The (non-central) location-scale Student t Distribution

Source: R/dist_student_t.R

[Stable]

The Student's T distribution is closely related to the Normal() distribution, but has heavier tails. As \(\nu\) increases to \(\infty\), the Student's T converges to a Normal. The T distribution appears repeatedly throughout classic frequentist hypothesis testing when comparing group means.

dist_student_t(df, mu = 0, sigma = 1, ncp = NULL)

Arguments

df

degrees of freedom (\(> 0\), maybe non-integer). df = Inf is allowed.

mu

The location parameter of the distribution. If ncp == 0 (or NULL), this is the median.

sigma

The scale parameter of the distribution.

ncp

non-centrality parameter \(\delta\); currently except for rt(), only for abs(ncp) <= 37.62. If omitted, use the central t distribution.

Details

We recommend reading this documentation on https://pkg.mitchelloharawild.com/distributional/, where the math will render nicely.

In the following, let \(X\) be a central Students T random variable with df = \(\nu\).

Support: \(R\), the set of all real numbers

Mean: Undefined unless \(\nu \ge 2\), in which case the mean is zero.

Variance:

$$ \frac{\nu}{\nu - 2} $$

Undefined if \(\nu < 1\), infinite when \(1 < \nu \le 2\).

Probability density function (p.d.f):

$$ f(x) = \frac{\Gamma(\frac{\nu + 1}{2})}{\sqrt{\nu \pi} \Gamma(\frac{\nu}{2})} (1 + \frac{x^2}{\nu} )^{- \frac{\nu + 1}{2}} $$

See also

stats::TDist

Examples

dist <- dist_student_t(df = c(1,2,5), mu = c(0,1,2), sigma = c(1,2,3))

dist
#> <distribution[3]>
#> [1] t(1, 0, 1) t(2, 1, 2) t(5, 2, 3)
mean(dist)
#> [1] NA  1  2
variance(dist)
#> [1]  NA Inf  15

generate(dist, 10)
#> [[1]]
#>  [1]   0.53769647  -1.12161683  -0.03008724  -0.23799141   1.40382052
#>  [6]  10.68439819  -4.59709841  -0.74840329 -10.61638467  -0.49604079
#> 
#> [[2]]
#>  [1]  4.50175371  2.63206675  7.82914627  0.04973132  2.49117596  0.23530374
#>  [7]  2.90607445 -1.14776366  1.08072715  1.09743804
#> 
#> [[3]]
#>  [1]  2.8579755  4.3154884 -0.2946130  4.7344929  3.1980660  8.1827146
#>  [7]  0.8940991  5.1130355 -1.1075269 -2.4289302
#> 

density(dist, 2)
#> [1] 0.06366198 0.14814815 0.12653556
density(dist, 2, log = TRUE)
#> [1] -2.754168 -1.909543 -2.067232

cdf(dist, 4)
#> [1] 0.9220209 0.8638034 0.7327454

quantile(dist, 0.7)
#> [1] 0.7265425 2.2344268 3.6782889