ForwardSearch.pointwise.asymptotics: Functions for asymptotic theory of Forward Search

In ForwardSearch: Forward Search using asymptotic theory

Description

Computes functions appearing in asymptotic theory of Forward Search based on Johansen and Nielsen (2013).

Usage

ForwardSearch.pointwise.asymptotics(psi, ref.dist = "normal")

Arguments

psi	Number or vector. Takes value(s) in interval 0,1.
ref.dist	Character. Reference distribution "normal" Standard normal distribution

Details

The asymptotic theory is developed in Johansen and Nielsen (2013), see Section 2.2.

c and ψ are linked through P(|ε|<c)=ψ, where ε is a random variable with the chosen reference distribution.

ζ is a consistency factor. Its square is defined as the truncated second moment τ = \int_{-c}^{c} x^2 f(x) dx divided by ψ.

\varpi is the asymptotic standard deviation resulting from Theorem 3.3.

Value

varpi	Number or vector. sdv for forward residuals normalized by variance estimator and multiplied by twice the reference densisty.
zeta	Number or vector. Consistency correction factor.
sdv.unbiased	Number or vector. varpi/2/f.
sdv.biased	Number or vector. varpi/2/f/zeta.
c	Number or vector. c (median in unbiased case).
median.biased	Number or vector. median (in biased case).

Author(s)

Bent Nielsen <[e-mail]> 9 Sep 2014

References

Johansen, S. and Nielsen, B. (2013) Asymptotic analysis of the Forward Search. Download: Nuffield DP.

Examples

#####################
#	EXAMPLE 1
#	Suppose n=100. Get asymptotic values for grid psi = (1, ... ,n)/n

n	<- 100
psi	<- seq(1,n-1)/n
FS	<- ForwardSearch.pointwise.asymptotics(psi)

#	Plot for biased normalisation
#	- matching choice of Atkinson and Riani (2000)

main <- "Pointwise confidence bands for n=100\n Biased normalisation"
ylab <-	"forward residual asymptotics"
 plot(psi,FS$median.biased,ylim=c(0,3),ylab=ylab,main=main,type="l")
lines(psi,FS$median.biased-2*FS$sdv.biased/sqrt(n))
lines(psi,FS$median.biased+2*FS$sdv.biased/sqrt(n))

#	Plot for unbiased normalisation

main <- "Pointwise confidence bands for n=100\n Unbiased normalisation"
ylab <-	"forward residual asymptotics"
 plot(psi,FS$c,ylim=c(0,3),ylab=ylab,main=main,type="l")
lines(psi,FS$c-2*FS$sdv.unbiased/sqrt(n))
lines(psi,FS$c+2*FS$sdv.unbiased/sqrt(n))

ForwardSearch documentation built on May 1, 2019, 6:51 p.m.