PlugInBand {NPHazardRate}

Simple Plug in badnwidth selector

Description

Provides the asymptotic MISE optimal plug-in bandwidth for the hazard rate estimator HazardRateEst, see Hua, Patil and Bagkavos (2018). The bandwidth is also suitable for use as a pilot bandwidth in TransHazRateEst and VarBandHazEst.

Usage

PlugInBand(xin, xout,   cens, kfun ) 

Arguments

xin

A vector of data points

xout

The point at which the estimates should be calculated.

cens

Censoring Indicators.

kfun

A kernel function.

Details

The asymptotic MISE optimal plug-in bandwidth selector for HazardRateEst is defined by

h_{ opt} = \left[\frac{R(K)}{nR(\lambda_T'')\mu_{2,K}^2}\int \frac{\lambda_T(x)}{1-F(x)}\,dx \right]^{1/5}

see (9) in Hua, Patil and Bagkavos (2018). The estimate of R(\lambda_T'') to be used in h_{opt} is

R(\hat \lambda_T'') = \int_0^\xi \left (\hat \lambda_T''(x|\hat b_n^\ast) \right )^2\,dx.

Also,

\int_0^T \frac{\lambda_T(x)}{1-F(x)}\,dx

is estimated by applying the extended Simpson's numerical integration rule, SimpsonInt, on

\frac{\hat \lambda_T(x|\hat b_n^\ast) }{1-F(x)}

where 1-F(x) is estimated by KMest. The estimation is implemented in the NP.M.Estimate function.

Currently b_n^\ast is estimated by bw.nrd. However according to (11) in Hua, Patil and Bagkavos (2018)., in future versions this package will support

b_n^\ast = \left \{ \frac{5R(K'')}{n \mu_{2,K}^2 R(\lambda_T^{(4)})} \int \frac{\lambda_T(x)}{1-F(x)}\,dx \right \}^{1/9}.

where

R(\hat \lambda_T^{(4)}) = \frac{(\hat a(\hat a-1)(\hat a-2)(\hat a-3)(\hat a-4))^2}{(2\hat a-9){\hat{b}}^{2\hat a} } (\xi^{2\hat a-9} - {p_\alpha}^{2\hat a-9}), \hat a\neq 9/2

and \hat M is already estimated by NP.M.Estimate as expalined above (it will be much more stable than using a Weibull reference model).

Value

A scalar with the value of the suggested bandwidth.

References

Hua, Patil and Bagkavos, An $L_1$ analysis of a kernel-based hazard rate estimator, Australian and New Zealand J. Statist., (60), 43-64, (2018).

See Also

HazardRateEst, LLHRPlugInBand

Examples

x<-seq(0, 5,length=100) #design points where the estimate will be calculated
SampleSize<-100 #amount of data to be generated
ti<- rweibull(SampleSize, .6, 1) # draw a random sample
ui<-rexp(SampleSize, .2)         # censoring sample
cat("\n AMOUNT OF CENSORING: ", length(which(ti>ui))/length(ti)*100, "\n")
x1<-pmin(ti,ui)                  # observed data
cen<-rep.int(1, SampleSize)      # initialize censoring indicators
cen[which(ti>ui)]<-0             # 0's correspond to censored indicators

huse1<- PlugInBand(x1, x, cen, Biweight)
huse1