# 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

 1 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 <bent.nielsen@nuffield.ox.ac.uk> 9 Sep 2014

## References

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

## Examples

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 ##################### # 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.