sbpsi               package:scaleboot               R Documentation

_M_o_d_e_l _S_p_e_c_i_f_i_c_a_t_i_o_n _F_u_n_c_t_i_o_n_s

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

     'sbpsi.poly' and 'sbpsi.sing' are psi functions to specify a
     polynomial model and a singular model, respectively.

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

     sbpsi.poly(beta,s=1,k=1,sp=-1,aux=NULL,check=FALSE)

     sbpsi.sing(beta,s=1,k=1,sp=-1,aux=NULL,check=FALSE)

     sbpsi.sphe(beta,s=1,k=1,sp=-1,aux=NULL,check=FALSE)

     sbpsi.generic(beta,s=1,k=1,sp=-1,aux=NULL,check=FALSE,zfun,eps=0.01)

     sbmodelnames(m=3,poly=m,sing=m,sphe=0)

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

    beta: numeric vector of parameters; beta_0='beta[1]',
          beta_1='beta[2]',... beta_{m-1}='beta[m]', where m is the
          number of parameters. 

       s: sigma_0^2.

       k: numeric to specify the order of derivatives.

      sp: sigma_p^2.

     aux: auxiliary parameter. Currently not used.

   check: logical for boundary check.

    zfun: z-value function with (s,beta) as parameters.

     eps: delta for numerical computation of derivatives.

       m: numeric to specify the maximum number of parameters.

    poly: maximum number of parameters in poly models.

    sing: maximum number of parameters in sing models.

    sphe: maximum number of parameters in sphe models.

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

     For k=1, the 'sbpsi' functions return their psi function values at
     sigma^2=sigma_0^2.  Currently, four types of 'sbpsi' functions are
     implemented. 'sbpsi.poly' defines the polynomial model;

       psi(sigma^2 | beta) = sum_{j=0}^{m-1} beta_j sigma^{2j}

     for m>=1. 'sbpsi.sing' defines the singular model;

 psi(sigma^2 | beta) = beta_0 + sum_{j=1}^{m-2} frac{beta_j sigma^{2j}}{1 + beta_{m-1}(sigma-1)}

     for m>=3 and 0<=beta_{m-1}<=1. 'sbpsi.sphe' defines the spherical
     model; currently the number of parameters must be $m=3$.
     'sbpsi.generic' is to calculate psi value and extrapolation from a
     given z-function.

     For k>1, the 'sbpsi' functions return values extrapolated at
     sigma^2=sigma_p^2 using derivatives up to order k-1 evaluated at
     sigma^2=sigma_0^2;

 q_k = sum_{j=0}^{k-1} frac{(sigma_p^2-sigma_0^2)^j}{j!} frac{d^j psi(x|beta)}{d x^j}Bigr|_{sigma_0^2},

     which reduces to psi(sigma_0^2|beta) for k=1. In the
     'summary.scaleboot', the AU p-values are defined by p_k =
     1-Phi(q_k) for k>=1.

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

     'sbpsi.poly' and 'sbpsi.sing' are examples of a sbpsi function;
     users can develop their own sbpsi functions for better model
     fitting by preparing 'sbpsi.foo' and 'sbini.foo' functions for
     model 'foo'. If check=FALSE, a sbpsi function returns the psi
     function value or the extrapolation value. If check=TRUE, a sbpsi
     function returns NULL when all the elements of beta are included
     in the their valid intervals. Otherwise, a 'sbpsi' function
     returns a list with components 'beta' for the parameter value
     being modified to be on a boundary of the interval and 'mask', a
     logical vector indicating which elements are not on the boundary.

     'sbmodelnames' returns a character vector of model names.

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

     Hidetoshi Shimodaira

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

     'sbfit'.

