gamlss.tr (version 5.1-9)

trun.d: Truncated Probability Density Function of a gamlss.family Distribution

Description

Creates a truncated probability density function version from a current GAMLSS family distribution

For continuous distributions left truncation at 3 means that the random variable can take the value 3. For discrete distributions left truncation at 3 means that the random variable can take values from 4 onwards. This is the same for right truncation. Truncation at 15 for a discrete variable means that 15 and greater values are not allowed but for continuous variable it mean values greater that 15 are not allowed (so 15 is a possible value).

Usage

trun.d(par, family = "NO", type = c("left", "right", "both"),
       varying = FALSE, ...)

Value

Returns a d family function

Arguments

par

a vector with one (for "left" or "right" truncation) or two elements for "both". When the argument varying = TRUE then par can be a vector or a matrix with two columns respectively.

family

a gamlss.family object, which is used to define the distribution and the link functions of the various parameters. The distribution families supported by gamlss() can be found in gamlss.family. Functions such as BI() (binomial) produce a family object.

type

whether left, right or in both sides truncation is required, (left is the default).

varying

whether the truncation varies for diferent observations. This can be usefull in regression analysis. If varying = TRUE then par should be an n-length vector for type equal "left" and "right" and an n by 2 matrix for type="both"

...

for extra arguments

Author

Mikis Stasinopoulos and Bob Rigby

References

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/).

See Also

trun.p, trun.q, trun.r, gen.trun

Examples

#------------------------------------------------------------------------------------------
# continuous distribution 
# left truncation 
test1<-trun.d(par=c(0), family="TF", type="left")
test1(1)
dTF(1)/(1-pTF(0))
if(abs(test1(1)-(dTF(1)/pTF(0)))>0.00001) stop("error in left trucation")
test1(1, log=TRUE)
log(dTF(1)/(1-pTF(0)))
if(abs(test1(1, log=TRUE)-log(dTF(1)/pTF(0)))>0.00001) 
                   stop("error in left trucation")
integrate(function(x) test1(x, mu=-2, sigma=1, nu=1),0,Inf) 
# the pdf is defined even with negative mu
integrate(function(x) test1(x, mu=0, sigma=10, nu=1),0,Inf) 
integrate(function(x) test1(x, mu=5, sigma=5, nu=10),0,Inf)
plot(function(x) test1(x, mu=-3, sigma=1, nu=1),0,10)
plot(function(x) test1(x, mu=3, sigma=5, nu=10),0,10)
#----------------------------------------------------------------------------------------
# right truncation
test2<-trun.d(par=c(10), family="BCT", type="right")
test2(1)
dBCT(1)/(pBCT(10))
#if(abs(test2(1)-(dBCT(1)/pBCT(10)))>0.00001) stop("error in right trucation")
test2(1, log=TRUE)
log(dBCT(1)/(pBCT(10)))
if(abs(test2(1, log=TRUE)-log(dBCT(1)/(pBCT(10))))>0.00001) 
                   stop("error in right trucation")
integrate(function(x) test2(x, mu=2, sigma=1, nu=1),0,10) 
integrate(function(x) test2(x, mu=2, sigma=.1, nu=1),0,10) 
integrate(function(x) test2(x, mu=2, sigma=.1, nu=10),0,10) 
plot(function(x) test2(x, mu=2, sigma=.1, nu=1),0,10)
plot(function(x) test2(x, mu=2, sigma=1, nu=1),0,10)
#-----------------------------------------------------------------------------------------
# both left and right truncation
test3<-trun.d(par=c(-3,3), family="TF", type="both")
test3(0)
dTF(0)/(pTF(3)-pTF(-3))
if(abs(test3(0)-dTF(0)/(pTF(3)-pTF(-3)))>0.00001) 
              stop("error in right trucation")
test3(0, log=TRUE)
log(dTF(0)/(pTF(3)-pTF(-3)))
if(abs(test3(0, log=TRUE)-log(dTF(0)/(pTF(3)-pTF(-3))))>0.00001) 
            stop("error in both trucation")
plot(function(x) test3(x, mu=0, sigma=1, nu=1),-3,3)
integrate(function(x) test3(x, mu=2, sigma=1, nu=1),-3,3)
#-----------------------------------------------------------------------------------------
# discrete distribution
# left 
# Poisson truncated at zero means zero is excluded
test4<-trun.d(par=c(0), family="PO", type="left")
test4(1)
dPO(1)/(1-pPO(0))
if(abs(test4(1)-dPO(1)/(1-pPO(0)))>0.00001) stop("error in left trucation")
test4(1, log=TRUE)
log(dPO(1)/(1-pPO(0)))
if(abs(test4(1, log=TRUE)-log(dPO(1)/(1-pPO(0))))>0.00001) 
               stop("error in left trucation")
sum(test4(x=1:20, mu=2)) # 
sum(test4(x=1:200, mu=80)) #
plot(function(x) test4(x, mu=20), from=1, to=51, n=50+1, type="h") # pdf 
# right
# right truncated at 10 means 10 is excluded
test5<-trun.d(par=c(10), family="NBI", type="right")
test5(2)
dNBI(2)/(pNBI(9))
if(abs(test5(1)-dNBI(1)/(pNBI(9)))>0.00001) stop("error in right trucation")
test5(1, log=TRUE)
log(dNBI(1)/(pNBI(9)))
if(abs(test5(1, log=TRUE)-log(dNBI(1)/(pNBI(9))))>0.00001) stop("error in right trucation")
sum(test5(x=0:9, mu=2,   sigma=2)) # 
sum(test5(x=0:9, mu=300, sigma=5)) # can have mu > parameter
plot(function(x) test5(x, mu=20, sigma=3), from=0, to=9, n=10, type="h") # pdf
plot(function(x) test5(x, mu=300, sigma=5), from=0, to=9, n=10, type="h") # pdf
#----------------------------------------------------------------------------------------
# both
test6<-trun.d(par=c(0,10), family="NBI", type="both")
test6(2)
dNBI(2)/(pNBI(9)-pNBI(0))
if(abs(test6(2)-dNBI(2)/(pNBI(9)-pNBI(0)))>0.00001) 
        stop("error in right trucation")
test6(1, log=TRUE)
log(dNBI(1)/(pNBI(9)-pNBI(0)))
if(abs(test6(1, log=TRUE)-log(dNBI(1)/(pNBI(9)-pNBI(0))))>0.00001) 
  stop("error in right trucation")
sum(test6(x=1:9, mu=2,   sigma=2)) # 
sum(test6(x=1:9, mu=100, sigma=5)) # can have mu > parameter
plot(function(x) test6(x, mu=20, sigma=3), from=1, to=9, n=9, type="h") # pdf
plot(function(x) test6(x, mu=300, sigma=.4), from=1, to=9, n=9, type="h") # pdf
#------------------------------------------------------------------------------------------
# now try when the trucated points varies for each observarion
# this will be appropriate for regression models only 
# continuous
#----------------------------------------------------------------------------------------
# left truncation
test7<-trun.d(par=c(0,1,2), family="TF", type="left", varying=TRUE)
test7(c(1,2,3))
dTF(c(1,2,3))/(1-pTF(c(0,1,2)))
test7(c(1,2,3), log=TRUE)
#----------------------------------------------------------------------------------------
# right truncation
test8<-trun.d(par=c(10,11,12), family="BCT", type="right", varying=TRUE)
test8(c(1,2,3))
dBCT(c(1,2,3))/(pBCT(c(10,11,12)))
test8(c(1,2,3), log=TRUE)
#----------------------------------------------------------------------------------------
# both left and right truncation
test9<-trun.d(par=cbind(c(0,1,2),c(10,11,12) ), family="TF", type="both", 
             varying=TRUE)
test9(c(1,2,3))
dTF(c(1,2,3))/ (pTF(c(10,11,12))-pTF(c(0,1,2)))
test3(c(1,2,3), log=TRUE)
#----------------------------------------------------------------------------------------
# discrete
# left
test10<-trun.d(par=c(0,1,2), family="PO", type="left", varying=TRUE)
test10(c(1,2,3))
dPO(c(1,2,3))/(1-pPO(c(0,1,2)))
# right
test11<-trun.d(par=c(10,11,12), family="NBI", type="right", varying=TRUE)
test11(c(1,2,3))
dNBI(c(1,2,3))/pNBI(c(9,10,11))
# both
test12<-trun.d(par=rbind(c(0,10), c(1,11), c(2,12)), family="NBI", type="both", varying=TRUE)
test12(c(2,3,4))
dNBI(c(2,3,4))/(pNBI(c(9,10,11))-pNBI(c(0,1,2)))