g01 Chapter Contents
g01 Chapter Introduction
NAG Library Manual

# NAG Library Function Documentnag_prob_beta_vector (g01sec)

## 1  Purpose

nag_prob_beta_vector (g01sec) computes a number of lower or upper tail probabilities for the beta distribution.

## 2  Specification

 #include #include
 void nag_prob_beta_vector (Integer ltail, const Nag_TailProbability tail[], Integer lbeta, const double beta[], Integer la, const double a[], Integer lb, const double b[], double p[], Integer ivalid[], NagError *fail)

## 3  Description

The lower tail probability, $P\left({B}_{i}\le {\beta }_{i}:{a}_{i},{b}_{i}\right)$ is defined by
 $P Bi ≤ βi :ai,bi = Γ ai + bi Γ ai Γ bi ∫ 0 βi Bi ai-1 1-Bi bi-1 dBi = Iβi ai,bi , 0 ≤ βi ≤ 1 ; ai , bi > 0 .$
The function ${I}_{{\beta }_{i}}\left({a}_{i},{b}_{i}\right)$, also known as the incomplete beta function is calculated using nag_incomplete_beta (s14ccc).
The input arrays to this function are designed to allow maximum flexibility in the supply of vector arguments by re-using elements of any arrays that are shorter than the total number of evaluations required. See Section 2.6 in the g01 Chapter Introduction for further information.

## 4  References

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
Majumder K L and Bhattacharjee G P (1973) Algorithm AS 63. The incomplete beta integral Appl. Statist. 22 409–411

## 5  Arguments

1:    $\mathbf{ltail}$IntegerInput
On entry: the length of the array tail.
Constraint: ${\mathbf{ltail}}>0$.
2:    $\mathbf{tail}\left[{\mathbf{ltail}}\right]$const Nag_TailProbabilityInput
On entry: indicates whether a lower or upper tail probabilities are required. For , for $\mathit{i}=1,2,\dots ,\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{ltail}},{\mathbf{lbeta}},{\mathbf{la}},{\mathbf{lb}}\right)$:
${\mathbf{tail}}\left[j\right]=\mathrm{Nag_LowerTail}$
The lower tail probability is returned, i.e., ${p}_{i}=P\left({B}_{i}\le {\beta }_{i}:{a}_{i},{b}_{i}\right)$.
${\mathbf{tail}}\left[j\right]=\mathrm{Nag_UpperTail}$
The upper tail probability is returned, i.e., ${p}_{i}=P\left({B}_{i}\ge {\beta }_{i}:{a}_{i},{b}_{i}\right)$.
Constraint: ${\mathbf{tail}}\left[\mathit{j}-1\right]=\mathrm{Nag_LowerTail}$ or $\mathrm{Nag_UpperTail}$, for $\mathit{j}=1,2,\dots ,{\mathbf{ltail}}$.
3:    $\mathbf{lbeta}$IntegerInput
On entry: the length of the array beta.
Constraint: ${\mathbf{lbeta}}>0$.
4:    $\mathbf{beta}\left[{\mathbf{lbeta}}\right]$const doubleInput
On entry: ${\beta }_{i}$, the value of the beta variate with ${\beta }_{i}={\mathbf{beta}}\left[j\right]$, .
Constraint: $0.0\le {\mathbf{beta}}\left[\mathit{j}-1\right]\le 1.0$, for $\mathit{j}=1,2,\dots ,{\mathbf{lbeta}}$.
5:    $\mathbf{la}$IntegerInput
On entry: the length of the array a.
Constraint: ${\mathbf{la}}>0$.
6:    $\mathbf{a}\left[{\mathbf{la}}\right]$const doubleInput
On entry: ${a}_{i}$, the first parameter of the required beta distribution with ${a}_{i}={\mathbf{a}}\left[j\right]$, .
Constraint: ${\mathbf{a}}\left[\mathit{j}-1\right]>0.0$, for $\mathit{j}=1,2,\dots ,{\mathbf{la}}$.
7:    $\mathbf{lb}$IntegerInput
On entry: the length of the array b.
Constraint: ${\mathbf{lb}}>0$.
8:    $\mathbf{b}\left[{\mathbf{lb}}\right]$const doubleInput
On entry: ${b}_{i}$, the second parameter of the required beta distribution with ${b}_{i}={\mathbf{b}}\left[j\right]$, .
Constraint: ${\mathbf{b}}\left[\mathit{j}-1\right]>0.0$, for $\mathit{j}=1,2,\dots ,{\mathbf{lb}}$.
9:    $\mathbf{p}\left[\mathit{dim}\right]$doubleOutput
Note: the dimension, dim, of the array p must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{ltail}},{\mathbf{lbeta}},{\mathbf{la}},{\mathbf{lb}}\right)$.
On exit: ${p}_{i}$, the probabilities for the beta distribution.
10:  $\mathbf{ivalid}\left[\mathit{dim}\right]$IntegerOutput
Note: the dimension, dim, of the array ivalid must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{ltail}},{\mathbf{lbeta}},{\mathbf{la}},{\mathbf{lb}}\right)$.
On exit: ${\mathbf{ivalid}}\left[i-1\right]$ indicates any errors with the input arguments, with
${\mathbf{ivalid}}\left[i-1\right]=0$
No error.
${\mathbf{ivalid}}\left[i-1\right]=1$
 On entry, invalid value supplied in tail when calculating ${p}_{i}$.
${\mathbf{ivalid}}\left[i-1\right]=2$
 On entry, ${\beta }_{i}<0.0$, or ${\beta }_{i}>1.0$.
${\mathbf{ivalid}}\left[i-1\right]=3$
 On entry, ${a}_{i}\le 0.0$, or ${b}_{i}\le 0.0$,
11:  $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

## 6  Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.2.1.2 in the Essential Introduction for further information.
NE_ARRAY_SIZE
On entry, $\text{array size}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{la}}>0$.
On entry, $\text{array size}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{lb}}>0$.
On entry, $\text{array size}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{lbeta}}>0$.
On entry, $\text{array size}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{ltail}}>0$.
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 3.6.6 in the Essential Introduction for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 3.6.5 in the Essential Introduction for further information.
NW_IVALID
On entry, at least one value of beta, a, b or tail was invalid.

## 7  Accuracy

The accuracy is limited by the error in the incomplete beta function. See Section 7 in nag_incomplete_beta (s14ccc) for further details.

Not applicable.

None.

## 10  Example

This example reads values from a number of beta distributions and computes the associated lower tail probabilities.

### 10.1  Program Text

Program Text (g01sece.c)

### 10.2  Program Data

Program Data (g01sece.d)

### 10.3  Program Results

Program Results (g01sece.r)