

envelope(boot)                               R Documentation

_C_o_n_f_i_d_e_n_c_e _E_n_v_e_l_o_p_e_s _f_o_r _C_u_r_v_e_s

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

     This function calculates overall and pointwise confi-
     dence envelopes for a curve based on bootstrap repli-
     cates of the curve evaluated at a number of fixed
     points.

_U_s_a_g_e_:

     envelope(boot.out=NULL, mat=boot.out$t, level=0.95, index=1:ncol(mat))

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

boot.out: An object of class `"boot"' for which `boot.out$t'
          contains the replicates of the curve at a number
          of fixed points.

     mat: A matrix of bootstrap replicates of the values of
          the curve at a number of fixed points.  This is a
          required argument if `boot.out' is not supplied
          and is set to `boot.out$t' otherwise.

   level: The confidence level of the envelopes required.
          The default is to find 95% confidence envelopes.
          It can be a scalar or a vector of length 2.  If it
          is scalar then both the pointwise and the overall
          envelopes are found at that level.  If is a vector
          then the first element gives the level for the
          pointwise envelope and the second gives the level
          for the overall envelope.

   index: The numbers of the columns of `mat' which contain
          the bootstrap replicates.  This can be used to
          ensure that other statistics which may have been
          calculated in the bootstrap are not considered as
          values of the function.

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

     The pointwise envelope is found by simply looking at
     the quantiles of the replicates at each point.  The
     overall error for that envelope is then calculated
     using equation (4.17) of Davison and Hinkley (1997).  A
     sequence of pointwise envelopes is then found until one
     of them has overall error approximately equal to the
     level required.  If no such envelope can be found then
     the envelope returned will just contain the extreme
     values of each column of `mat'.

_V_a_l_u_e_:

     A list with the following components :

   point: A matrix with two rows corresponding to the values
          of the upper and lower pointwise confidence enve-
          lope at the same points as the bootstrap repli-
          cates were calculated.

 overall: A matrix similar to `point' but containing the
          envelope which controls the overall error.

    k.pt: The quantiles used for the pointwise envelope.

  err.pt: A vector with two components, the first gives the
          pointwise error rate for the pointwise envelope,
          and the second the overall error rate for that
          envelope.

    k.ov: The quantiles used for the overall envelope.

  err.ov: A vector with two components, the first gives the
          pointwise error rate for the overall envelope, and
          the second the overall error rate for that enve-
          lope.

 err.nom: A vector of length 2 giving the nominal error
          rates for the pointwise and the overall envelopes.

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

     Davison, A.C. and Hinkley, D.V. (1997) Bootstrap Meth-
     ods and Their Application. Cambridge University Press.

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

     `boot', `boot.ci'

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

     # Testing whether the final series of measurements of the gravity data
     # may come from a normal distribution.  This is done in Examples 4.7
     # and 4.8 of Davison and Hinkley (1997).
     data(gravity)
     grav1 <- gravity$g[gravity$series==8]
     grav.z <- (grav1-mean(grav1))/sqrt(var(grav1))
     grav.gen <- function(dat,mle)
          rnorm(length(dat))
     grav.qqboot <- boot(grav.z,sort,R=999,sim="parametric",ran.gen=grav.gen)
     grav.qq <- qqnorm(grav.z,plot=F)
     grav.qq <- lapply(grav.qq,sort)
     plot(grav.qq,ylim=c(-3.5,3.5),ylab="Studentized Order Statistics",
          xlab="Normal Quantiles")
     grav.env <- envelope(grav.qqboot,level=0.9)
     lines(grav.qq$x,grav.env$point[1,],lty=4)
     lines(grav.qq$x,grav.env$point[2,],lty=4)
     lines(grav.qq$x,grav.env$overall[1,],lty=1)
     lines(grav.qq$x,grav.env$overall[2,],lty=1)

