# DefVarBandRule: Default adaptive bandwidth rule In NPHazardRate: Nonparametric Hazard Rate Estimation

## Description

Implements an adaptive variable bandwidth hazard rate rule for use with the VarBandHazEst based on the Weibull distribution, with parameters estimated by maximum likelihood

## Usage

 1 DefVarBandRule(xin, cens) 

## Arguments

 xin A vector of data points. Missing values not allowed. cens A vector of censoring indicators: 1's indicate uncensored observations, 0's correspond to censored obs.

## Details

The adaptive AMISE optimal bandwidth for the variable bandwidth hazard rate estimator VarBandHazEst is given by

h_2 = ≤ft [ \frac{R(K) M_2}{8nμ_4^2(K) R(g)} \right ]^{1/14}

where

M_2 = \int \frac{λ^{3/2}(x)}{1-F(x)} \,dx

and

g(x)=\frac{1}{24λ(x)^5} \Bigl (24{λ'(x)}^4-36{λ'(x)}^2{λ''(x)}^2λ(x)+6{λ''(x)}^2λ^2(x) + 8λ'(x)λ'''(x)λ^2(x) -λ^{(4)}(x)λ^3(x)\Bigr )

## Value

the value of the adaptive bandwidth

## References

HazardRateEst, TransHazRateEst, PlugInBand
  1 2 3 4 5 6 7 8 9 10 11 12 13 library(survival) x<-seq(0, 5,length=100) #design points where the estimate will be calculated SampleSize <- 100 ti<- rweibull(SampleSize, .6, 1)#draw a random sample from the actual distribution ui<-rexp(SampleSize, .05) #draw a random sample from the censoring distribution cat("\n AMOUNT OF CENSORING: ", length(which(ti>ui))/length(ti)*100, "\n") x1<-pmin(ti,ui) #this is the observed sample cen<-rep.int(1, SampleSize) #censoring indicators cen[which(ti>ui)]<-0 #censored values correspond to zero h2<-DefVarBandRule(ti, cen) #Deafult Band. Rule - Weibull Reference