and the location as last argument
             (in which case `shfn' must be set to TRUE). It may
             also be a language expression beginning with ~,
             specifying a linear regression function for the
             shape parameter. If neither 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.

     linear: Language expression beginning with ~, specifying
             the linear part of the regression function for the
             location parameter or list of two such expressions
             for the location and/or shape parameters.

      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.)

       shfn: If true, the supplied shape function depends on
             the location (function). The name of this location
             function must be the last argument of the shape
             function.

     others: Arguments controlling `nlm'.

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

        `gnlr' fits user-specified nonlinear regression equa-
        tions to one or both parameters of the common one and
        two parameter distributions (binomial, beta binomial,
        double binomial, Poisson, negative binomial, double
        Poisson, Consul generalized Poisson, logarithmic
        series, geometric, normal, inverse Gauss, logistic,
        exponential, gamma, Weibull, extreme value, Cauchy,
        Student t, and Laplace; all but the binomial and beta
        binomial may be right and/or left censored). A user-
        specified -log likelihood can also be supplied for the
        distribution.

   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.

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

        # linear regression with inverse Gauss distribution
        mu <- function(p) p[1]+p[2]*sex+p[3]*age
        gnlr(data, dist="inverse Gauss", pmu=rep(1,3), psh=1, mu=mu)
        # or equivalently
        gnlr(data, dist="inverse Gauss", pmu=rep(1,3), psh=1, mu=~sex+age)
        # or
        gnlr(data, dist="inverse Gauss", pmu=rep(1,3), psh=1,
             linear=~sex+age)
        #
        # nonlinear regression with inverse Gauss distribution
        mu <- function(p, linear) p[4]+exp(linear)
        gnlr(data, dist="inverse Gauss", pmu=rep(1,4), psh=1, mu=mu,
             linear=~sex+age)
        # one explicit parameter in mu, three in linear, one for shape
        #
        # include regression for the shape parameter with same mu function
        shape <- function(p) p[1]+p[2]*sex+p[3]*age
        gnlr(data, dist="inverse Gauss", pmu=rep(1,4), psh=rep(1,3), mu=mu,
             shape=shape)
        # or equivalently
        gnlr(data, dist="inverse Gauss", pmu=rep(1,4), psh=rep(1,3), mu=mu,
             linear=~sex+age, shape=~sex+age)
        # or
        gnlr(data, dist="inverse Gauss", p=rep(1,7), mu=mu,
             linear=list(~sex+age,~sex+age))
        # shape as a function of the mean
        shape <- function(p, mu) p[1]+p[2]*sex+p[3]*mu
        gnlr(data, dist="inverse Gauss", pmu=rep(1,4), psh=rep(1,3), mu=mu,
             shape=shape, linear=~sex+age, shfn=T)

