sbpsi.poly and sbpsi.sing are \(\psi\) functions to
specify a polynomial model and a singular model, respectively.
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)
numeric vector of parameters;
\(\beta_0\)=beta[1], \(\beta_1\)=beta[2],...
\(\beta_{m-1}\)=beta[m], where \(m\) is the number of
parameters.
\(\sigma_0^2\).
numeric to specify the order of derivatives.
\(\sigma_p^2\).
a numeric of specifying the type of p-values; Bayesian (lambda=0) Frequentist (lambda=1).
auxiliary parameter. Currently not used.
logical for boundary check.
z-value function with (s,beta) as parameters.
delta for numerical computation of derivatives.
numeric vector to specify the numbers of parameters.
logical to include poly and sing models.
logical to include poa and sia models.
logical to include pob and sib models.
maximum number of parameters in poly models.
maximum number of parameters in sing models.
maximum number of parameters in sphe models.
maximum number of parameters in poa models.
maximum number of parameters in pob models.
maximum number of parameters in poc models.
maximum number of parameters in pod models.
maximum number of parameters in sia models.
maximum number of parameters in sib models.
maximum number of parameters in sic models.
maximum number of parameters in sid models.
maximum number of parameters in pom models.
maximum number of parameters in sim models.
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.
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 \(k\ge1\).