# find_h_disc

##### Finding optimal instrumental mass function.

Finds the optimal probability mass function `h`

to be used in the bidirectional acceptance sampling.

##### Usage

```
find_h_disc(data,g,dhat,lattice=NULL,M_0=NULL,size,par0=NULL,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.- lattice
Support of the discrete data distribution.

- size
A postive value corresponding to the target for number of successful trials.

- M_0
Starting point for optimization. See details.

- par0
A vector of starting values of the parameters to be estimated. See details.

- 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 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 could choose the `M_0`

to be the central point of the `lattice`

. For example, if the range is from `0`

to `30`

, we could choose `15`

as the 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 probability mass function. For instance, if `h`

is a mixuture of `p`

negative binomials, 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 probablities of success of the `p`

negative binomials 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

The reciprocal of the acceptance rate.

The optimal set of mixture weights.

The optimal set of probabilities of success.

Function corresponding to the optimal instrumental probability mass function.

##### References

Algeri S. and Zhang X. (2020). Smoothed inference and graphics via LP modeling, arXiv:2005.13011.

##### See Also

##### Examples

```
# NOT RUN {
lattice=seq(0,20,length=21)
n=200
data<-rbinom(n,size=20,prob=0.5)
g<-function(x)dpois(x,10)/(ppois(20,10)-ppois(0,10))
ddhat<-d_hat(data,m=1,g=g,lattice=lattice,selection=TRUE)$dx
find_h_disc(data=data,g=g,dhat=ddhat,lattice,M_0=10,size=15,par0=c(0.3,0.5,0.6),
check.plot=TRUE,ylim.f=c(0,0.4),ylim.d=c(-3,2.5),global=FALSE)
# }
```

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