sbpsi {scaleboot} | R Documentation |
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,sia,sib,sphe)
beta |
numeric vector of parameters;
β_0=beta[1] , β_1=beta[2] ,...
β_{m-1}=beta[m] , where m is the number of
parameters.
|
s |
σ_0^2. |
k |
numeric to specify the order of derivatives. |
sp |
σ_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. |
sia |
maximum number of parameters in sia models. |
sib |
maximum number of parameters in sib models. |
For k=1, the sbpsi
functions return their psi function
values at σ^2=σ_0^2. Currently, four types of
sbpsi
functions are
implemented. sbpsi.poly
defines the polynomial model;
psi(σ^2 | β) = sum_{j=0}^{m-1} β_j σ^{2j}
for m>=1.
sbpsi.sing
defines the singular model;
psi(σ^2 | β) = β_0 + sum_{j=1}^{m-2} frac{β_j σ^{2j}}{1 + β_{m-1}(σ-1)}
for m>=3 and 0<=β_{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
σ^2=σ_p^2 using derivatives up to order k-1
evaluated at σ^2=σ_0^2;
q_k = sum_{j=0}^{k-1} frac{(σ_p^2-σ_0^2)^j}{j!} frac{d^j psi(x|β)}{d x^j}Bigr|_{σ_0^2},
which reduces to psi(σ_0^2|β) for k=1. In the
summary.scaleboot
, the AU p-values are defined
by p_k = 1-Phi(q_k) for k>=1.
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.
Hidetoshi Shimodaira