This functions plots the profile deviance of one of the (four) parameters in a GAMLSS model. It can be used if one
of the parameters `mu`

, `sigma`

, `nu`

or `tau`

is a constant (not a function of explanatory variables) to obtain
a profile confidence intervals.

```
prof.dev(object, which = NULL, min = NULL, max = NULL,
step = NULL, length = 7, startlastfit = TRUE,
plot = TRUE, perc = 95, col="darkgreen")
```

Return a profile plot (if the argument `plot=TRUE`

) and an `ProfLikelihood.gamlss`

object if saved. The object contains:

- values
the values at the grid where the parameter was evaluated

- fun
the function which approximates the points using splines

- min
the minimum values in the grid

- max
te maximum values in the grid

- max.value
the value of the parameter maximising the Profile deviance (or GAIC)

- CI
the profile confidence interval (if global deviance is used)

- criterion
which criterion was used

- object
A fitted GAMLSS model

- which
which parameter to get the profile deviance e.g.

`which="tau"`

- min
the minimum value for the parameter e.g.

`min=1`

- max
the maximum value for the parameter e.g.

`max=20`

- step
how often to evaluate the global deviance (defines the step length of the grid for the parameter) e.g.

`step=1`

- length
the length if step is not set, default equal 7

- startlastfit
whether to start fitting from the last fit or not, default value is

`startlastfit=TRUE`

- plot
whether to plot,

`plot=TRUE`

or save the results,`plot=FALSE`

- perc
what % confidence interval is required

- col
The colour of the profile line

Calliope Akantziliotou, Mikis Stasinopoulos d.stasinopoulos@londonmet.ac.uk and Bob Rigby

A dense grid (i.e. small step) evaluation of the global deviance can take a long time, so start with a sparse grid (i.e. large step) and decrease gradually the step length for more accuracy.

This function can be use to provide likelihood based confidence intervals for a parameter for which a constant model (i.e. no explanatory model) is fitted and
consequently for checking the adequacy of a particular values of the parameter. This can be used to check the adequacy of one distribution (e.g. Box-Cox Cole and Green)
nested within another (e.g. Box-Cox power exponential). For example one can test whether a Box-Cox Cole and Green (Box-Cox-normal) distribution
or a Box-Cox power exponential is appropriate by plotting the profile of the parameter `tau`

.
A profile deviance showing support for `tau=2`

indicates adequacy of the Box-Cox Cole and Green (i.e. Box-Cox normal) distribution.

Rigby, R. A. and Stasinopoulos D. M. (2005). Generalized additive models for location, scale and shape,(with discussion),
*Appl. Statist.*, **54**, part 3, pp 507-554.

Rigby, R. A., Stasinopoulos, D. M., Heller, G. Z., and De Bastiani, F. (2019)
*Distributions for modeling location, scale, and shape: Using GAMLSS in R*, Chapman and Hall/CRC. An older version can be found in https://www.gamlss.com/.

Stasinopoulos D. M. Rigby R.A. (2007) Generalized additive models for location scale and shape (GAMLSS) in R.
*Journal of Statistical Software*, Vol. **23**, Issue 7, Dec 2007, https://www.jstatsoft.org/v23/i07/.

Stasinopoulos D. M., Rigby R.A., Heller G., Voudouris V., and De Bastiani F., (2017)
*Flexible Regression and Smoothing: Using GAMLSS in R*, Chapman and Hall/CRC.

(see also https://www.gamlss.com/).

`gamlss`

, `prof.term`

```
if (FALSE) {
data(abdom)
h<-gamlss(y~pb(x), sigma.formula=~pb(x), family=BCT, data=abdom)
prof.dev(h,"nu",min=-2.000,max=2)
rm(h)}
```

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