g07ab: FL CL CPP AD

NAG FL Interface
g07abf (ci_poisson)

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NAG Library Manual, Mark 27.2

Interfaces: FL CL CPP AD

NAG FL Interface Introduction

G07 (Univar) Chapter Contents

G07 (Univar) Chapter Introduction

g07ab: FL CL CPP AD

▸▿ 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

g07abf computes a confidence interval for the mean parameter of the Poisson distribution.

2 Specification

Fortran Interface

Subroutine g07abf (

n, xmean, clevel, tl, tu, ifail)

Integer, Intent (In)	::	n
Integer, Intent (Inout)	::	ifail
Real (Kind=nag_wp), Intent (In)	::	xmean, clevel
Real (Kind=nag_wp), Intent (Out)	::	tl, tu

C Header Interface

#include <nag.h>

void	g07abf_ (const Integer n, const double xmean, const double clevel, double tl, double tu, Integer ifail)

The routine may be called by the names g07abf or nagf_univar_ci_poisson.

3 Description

Given a random sample of size

n

, denoted by

x_{1}, x_{2}, \dots, x_{n}

, from a Poisson distribution with probability function

p (x) = e^{- θ} \frac{θ^{x}}{x!}, x = 0, 1, 2, \dots

the point estimate,

\hat{θ}

, for

θ

is the sample mean,

\bar{x}

Given

n

and

\bar{x}

this routine computes a

100 (1 - α) %

confidence interval for the parameter

θ

, denoted by [

θ_{l}, θ_{u}

], where

α

is in the interval

(0, 1)

The lower and upper confidence limits are estimated by the solutions to the equations

\begin{matrix} e^{- n θ_{l}} \sum_{x = T}^{\infty} \frac{{(n θ_{l})}^{x}}{x!} = \frac{α}{2}, \\ e^{- n θ_{u}} \sum_{x = 0}^{T} \frac{{(n θ_{u})}^{x}}{x!} = \frac{α}{2}, \end{matrix}

where

T = \sum_{i = 1}^{n} x_{i} = n \hat{θ}

The relationship between the Poisson distribution and the

χ^{2}

-distribution (see page 112 of Hastings and Peacock (1975)) is used to derive the equations

\begin{array}{l} θ_{l} = \frac{1}{2 n} χ_{2 T, α / 2}^{2}, \\ θ_{u} = \frac{1}{2 n} χ_{2 T + 2, 1 - α / 2}^{2}, \end{array}

where

χ_{ν, p}^{2}

is the deviate associated with the lower tail probability

p

of the

χ^{2}

-distribution with

ν

degrees of freedom.

In turn the relationship between the

χ^{2}

-distribution and the gamma distribution (see page 70 of Hastings and Peacock (1975)) yields the following equivalent equations;

\begin{array}{l} θ_{l} = \frac{1}{2 n} γ_{T, 2; α / 2}, \\ θ_{u} = \frac{1}{2 n} γ_{T + 1, 2; 1 - α / 2}, \end{array}

where

γ_{α, β; δ}

is the deviate associated with the lower tail probability,

δ

, of the gamma distribution with shape parameter

α

and scale parameter

β

. These deviates are computed using g01fff.

4 References

Hastings N A J and Peacock J B (1975) Statistical Distributions Butterworth

Snedecor G W and Cochran W G (1967) Statistical Methods Iowa State University Press

5 Arguments

1: $n$ – Integer Input: On entry: $n$ , the sample size.

Constraint: $n \geq 1$ .
2: $xmean$ – Real (Kind=nag_wp) Input: On entry: the sample mean, $\bar{x}$ .

Constraint: $xmean \geq 0.0$ .
3: $clevel$ – Real (Kind=nag_wp) Input: On entry: the confidence level, $(1 - α)$ , for two-sided interval estimate. For example $clevel = 0.95$ gives a $95 %$ confidence interval.

Constraint: $0.0 < clevel < 1.0$ .
4: $tl$ – Real (Kind=nag_wp) Output: On exit: the lower limit, $θ_{l}$ , of the confidence interval.
5: $tu$ – Real (Kind=nag_wp) Output: On exit: the upper limit, $θ_{u}$ , of the confidence interval.
6: $ifail$ – Integer Input/Output: On entry: ifail must be set to $0$ , $−1$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $−1$ means that an error message is printed while a value of $1$ means that it is not.

If halting is not appropriate, the value $−1$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $0$ is recommended. 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, $clevel = ⟨ value ⟩$ .
Constraint: $0.0 < clevel < 1.0$ .

On entry, $n = ⟨ value ⟩$ .
Constraint: $n > 0$ .

On entry, $xmean = ⟨ value ⟩$ .
Constraint: $xmean \geq 0.0$ .

$ifail = 2$: When using the relationship with the gamma distribution the series to calculate the gamma probabilities has failed to converge. Both tl and tu are set to zero. This is an unlikely error exit.

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

7 Accuracy

For most cases the results should have a relative accuracy of

\max (0.5E - 12, 50.0 \times ε)

where

ε

is the machine precision (see x02ajf). Thus on machines with sufficiently high precision the results should be accurate to

12

significant digits. Some accuracy may be lost when

α / 2

1 - α / 2

is very close to

0.0

, which will occur if clevel is very close to

1.0

. This should not affect the usual confidence intervals used.

8 Parallelism and Performance

g07abf is not threaded in any implementation.

9 Further Comments

None.

10 Example

The following example reads in data showing the number of noxious weed seeds and the frequency with which that number occurred in

98

subsamples of meadow grass. The data is taken from page 224 of Snedecor and Cochran (1967). The sample mean is computed as the point estimate of the Poisson parameter

θ

. g07abf is then called to compute both a 95% and a 99% confidence interval for the parameter

θ

g07ab: FL CL CPP AD

NAG FL Interfaceg07abf (ci_​poisson)

▸▿ 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

NAG FL Interface
g07abf (ci_poisson)