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] -0.08808012 -1.93113430 -3.46182652  0.74689224  1.54425503 -0.19301405
#>  [7] -0.67914176  0.96010252  0.19686454 -3.10147957
#> 
#> [[2]]
#>  [1] 5.33659937 0.66755900 1.06674924 0.88246880 2.63206675 7.82914627
#>  [7] 0.04973132 2.49117596 0.23530374 2.90607445
#> 
#> [[3]]
#>  [1] -3.8580694 -0.5277899  0.5172049 -2.7594600  2.1386771  2.1126339
#>  [7]  2.8579755  4.3154884 -0.2946130  4.7344929
#> 

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