amlpoisson               package:VGAM               R Documentation

_P_o_i_s_s_o_n _R_e_g_r_e_s_s_i_o_n _b_y _A_s_y_m_m_e_t_r_i_c _M_a_x_i_m_u_m _L_i_k_e_l_i_h_o_o_d _E_s_t_i_m_a_t_i_o_n

_D_e_s_c_r_i_p_t_i_o_n:

     Poisson quantile regression estimated by maximizing an asymmetric
     likelihood function.

_U_s_a_g_e:

     amlpoisson(w.aml=1, parallel=FALSE, method.init=1, digw=4,
                link="loge", earg=list())

_A_r_g_u_m_e_n_t_s:

     In this documentation the word _quantile_ can often be
     interchangeably replaced by _expectile_ (things are informal
     here).

   w.aml: Numeric, a vector of positive constants controlling the
          percentiles. The larger the value the larger the fitted
          percentile value (the proportion of points below the
          ``w-regression plane''). The default value of unity results
          in the ordinary maximum likelihood (MLE) solution.

parallel: If 'w.aml' has more than one value then this argument allows
          the quantile curves to differ by the same amount as a
          function of the covariates. Setting this to be 'TRUE' should
          force the quantile curves to not cross (although they may not
          cross anyway). See 'CommonVGAMffArguments' for more
          information.

method.init: Integer, either 1 or 2 or 3. Initialization method. Choose
          another value if convergence fails.

   digw : Passed into 'Round' as the 'digits' argument for the 'w.aml'
          values; used cosmetically for labelling.

link, earg: See 'poissonff'.

_D_e_t_a_i_l_s:

     This method was proposed by Efron (1992) and full details can be
     obtained there. The model is essentially a Poisson regression
     model (see 'poissonff') but the usual deviance is replaced by an
     asymmetric squared error loss function; it is multiplied by w.aml
     for positive residuals. The solution is the set of regression
     coefficients that minimize the sum of these deviance-type values
     over the data set, weighted by the 'weights' argument (so that it
     can contain frequencies). Newton-Raphson estimation is used here.

_V_a_l_u_e:

     An object of class '"vglmff"' (see 'vglmff-class'). The object is
     used by modelling functions such as 'vglm' and 'vgam'.

_W_a_r_n_i_n_g:

     If 'w.aml' has more than one value then the value returned by
     'deviance' is the sum of all the (weighted) deviances taken over
     all the 'w.aml' values. See Equation (1.6) of Efron (1992).

_N_o_t_e:

     On fitting, the 'extra' slot has list components '"w.aml"' and
     '"percentile"'. The latter is the percent of observations below
     the ``w-regression plane'', which is the fitted values.  Also, the
     individual deviance values corresponding to each element of the
     argument 'w.aml' is stored in the 'extra' slot.

     For 'amlpoisson' objects, methods functions for the generic
     functions 'qtplot' and 'cdf' have not been written yet.

     About the jargon, Newey and Powell (1987) used the name
     _expectiles_ for regression surfaces obtained by asymmetric least
     squares. This was deliberate so as to distinguish them from the
     original _regression quantiles_ of Koenker and Bassett (1978).
     Efron (1991) and Efron (1992) use the general name _regression
     percentile_ to apply to all forms of asymmetric fitting. Although
     the asymmetric maximum likelihood method very nearly gives
     regression percentiles in the strictest sense for the normal and
     Poisson cases, the phrase _quantile regression_ is used loosely in
     this 'VGAM' documentation.

_A_u_t_h_o_r(_s):

     Thomas W. Yee

_R_e_f_e_r_e_n_c_e_s:

     Efron, B. (1991) Regression percentiles using asymmetric squared
     error loss. _Statistica Sinica_, *1*, 93-125.

     Efron, B. (1992) Poisson overdispersion estimates based on the
     method of asymmetric maximum likelihood. _Journal of the American
     Statistical Association_, *87*, 98-107.

     Koenker, R. and Bassett, G. (1978) Regression quantiles.
     _Econometrica_, *46*, 33-50.

     Newey, W. K. and Powell, J. L. (1987) Asymmetric least squares
     estimation and testing. _Econometrica_, *55*, 819-847.

_S_e_e _A_l_s_o:

     'amlnormal', 'amlbinomial', 'alaplace1'.

_E_x_a_m_p_l_e_s:

     set.seed(1234)
     mydat = data.frame(x = sort(runif(n <- 200)))
     mydat = transform(mydat, y = rpois(n, exp(0-sin(8*x))))
     (fit = vgam(y ~ s(x), fam=amlpoisson(w.aml=c(0.02, 0.2, 1, 5, 50)),
                data=mydat, trace=TRUE))
     fit@extra

     ## Not run: 
     # Quantile plot
     with(mydat, plot(x, jitter(y), col="blue", las=1, main=
          paste(paste(round(fit@extra$percentile, dig=1), collapse=", "),
                "percentile-expectile curves")))
     with(mydat, matlines(x, fitted(fit), lwd=2))
     ## End(Not run)

