sbpsi: Model Specification Functions
In scaleboot: Approximately Unbiased P-Values via Multiscale Bootstrap

Description Usage Arguments Details Value Author(s) See Also

sbpsi.poly and sbpsi.sing are ψ 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)

`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.
`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.

For k=1, the sbpsi functions return their ψ function values at σ^2=σ_0^2. Currently, four types of sbpsi functions are implemented. sbpsi.poly defines the polynomial model;

ψ(σ^2 | β) = ∑_{j=0}^{m-1} β_j σ^{2j}

for m≥1. sbpsi.sing defines the singular model;

ψ(σ^2 | β) = β_0 + ∑_{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 a generic sbpsi function for specified zfun.

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 = ∑_{j=0}^{k-1} \frac{(σ_p^2-σ_0^2)^j}{j!} \frac{d^j ψ(x|β)}{d x^j}\Bigr|_{σ_0^2},

which reduces to ψ(σ_0^2|β) for k=1. In the summary.scaleboot, the AU p-values are defined by p_k = 1-Φ(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 ψ 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

sbfit.

scaleboot documentation built on Dec. 4, 2019, 5:07 p.m.