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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_stat_prob_students_t_noncentral (g01gb)

## Purpose

nag_stat_prob_students_t_noncentral (g01gb) returns the lower tail probability for the noncentral Student's t$t$-distribution.

## Syntax

[result, ifail] = g01gb(t, df, delta, 'tol', tol, 'maxit', maxit)
[result, ifail] = nag_stat_prob_students_t_noncentral(t, df, delta, 'tol', tol, 'maxit', maxit)
Note: the interface to this routine has changed since earlier releases of the toolbox:
Mark 23: tol now optional (default 0)
.

## Description

The lower tail probability of the noncentral Student's t$t$-distribution with ν$\nu$ degrees of freedom and noncentrality parameter δ$\delta$, P(Tt : ν;δ)$P\left(T\le t:\nu \text{;}\delta \right)$, is defined by
P(Tt : ν;δ) = Cν
 ( αu − δ ) 1/(sqrt(2π)) ∫ e − x2 / 2dx − ∞
uν1eu2 / 2du,  ν > 0.0
0
$P(T≤t:ν;δ)=Cν∫0∞ (12π∫-∞ αu-δe-x2/2dx) uν-1e-u2/2du, ν>0.0$
with
 Cν = 1/(Γ ((1/2)ν)2(ν − 2) / 2),   α = t/(sqrt(ν)). $Cν=1Γ (12ν )2(ν- 2)/2 , α=tν.$
The probability is computed in one of two ways.
(i) When t = 0.0$t=0.0$, the relationship to the normal is used:
 ∞ P(T ≤ t : ν;δ) = 1/(sqrt(2π)) ∫ e − u2 / 2du. δ
$P(T≤t:ν;δ)=12π∫δ∞e-u2/2du.$
(ii) Otherwise the series expansion described in Equation 9 of Amos (1964) is used. This involves the sums of confluent hypergeometric functions, the terms of which are computed using recurrence relationships.

## References

Amos D E (1964) Representations of the central and non-central t$t$-distributions Biometrika 51 451–458

## Parameters

### Compulsory Input Parameters

1:     t – double scalar
t$t$, the deviate from the Student's t$t$-distribution with ν$\nu$ degrees of freedom.
2:     df – double scalar
ν$\nu$, the degrees of freedom of the Student's t$t$-distribution.
Constraint: df1.0${\mathbf{df}}\ge 1.0$.
3:     delta – double scalar
δ$\delta$, the noncentrality parameter of the Students t$t$-distribution.

### Optional Input Parameters

1:     tol – double scalar
The absolute accuracy required by you in the results. If nag_stat_prob_students_t_noncentral (g01gb) is entered with tol greater than or equal to 1.0$1.0$ or less than 10 × machine precision (see nag_machine_precision (x02aj)), then the value of 10 × machine precision is used instead.
Default: 0.0$0.0$
2:     maxit – int64int32nag_int scalar
The maximum number of terms that are used in each of the summations.
Default: 100$100$. See Section [Further Comments] for further comments.
Constraint: maxit1${\mathbf{maxit}}\ge 1$.

None.

### Output Parameters

1:     result – double scalar
The result of the function.
2:     ifail – int64int32nag_int scalar
${\mathrm{ifail}}={\mathbf{0}}$ unless the function detects an error (see [Error Indicators and Warnings]).

## Error Indicators and Warnings

Errors or warnings detected by the function:
If on exit ${\mathbf{ifail}}\ne {\mathbf{0}}$, then nag_stat_prob_students_t_noncentral (g01gb) returns 0.0$0.0$.
ifail = 1${\mathbf{ifail}}=1$
 On entry, df < 1.0${\mathbf{df}}<1.0$.
ifail = 2${\mathbf{ifail}}=2$
 On entry, maxit < 1${\mathbf{maxit}}<1$.
ifail = 3${\mathbf{ifail}}=3$
One of the series has failed to converge. Reconsider the requested tolerance and/or maximum number of iterations.
ifail = 4${\mathbf{ifail}}=4$
The probability is too small to calculate accurately.

## Accuracy

The series described in Amos (1964) are summed until an estimated upper bound on the contribution of future terms to the probability is less than tol. There may also be some loss of accuracy due to calculation of gamma functions.

The rate of convergence of the series depends, in part, on the quantity t2 / (t2 + ν)${t}^{2}/\left({t}^{2}+\nu \right)$. The smaller this quantity the faster the convergence. Thus for large t$t$ and small ν$\nu$ the convergence may be slow. If ν$\nu$ is an integer then one of the series to be summed is of finite length.
If two tail probabilities are required then the relationship of the t$t$-distribution to the F$F$-distribution can be used:
 F = T2,λ = δ2,ν1 = 1  and  ν2 = ν, $F=T2,λ=δ2,ν1=1 and ν2=ν,$
and a call made to nag_stat_prob_f_noncentral (g01gd).
Note that nag_stat_prob_students_t_noncentral (g01gb) only allows degrees of freedom greater than or equal to 1$1$ although values between 0$0$ and 1$1$ are theoretically possible.

## Example

```function nag_stat_prob_students_t_noncentral_example
t = -1.528;
df = 20;
delta = 2;
[result, ifail] = nag_stat_prob_students_t_noncentral(t, df, delta)
```
```

result =

3.1800e-04

ifail =

0

```
```function g01gb_example
t = -1.528;
df = 20;
delta = 2;
[result, ifail] = g01gb(t, df, delta)
```
```

result =

3.1800e-04

ifail =

0

```