sbpsi {scaleboot}

Model Specification Functions

Description

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

Usage

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

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

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

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

sbmodelnames(m=1:3,one.sided=TRUE,two.sided=FALSE,rev.sided=FALSE,
  poly,sing,poa,pob,poc,pod,sia,sib,sic,sid,sphe,pom,sim)

Arguments

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.

lambda

a numeric of specifying the type of p-values; Bayesian (lambda=0) Frequentist (lambda=1).

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 vector to specify the numbers of parameters.

one.sided

logical to include poly and sing models.

two.sided

logical to include poa and sia models.

rev.sided

logical to include pob and sib models.

poly

maximum number of parameters in poly models.

sing

maximum number of parameters in sing models.

sphe

maximum number of parameters in sphe models.

poa

maximum number of parameters in poa models.

pob

maximum number of parameters in pob models.

poc

maximum number of parameters in poc models.

pod

maximum number of parameters in pod models.

sia

maximum number of parameters in sia models.

sib

maximum number of parameters in sib models.

sic

maximum number of parameters in sic models.

sid

maximum number of parameters in sid models.

pom

maximum number of parameters in pom models.

sim

maximum number of parameters in sim models.

Details

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\ge1. 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\ge3 and 0\le\beta_{m-1}\le1. sbpsi.sphe defines the spherical model; currently the number of parameters must be $m=3$. sbpsi.generic is a generic sbpsi function for specified zfun.

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\ge1.

Value

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.

Author(s)

Hidetoshi Shimodaira

See Also

sbfit.