ForwardSearch (version 1.0)

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

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

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).

Details

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

$c$ and $\psi$ are linked through $P(|\epsilon|

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

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

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))