Class and methods for left-, right-, and interval-truncated t distributions using the workflow from the distributions3 package.
TruncatedStudentsT(df, location = 0, scale = 1, left = -Inf, right = Inf)
numeric. The degrees of freedom of the underlying untruncated
t distribution. Can be any positive number, with
numeric. The location parameter of the underlying untruncated
t distribution, typically written μ in textbooks.
Can be any real number, defaults to
numeric. The scale parameter (standard deviation) of
the underlying untruncated t distribution,
typically written σ in textbooks.
Can be any positive number, defaults to
numeric. The left truncation point. Can be any real number,
numeric. The right truncation point. Can be any real number,
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),
left truncation point (with
-Inf corresponding to untruncated),
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.
TruncatedStudentsT distribution objects there is a wide range
of standard methods available to the generics provided in the distributions3
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
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
dtt, and the
function from the scoringRules package.
can be used to query whether the distributions are discrete on the entire support
FALSE) or continuous on the entire support (always
See the examples below for an illustration of the workflow for the class and methods.
TruncatedStudentsT distribution object.
## 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)
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