NAG Library Routine Document

Integer, Intent (In)	::	ln, n(ln), lp, lk, k(lk)
Integer, Intent (Inout)	::	ifail
Integer, Intent (Out)	::	ivalid(*)
Real (Kind=nag_wp), Intent (In)	::	p(lp)
Real (Kind=nag_wp), Intent (Out)	::	plek(), pgtk(), peqk(*)

C Header Interface

#include <nagmk26.h>

void	g01sjf_ (const Integer ln, const Integer n[], const Integer lp, const double p[], const Integer lk, const Integer k[], double plek[], double pgtk[], double peqk[], Integer ivalid[], Integer ifail)

3

Description

Let

X = \{X_{i} : i = 1, 2, \dots, m\}

denote a vector of random variables each having a binomial distribution with parameters

n_{i}

and

p_{i}

(

n_{i} \geq 0

and

0 < p_{i} < 1

). Then

Prob \{X_{i} = k_{i}\} = (\begin{array}{l} n_{i} \\ k_{i} \end{array}) {p_{i}}^{k_{i}} {(1 - p_{i})}^{n_{i} - k_{i}}, k_{i} = 0, 1, \dots, n_{i} .

The mean of the each distribution is given by

n_{i} p_{i}

and the variance by

n_{i} p_{i} (1 - p_{i})

g01sjf computes, for given

n_{i}

p_{i}

and

k_{i}

, the probabilities:

Prob \{X_{i} \leq k_{i}\}

Prob \{X_{i} > k_{i}\}

and

Prob \{X_{i} = k_{i}\}

using an algorithm similar to that described in Knüsel (1986) for the Poisson distribution.

The input arrays to this routine 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

Knüsel L (1986) Computation of the chi-square and Poisson distribution SIAM J. Sci. Statist. Comput. 7 1022–1036

5

Arguments

1: $\ln$ – IntegerInput

On entry: the length of the array n.

Constraint:

\ln > 0

2: $n (\ln)$ – Integer arrayInput

On entry:

n_{i}

, the first parameter of the binomial distribution with

n_{i} = n (j)

j = ((i - 1) mod \ln) + 1

, for

i = 1, 2, \dots, \max (\ln, lp, lk)

Constraint:

n (j) \geq 0

, for

j = 1, 2, \dots, \ln

3: $lp$ – IntegerInput

On entry: the length of the array p.

Constraint:

lp > 0

4: $p (lp)$ – Real (Kind=nag_wp) arrayInput

On entry:

p_{i}

, the second parameter of the binomial distribution with

p_{i} = p (j)

j = ((i - 1) mod lp) + 1

Constraint:

0.0 < p (j) < 1.0

, for

j = 1, 2, \dots, lp

5: $lk$ – IntegerInput

On entry: the length of the array k.

Constraint:

lk > 0

6: $k (lk)$ – Integer arrayInput

On entry:

k_{i}

, the integer which defines the required probabilities with

k_{i} = k (j)

j = ((i - 1) mod lk) + 1

Constraint:

0 \leq k_{i} \leq n_{i}

7: $plek (*)$ – Real (Kind=nag_wp) arrayOutput

Note: the dimension of the array plek must be at least

\max (\ln, lp, lk)

On exit:

Prob \{X_{i} \leq k_{i}\}

, the lower tail probabilities.

8: $pgtk (*)$ – Real (Kind=nag_wp) arrayOutput

Note: the dimension of the array pgtk must be at least

\max (\ln, lp, lk)

On exit:

Prob \{X_{i} > k_{i}\}

, the upper tail probabilities.

9: $peqk (*)$ – Real (Kind=nag_wp) arrayOutput

Note: the dimension of the array peqk must be at least

\max (\ln, lp, lk)

On exit:

Prob \{X_{i} = k_{i}\}

, the point probabilities.

10: $ivalid (*)$ – Integer arrayOutput

Note: the dimension of the array ivalid must be at least

\max (\ln, lp, lk)

On exit:

ivalid (i)

indicates any errors with the input arguments, with

$ivalid (i) = 0$

No error.

$ivalid (i) = 1$

On entry,

n_{i} < 0

$ivalid (i) = 2$

On entry,	$p_{i} \leq 0.0$ ,
or	$p_{i} \geq 1.0$ .

$ivalid (i) = 3$

On entry,	$k_{i} < 0$ ,
or	$k_{i} > n_{i}$ .

$ivalid (i) = 4$

On entry,

n_{i}

is too large to be represented exactly as a real number.

$ivalid (i) = 5$

On entry,

the variance (

= n_{i} p_{i} (1 - p_{i})

) exceeds

10^{6}

11: $ifail$ – IntegerInput/Output

On entry: ifail must be set to

0

- 1 or 1

. If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.

For environments where it might be inappropriate to halt program execution when an error is detected, the value

- 1 or 1

is recommended. If the output of error messages is undesirable, then the value

1

is recommended. Otherwise, if you are not familiar with this argument, the recommended value is

0

. When the value $- 1 or 1$ is used it is essential to test the value of ifail on exit.

On exit:

ifail = 0

unless the routine detects an error or a warning has been flagged (see Section 6).

6

Error Indicators and Warnings

If on entry

ifail = 0

- 1

, explanatory error messages are output on the current error message unit (as defined by x04aaf).

Errors or warnings detected by the routine:

$ifail = 1$: On entry, at least one value of n, p or k was invalid.
Check ivalid for more information.

$ifail = 2$: On entry, $array size = 〈value〉$ .
Constraint: $\ln > 0$ .

$ifail = 3$: On entry, $array size = 〈value〉$ .
Constraint: $lp > 0$ .

$ifail = 4$: On entry, $array size = 〈value〉$ .
Constraint: $lk > 0$ .

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

7

Accuracy

Results are correct to a relative accuracy of at least

10^{- 6}

on machines with a precision of

9

or more decimal digits, and to a relative accuracy of at least

10^{- 3}

on machines of lower precision (provided that the results do not underflow to zero).

8

Parallelism and Performance

g01sjf is not threaded in any implementation.

9

Further Comments

The time taken by g01sjf to calculate each probability depends on the variance (

= n_{i} p_{i} (1 - p_{i})

) and on

k_{i}

. For given variance, the time is greatest when

k_{i} \approx n_{i} p_{i}

(

= the mean

), and is then approximately proportional to the square-root of the variance.

10

Example

This example reads a vector of values for

n

p

and

k

, and prints the corresponding probabilities.

NAG Library Routine Document

g01sjf (prob_binomial_vector)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

1

Purpose

2

Specification

3

Description

4

References

5

Arguments

6

Error Indicators and Warnings

7

Accuracy

8

Parallelism and Performance

9

Further Comments

10

Example

10.1

Program Text

10.2

Program Data

10.3

Program Results