# find_h_cont

0th

Percentile

##### Finding optimal instrumental density.

Finds the optimal instrumental density h to be used in the bidirectional acceptance sampling.

Keywords
Optimal instrumental density for bidirectional acceptance sampling
##### Usage
find_h_cont(data,g,dhat,range=NULL,M_0=NULL,par0=NULL,lbs,ubs,check.plot=TRUE,
ylim.f=c(0,2),ylim.d=c(0,2),global=FALSE)
##### Arguments
data

A data vector.

g

Function corresponding to the parametric start or postulated model. See details.

dhat

Function corresponding to the estimated comparison density in the x domain. See details.

range

Interval corresponding to the support of the continuous data distribution.

M_0

Starting point for optimization. See details.

par0

A vector of starting values of the parameters to be estimated. See details.

lbs

A vector of the lower bounds of the parameters to be estimated.

ubs

A vector of the upper bounds of the parameters to be estimated.

check.plot

A logical argument indicating if the plot comparing the densities involved should be displayed or not. The default is TRUE.

ylim.f

If check.plot=TRUE, the range of the y-axis of the plot for the probability density functions.

ylim.d

If check.plot=TRUE, the range of the y-axis of the plot for the comparison densities.

global

A logical argument indicating if a global optimization is needed to find the instrumental probability function h. See details.

##### Details

The parametric start specified in g is assumed to be fully specified and takes x as the only argument. The argument dhat is the estimated comparison density in the x domain. We usually get the argument dhat by means of the function d_hat within our package. The value M_0 and the vector par0 are used for the optimization process for finding the optimal instrumental density h. Usually, we choose M_0 to be the central point of the range. For example, if the range is from 0 to 30, we choose 15 as starting point. The choice of M_0 is not expected to affect substantially the accuracy of the solution. The vector par0 collects initial values for the parameters which characterize the instrumental density. For instance, if h is a mixuture of p truncated normals, the first p-1 elements of pis correspond to the starting values for the first p-1 mixture weights. The following p elements are the initial values for the means of the p truncated normals contributing to the mixture. Finally, the last p elements of par0 correspond to the starting values for the standard deviations of the p truncated normals contributing to the mixture. The argument global controls whether to use a global optimization or not. A local method allows to reduce the optimization time but the solution is particularly sensible to the choice of par0. Conversely, setting global=TRUE leads to more accurate result.

##### Value

Mstar

The reciprocal of the acceptance rate.

pis

The optimal set of mixture weights.

means

The optimal mean vector.

sds

The optimal set of standard deviations.

h

Function corresponding to the optimal instrumental density.

##### References

Algeri S. and Zhang X. (2020). Exhaustive goodness-of-fit via smoothed inference and graphics. arXiv:2005.13011.

d_hat, find_h_disc, rmixtruncnorm, dmixtruncnorm

• find_h_cont
##### Examples
# NOT RUN {
library("truncnorm")
library("LPBkg")
L=0
U=30
range=c(L,U)
set.seed(12395)
meant=-15
sdt=15
n=300
data<-rtruncnorm(n,a=L,b=U,mean=meant,sd=sdt)
poly2_num<-function(x){4.576-0.317*x+0.00567*x^2}
poly2_den<-integrate(poly2_num,lower=L,upper=U)$value g<-function(x){poly2_num(x)/poly2_den} ddhat<-d_hat(data,m=2,g, range=c(L,U), selection=FALSE)$dx
lb=c(0,-20,0,0,0)
ub=c(1,10,rep(30,3))
par0=c(0.3,-17,1,10,15)
range=c(L,U)
find_h_cont(data,g,ddhat,range,M_0=10,par0,lb,ub,ylim.f=c(0,0.25),ylim.d=c(-1,2))
# }

Documentation reproduced from package LPsmooth, version 0.1.0, License: GPL-3

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