# TruncatedStudentsT: Create a Truncated Student's T Distribution In crch: Censored Regression with Conditional Heteroscedasticity

 TruncatedStudentsT R Documentation

## Create a Truncated Student's T Distribution

### Description

Class and methods for left-, right-, and interval-truncated t distributions using the workflow from the distributions3 package.

### Usage

```TruncatedStudentsT(df, location = 0, scale = 1, left = -Inf, right = Inf)
```

### Arguments

 `df` numeric. The degrees of freedom of the underlying untruncated t distribution. Can be any positive number, with `df = Inf` corresponding to the normal distribution. `location` numeric. The location parameter of the underlying untruncated t distribution, typically written μ in textbooks. Can be any real number, defaults to `0`. `scale` numeric. The scale parameter (standard deviation) of the underlying untruncated t distribution, typically written σ in textbooks. Can be any positive number, defaults to `1`. `left` numeric. The left truncation point. Can be any real number, defaults to `-Inf` (untruncated). If set to a finite value, the distribution has a point mass at `left` whose probability corresponds to the cumulative probability function of the untruncated t distribution at this point. `right` numeric. The right truncation point. Can be any real number, defaults to `Inf` (untruncated). If set to a finite value, the distribution has a point mass at `right` whose probability corresponds to 1 minus the cumulative probability function of the untruncated t distribution at this point.

### Details

The constructor function `TruncatedStudentsT` sets up a distribution object, representing the truncated t probability distribution by the corresponding parameters: the degrees of freedom `df`, the latent mean `location` = μ and latent scale parameter `scale` = σ (i.e., the parameters of the underlying untruncated t variable), the `left` truncation point (with `-Inf` corresponding to untruncated), and the `right` truncation point (with `Inf` corresponding to untruncated).

The truncated t distribution has probability density function (PDF) f(x):

f(x) = 1/σ τ((x - μ)/σ) / (T((right - μ)/σ) - T((left - μ)/σ))

for left ≤ x ≤ right, and 0 otherwise, where T and τ are the cumulative distribution function and probability density function of the standard t distribution with `df` degrees of freedom, respectively.

All parameters can also be vectors, so that it is possible to define a vector of truncated t distributions with potentially different parameters. All parameters need to have the same length or must be scalars (i.e., of length 1) which are then recycled to the length of the other parameters.

For the `TruncatedStudentsT` distribution objects there is a wide range of standard methods available to the generics provided in the distributions3 package: `pdf` and `log_pdf` for the (log-)density (PDF), `cdf` for the probability from the cumulative distribution function (CDF), `quantile` for quantiles, `random` for simulating random variables, `crps` for the continuous ranked probability score (CRPS), and `support` for the support interval (minimum and maximum). Internally, these methods rely on the usual d/p/q/r functions provided for the truncated t distributions in the crch package, see `dtt`, and the `crps_tt` function from the scoringRules package. The methods `is_discrete` and `is_continuous` can be used to query whether the distributions are discrete on the entire support (always `FALSE`) or continuous on the entire support (always `TRUE`).

See the examples below for an illustration of the workflow for the class and methods.

### Value

A `TruncatedStudentsT` distribution object.

`dtt`, `StudentsT`, `CensoredStudentsT`, `TruncatedNormal`, `TruncatedLogistic`

### Examples

```
## package and random seed
library("distributions3")
set.seed(6020)

## three truncated t distributions:
## - untruncated standard t with 5 degrees of freedom
## - left-truncated at zero with 5 df, latent location = 1 and scale = 1
## - interval-truncated in [0, 5] with 5 df, latent location = 2 and scale = 2
X <- TruncatedStudentsT(
df       = c(   5,   5, 5),
location = c(   0,   1, 2),
scale    = c(   1,   1, 2),
left     = c(-Inf,   0, 0),
right    = c( Inf, Inf, 5)
)
X

## compute mean of the truncated distribution
mean(X)
## higher moments (variance, skewness, kurtosis) are not implemented yet

## support interval (minimum and maximum)
support(X)

## simulate random variables
random(X, 5)

## histograms of 1,000 simulated observations
x <- random(X, 1000)
hist(x[1, ], main = "untruncated")
hist(x[2, ], main = "left-truncated at zero")
hist(x[3, ], main = "interval-truncated in [0, 5]")

## probability density function (PDF) and log-density (or log-likelihood)
x <- c(0, 0, 1)
pdf(X, x)
pdf(X, x, log = TRUE)
log_pdf(X, x)

## cumulative distribution function (CDF)
cdf(X, x)

## quantiles
quantile(X, 0.5)

## cdf() and quantile() are inverses (except at truncation points)
cdf(X, quantile(X, 0.5))
quantile(X, cdf(X, 1))

## all methods above can either be applied elementwise or for
## all combinations of X and x, if length(X) = length(x),
## also the result can be assured to be a matrix via drop = FALSE
p <- c(0.05, 0.5, 0.95)
quantile(X, p, elementwise = FALSE)
quantile(X, p, elementwise = TRUE)
quantile(X, p, elementwise = TRUE, drop = FALSE)

## compare theoretical and empirical mean from 1,000 simulated observations
cbind(
"theoretical" = mean(X),
"empirical" = rowMeans(random(X, 1000))
)

## evaluate continuous ranked probability score (CRPS) using scoringRules
library("scoringRules")
crps(X, x)
```

crch documentation built on Sept. 10, 2022, 1:06 a.m.