# trun.d: Truncated Probability Density Function of a gamlss.family... In gamlss.tr: Generating and Fitting Truncated `gamlss.family' Distributions

## 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

 ```1 2``` ```trun.d(par, family = "NO", type = c("left", "right", "both"), varying = FALSE, ...) ```

## 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

## Value

Returns a d family function

## Author(s)

Mikis Stasinopoulos [email protected] 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.

Stasinopoulos D. M., Rigby R.A. and Akantziliotou C. (2003) Instructions on how to use the GAMLSS package in R. Accompanying documentation in the current GAMLSS help files, (see also http://www.gamlss.org/).

`trun.p`, `trun.q`, `trun.r`, `gen.trun`
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125``` ```#------------------------------------------------------------------------------------------ # 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))) ```