[SOLVED] Student's t-distribution CDF R base documentation

Student's t-distribution CDF R base documentation

In the context of the Student's t-distribution cumulative distribution function, R Version 4.3.1's ?dt documentation highlights the following result:

$F_\nu(t) = 1- I_{\frac{\nu}{\nu + t^2}}\biggr(\frac{\nu}{2}, \frac{1}{2}\biggr)$

However, upon attempting to verify the accuracy of this formula, an inconsistency arises, as illustrated in the following code snippet:

v <- 5
t <- -1

## Student's t-distribution cumulative distribution function
pt(q = t, df = v, lower.tail = TRUE, ncp = 0)
#> [1] 0.1816087

## Application of the theorical result where there is a discrepancy 
## based on what is mentioned in R Version 4.3.1's ?dt documentation
1 - pbeta(q = v / (v + t^2), shape1 = v/2, shape2 = 1/2, 
          ncp = 0,lower.tail = TRUE)
#> [1] 0.6367825

^{Created on 2023-10-09 with reprex v2.0.2}

This issue raises questions about the accuracy of the documentation. I am seeking clarification to determine whether the problem lies in the documentation itself before reporting a potential mistake to the R project. This inquiry is related to a theoretical concept, where a detailed explanation can be found here

Solution

Hmm looks like an error. Here is a valid identity:

v <- 5
t <- -1

## Student's t-distribution cumulative distribution function
pt(q = t, df = v, lower.tail = TRUE, ncp = 0)
#> [1] 0.1816087

x = (t + sqrt(t * t + v)) / (2.0 * sqrt(t * t + v))
pbeta(q = x, shape1 = v/2, shape2 = v/2, ncp = 0, lower.tail = TRUE)
#> [1] 0.1816087

And another one, closer to the claim of the R doc:

pbeta(q = v / (v + t^2), shape1 = v/2, shape2 = 1/2, 
            ncp = 0,lower.tail = TRUE) / 2
#> [1] 0.1816087