# pnt

##### Non-central t Probability Distribution - Algorithms and Approximations

Compute different approximations for the non-central t-Distribution cumulative probability distribution function.

- Keywords
- distribution, math

##### Usage

```
pntR (t, df, ncp, lower.tail = TRUE, log.p = FALSE,
itrmax = 1000, errmax = 1e-12, verbose = TRUE)
pntR1 (t, df, ncp, lower.tail = TRUE, log.p = FALSE,
itrmax = 1000, errmax = 1e-12, verbose = TRUE)
pnt3150 (t, df, ncp, lower.tail = TRUE, log.p = FALSE, M = 1000, verbose = TRUE)
pnt3150.1 (t, df, ncp, lower.tail = TRUE, log.p = FALSE, M = 1000, verbose = TRUE)
pntP94 (t, df, ncp, lower.tail = TRUE, log.p = FALSE,
itrmax = 1000, errmax = 1e-12, verbose = TRUE)
pntP94.1 (t, df, ncp, lower.tail = TRUE, log.p = FALSE,
itrmax = 1000, errmax = 1e-12, verbose = TRUE)
pntLrg (t, df, ncp, lower.tail = TRUE, log.p = FALSE)
pntJW39 (t, df, ncp, lower.tail = TRUE, log.p = FALSE)
pntJW39.0 (t, df, ncp, lower.tail = TRUE, log.p = FALSE)
pntChShP94 (t, df, ncp, lower.tail = TRUE, log.p = FALSE,
itrmax = 1000, errmax = 1e-12, verbose = TRUE)
pntChShP94.1(t, df, ncp, lower.tail = TRUE, log.p = FALSE,
itrmax = 1000, errmax = 1e-12, verbose = TRUE)
```

##### Arguments

- t
vector of quantiles (called

`q`

in`pt(..)`

.- df
degrees of freedom (\(> 0\), maybe non-integer).

`df = Inf`

is allowed.- ncp
non-centrality parameter \(\delta\); currently except for

`rt()`

, only for`abs(ncp) <= 37.62`

. If omitted, use the central t distribution.- log, log.p
logical; if TRUE, probabilities p are given as log(p).

- lower.tail
logical; if TRUE (default), probabilities are \(P[X \le x]\), otherwise, \(P[X > x]\).

- itrmax
...

- errmax
...

- verbose
...

- M
positive integer specifying the number of terms to use in the series.

##### Details

`.`

:..

`.`

:..

`pnt3150()`

,`pnt3150.1()`

:Simple inefficient but hopefully correct version of pntP94..() This is really a direct implementation of formula (31.50), p.532 of Johnson, Kotz and Balakrishnan (1995)

`pntP94()`

,`pntP94.1()`

:New versions of

`pntR1()`

,`pntR()`

; using the Posten (1994) algorithm.`pntP94()`

is the`Vectorize()`

d version of`pntP94.1()`

.`.`

:..

`.`

:..

##### Value

a number for `pntJKBf1()`

and `.pntJKBch1()`

.

a numeric vector of the same length as the maximum of the lengths of
`x, df, ncp`

for `pntJKBf()`

and `.pntJKBch()`

.

##### References

Johnson, N.L., Kotz, S. and Balakrishnan, N. (1995)
Continuous Univariate Distributions Vol~2, 2nd ed.; Wiley.
Chapter 31, Section *5 Distribution Function*, p.514 ff

##### See Also

`pt`

, for R's version of non-central t probabilities.

##### Examples

```
# NOT RUN {
tt <- seq(0, 10, len = 21)
ncp <- seq(0, 6, len = 31)
dt3R <- outer(tt, ncp, pt, , df = 3)
dt3JKB <- outer(tt, ncp, pntR, df = 3)# currently verbose
stopifnot(all.equal(dt3R, dt3JKB, tolerance = 4e-15))# 64-bit Lnx: 2.78e-16
# }
```

*Documentation reproduced from package DPQ, version 0.3-3, License: GPL (>= 2)*