# NAG Library Function Document

## 1Purpose

nag_prob_students_t (g01ebc) returns the lower tail, upper tail or two tail probability for the Student's $t$-distribution with real degrees of freedom.

## 2Specification

 #include #include
 double nag_prob_students_t (Nag_TailProbability tail, double t, double df, NagError *fail)

## 3Description

The lower tail probability for the Student's $t$-distribution with $\nu$ degrees of freedom, $P\left(T\le t:\nu \right)$ is defined by:
 $P T≤t:ν = Γ ν+1 / 2 πν Γν/2 ∫ -∞ t 1+ T2ν -ν+1 / 2 dT , ν≥1 .$
Computationally, there are two situations:
(i) when $\nu <20$, a transformation of the beta distribution, ${P}_{\beta }\left(B\le \beta :a,b\right)$ is used
 $P T≤t:ν = 12 Pβ B≤ ν ν+t2 : ν/2, 12 when ​ t<0.0$
or
 $P T≤t:ν = 12 + 12 Pβ B≥ ν ν+t2 : ν/2, 12 when ​ t>0.0 ;$
(ii) when $\nu \ge 20$, an asymptotic normalizing expansion of the Cornish–Fisher type is used to evaluate the probability, see Hill (1970).

## 4References

Abramowitz M and Stegun I A (1972) Handbook of Mathematical Functions (3rd Edition) Dover Publications
Hastings N A J and Peacock J B (1975) Statistical Distributions Butterworth
Hill G W (1970) Student's $t$-distribution Comm. ACM 13(10) 617–619

## 5Arguments

1:    $\mathbf{tail}$Nag_TailProbabilityInput
On entry: indicates which tail the returned probability should represent.
${\mathbf{tail}}=\mathrm{Nag_UpperTail}$
The upper tail probability is returned, i.e., $P\left(T\ge t:\nu \right)$.
${\mathbf{tail}}=\mathrm{Nag_TwoTailSignif}$
The two tail (significance level) probability is returned, i.e., $P\left(T\ge \left|t\right|:\nu \right)+P\left(T\le -\left|t\right|:\nu \right)$.
${\mathbf{tail}}=\mathrm{Nag_TwoTailConfid}$
The two tail (confidence interval) probability is returned, i.e., $P\left(T\le \left|t\right|:\nu \right)-P\left(T\le -\left|t\right|:\nu \right)$.
${\mathbf{tail}}=\mathrm{Nag_LowerTail}$
The lower tail probability is returned, i.e., $P\left(T\le t:\nu \right)$.
Constraint: ${\mathbf{tail}}=\mathrm{Nag_UpperTail}$, $\mathrm{Nag_TwoTailSignif}$, $\mathrm{Nag_TwoTailConfid}$ or $\mathrm{Nag_LowerTail}$.
2:    $\mathbf{t}$doubleInput
On entry: $t$, the value of the Student's $t$ variate.
3:    $\mathbf{df}$doubleInput
On entry: $\nu$, the degrees of freedom of the Student's $t$-distribution.
Constraint: ${\mathbf{df}}\ge 1.0$.
4:    $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

## 6Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
NE_REAL_ARG_LT
On entry, ${\mathbf{df}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{df}}\ge 1.0$.

## 7Accuracy

The computed probability should be accurate to five significant places for reasonable probabilities but there will be some loss of accuracy for very low probabilities (less than ${10}^{-10}$), see Hastings and Peacock (1975).

## 8Parallelism and Performance

nag_prob_students_t (g01ebc) is not threaded in any implementation.

The probabilities could also be obtained by using the appropriate transformation to a beta distribution (see Abramowitz and Stegun (1972)) and using nag_prob_beta_dist (g01eec). This function allows you to set the required accuracy.

## 10Example

This example reads values from, and degrees of freedom for Student's $t$-distributions along with the required tail. The probabilities are calculated and printed until the end of data is reached.

### 10.1Program Text

Program Text (g01ebce.c)

### 10.2Program Data

Program Data (g01ebce.d)

### 10.3Program Results

Program Results (g01ebce.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017