Integer type:  int32  int64  nag_int  show int32  show int32  show int64  show int64  show nag_int  show nag_int

Chapter Contents
Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_stat_prob_beta_vector (g01se)

Purpose

nag_stat_prob_beta_vector (g01se) computes a number of lower or upper tail probabilities for the beta distribution.

Syntax

[p, ivalid, ifail] = g01se(tail, beta, a, b, 'ltail', ltail, 'lbeta', lbeta, 'la', la, 'lb', lb)
[p, ivalid, ifail] = nag_stat_prob_beta_vector(tail, beta, a, b, 'ltail', ltail, 'lbeta', lbeta, 'la', la, 'lb', lb)

Description

The lower tail probability, P( Bi βi : ai,bi) $P\left({B}_{i}\le {\beta }_{i}:{a}_{i},{b}_{i}\right)$ is defined by
 βi P( Bi ≤ βi : ai,bi) = ( Γ (ai + bi) )/( Γ (ai) Γ (bi) ) ∫ Biai − 1(1 − Bi)bi − 1dBi = Iβi(ai,bi),  0 ≤ βi ≤ 1; ai,bi > 0. 0
$P( Bi ≤ βi :ai,bi) = Γ ( ai + bi ) Γ (ai) Γ (bi) ∫ 0 βi Bi ai-1 (1-Bi) bi-1 dBi = Iβi (ai,bi) , 0 ≤ βi ≤ 1 ; ai , bi > 0 .$
The function Iβi(ai,bi)${I}_{{\beta }_{i}}\left({a}_{i},{b}_{i}\right)$, also known as the incomplete beta function is calculated using nag_specfun_beta_incomplete (s14cc).
The input arrays to this function are designed to allow maximum flexibility in the supply of vector parameters by re-using elements of any arrays that are shorter than the total number of evaluations required. See Section [Vectorized s] in the G01 Chapter Introduction for further information.

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

Parameters

Compulsory Input Parameters

1:     tail(ltail) – cell array of strings
ltail, the dimension of the array, must satisfy the constraint ltail > 0${\mathbf{ltail}}>0$.
Indicates whether a lower or upper tail probabilities are required. For j = ((i1)  mod  ltail) + 1 , for i = 1,2,,max (ltail,lbeta,la,lb)$\mathit{i}=1,2,\dots ,\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{ltail}},{\mathbf{lbeta}},{\mathbf{la}},{\mathbf{lb}}\right)$:
tail(j) = 'L'${\mathbf{tail}}\left(j\right)=\text{'L'}$
The lower tail probability is returned, i.e., pi = P( Bi βi : ai,bi) ${p}_{i}=P\left({B}_{i}\le {\beta }_{i}:{a}_{i},{b}_{i}\right)$.
tail(j) = 'U'${\mathbf{tail}}\left(j\right)=\text{'U'}$
The upper tail probability is returned, i.e., pi = P( Bi βi : ai,bi) ${p}_{i}=P\left({B}_{i}\ge {\beta }_{i}:{a}_{i},{b}_{i}\right)$.
Constraint: tail(j) = 'L'${\mathbf{tail}}\left(\mathit{j}\right)=\text{'L'}$ or 'U'$\text{'U'}$, for j = 1,2,,ltail$\mathit{j}=1,2,\dots ,{\mathbf{ltail}}$.
2:     beta(lbeta) – double array
lbeta, the dimension of the array, must satisfy the constraint lbeta > 0${\mathbf{lbeta}}>0$.
βi${\beta }_{i}$, the value of the beta variate with βi = beta(j)${\beta }_{i}={\mathbf{beta}}\left(j\right)$, j = ((i1)  mod  lbeta) + 1.
Constraint: 0.0beta(j)1.0$0.0\le {\mathbf{beta}}\left(\mathit{j}\right)\le 1.0$, for j = 1,2,,lbeta$\mathit{j}=1,2,\dots ,{\mathbf{lbeta}}$.
3:     a(la) – double array
la, the dimension of the array, must satisfy the constraint la > 0${\mathbf{la}}>0$.
ai${a}_{i}$, the first parameter of the required beta distribution with ai = a(j)${a}_{i}={\mathbf{a}}\left(j\right)$, j = ((i1)  mod  la) + 1.
Constraint: a(j) > 0.0${\mathbf{a}}\left(\mathit{j}\right)>0.0$, for j = 1,2,,la$\mathit{j}=1,2,\dots ,{\mathbf{la}}$.
4:     b(lb) – double array
lb, the dimension of the array, must satisfy the constraint lb > 0${\mathbf{lb}}>0$.
bi${b}_{i}$, the second parameter of the required beta distribution with bi = b(j)${b}_{i}={\mathbf{b}}\left(j\right)$, j = ((i1)  mod  lb) + 1.
Constraint: b(j) > 0.0${\mathbf{b}}\left(\mathit{j}\right)>0.0$, for j = 1,2,,lb$\mathit{j}=1,2,\dots ,{\mathbf{lb}}$.

Optional Input Parameters

1:     ltail – int64int32nag_int scalar
Default: The dimension of the array tail.
The length of the array tail.
Constraint: ltail > 0${\mathbf{ltail}}>0$.
2:     lbeta – int64int32nag_int scalar
Default: The dimension of the array beta.
The length of the array beta.
Constraint: lbeta > 0${\mathbf{lbeta}}>0$.
3:     la – int64int32nag_int scalar
Default: The dimension of the array a.
The length of the array a.
Constraint: la > 0${\mathbf{la}}>0$.
4:     lb – int64int32nag_int scalar
Default: The dimension of the array b.
The length of the array b.
Constraint: lb > 0${\mathbf{lb}}>0$.

None.

Output Parameters

1:     p( : $:$) – double array
Note: the dimension of the array p must be at least max (ltail,lbeta,la,lb)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{ltail}},{\mathbf{lbeta}},{\mathbf{la}},{\mathbf{lb}}\right)$.
pi${p}_{i}$, the probabilities for the beta distribution.
2:     ivalid( : $:$) – int64int32nag_int array
Note: the dimension of the array ivalid must be at least max (ltail,lbeta,la,lb)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{ltail}},{\mathbf{lbeta}},{\mathbf{la}},{\mathbf{lb}}\right)$.
ivalid(i)${\mathbf{ivalid}}\left(i\right)$ indicates any errors with the input arguments, with
ivalid(i) = 0${\mathbf{ivalid}}\left(i\right)=0$
No error.
ivalid(i) = 1${\mathbf{ivalid}}\left(i\right)=1$
 On entry, invalid value supplied in tail when calculating pi${p}_{i}$.
ivalid(i) = 2${\mathbf{ivalid}}\left(i\right)=2$
 On entry, βi < 0.0${\beta }_{i}<0.0$, or βi > 1.0${\beta }_{i}>1.0$.
ivalid(i) = 3${\mathbf{ivalid}}\left(i\right)=3$
 On entry, ai ≤ 0.0${a}_{i}\le 0.0$, or bi ≤ 0.0${b}_{i}\le 0.0$,
3:     ifail – int64int32nag_int scalar
${\mathrm{ifail}}={\mathbf{0}}$ unless the function detects an error (see [Error Indicators and Warnings]).

Error Indicators and Warnings

Note: nag_stat_prob_beta_vector (g01se) may return useful information for one or more of the following detected errors or warnings.
Errors or warnings detected by the function:

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

W ifail = 1${\mathbf{ifail}}=1$
On entry, at least one value of beta, a, b or tail was invalid.
ifail = 2${\mathbf{ifail}}=2$
Constraint: ltail > 0${\mathbf{ltail}}>0$.
ifail = 3${\mathbf{ifail}}=3$
Constraint: lbeta > 0${\mathbf{lbeta}}>0$.
ifail = 4${\mathbf{ifail}}=4$
Constraint: la > 0${\mathbf{la}}>0$.
ifail = 5${\mathbf{ifail}}=5$
Constraint: lb > 0${\mathbf{lb}}>0$.
ifail = 999${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.

Accuracy

The accuracy is limited by the error in the incomplete beta function. See Section [Accuracy] in (s14cc) for further details.

None.

Example

```function nag_stat_prob_beta_vector_example
tail = {'L'; 'L'; 'L'};
x = [0.25; 0.75; 0.5];
a = [1.0; 1.5; 2.0];
b = [2.0; 1.5; 1.0];
% calculate probability
[p, ivalid, ifail] = nag_stat_prob_beta_vector(tail, x, a, b);

fprintf('\n   X       A       B       P\n');
lx = numel(x);
la = numel(a);
lb = numel(b);
ltail = numel(tail);
len = max ([lx, la, lb, ltail]);
for i=0:len-1
fprintf('%6.4f%8.4f%8.4f%8.4f  %5d\n', x(mod(i,lx)+1), a(mod(i,la)+1), ...
b(mod(i,lb)+1), p(i+1), ivalid(i+1));
end
```
```

X       A       B       P
0.2500  1.0000  2.0000  0.4375      0
0.7500  1.5000  1.5000  0.8045      0
0.5000  2.0000  1.0000  0.2500      0

```
```function g01se_example
tail = {'L'; 'L'; 'L'};
x = [0.25; 0.75; 0.5];
a = [1.0; 1.5; 2.0];
b = [2.0; 1.5; 1.0];
% calculate probability
[p, ivalid, ifail] = g01se(tail, x, a, b);

fprintf('\n   X       A       B       P\n');
lx = numel(x);
la = numel(a);
lb = numel(b);
ltail = numel(tail);
len = max ([lx, la, lb, ltail]);
for i=0:len-1
fprintf('%6.4f%8.4f%8.4f%8.4f  %5d\n', x(mod(i,lx)+1), a(mod(i,la)+1), ...
b(mod(i,lb)+1), p(i+1), ivalid(i+1));
end
```
```

X       A       B       P
0.2500  1.0000  2.0000  0.4375      0
0.7500  1.5000  1.5000  0.8045      0
0.5000  2.0000  1.0000  0.2500      0

```