POIES
Estimates the parameter of the Poisson distribution.
Required Arguments
IX — Vector of length NOBS containing the data. (Input)
The data are assumed to be a random sample from a Poisson distribution; hence, all elements of IX must be nonnegative.
CONPER — Confidence level for two-sided interval estimate, in percent. (Input)
An approximate CONPER percent confidence interval is computed; hence, CONPER must be between 0.0 and 100.0. CONPER often will be 90.0, 95.0, or 99.0. For a one sided confidence interval with confidence level ONECL, set
CONPER = 100.0 ‑ 2.0 * (100.0 ‑ ONECL).
THAT — Estimate of the parameter, theta (the mean). (Output)
TLOWER — Lower confidence limit for theta. (Output)
TUPPER — Upper confidence limit for theta. (Output)
Optional Arguments
NOBS — Number of observations. (Input)
Default: NOBS = size (IX,1).
FORTRAN 90 Interface
Generic: CALL POIES (IX, CONPER, THAT, TLOWER, TUPPER [, …])
Specific: The specific interface names are S_POIES and D_POIES.
FORTRAN 77 Interface
Single: CALL POIES (NOBS, IX, CONPER, THAT, TLOWER, TUPPER)
Double: The double precision name is DPOIES.
Description
The routine POIES computes a point estimate and a confidence interval for the parameter, θ, of a Poisson distribution. It is assumed that the vector IX contains a random sample of size NOBS from a Poisson distribution with probability function
The point estimate for θ corresponds to the sample mean.
By exploiting the relationship between the Poisson distribution and the chi-squared distribution (see Johnson and Kotz, 1969, Chapter 4), the equations in Comment 2 can be written as
where
is the chi-squared 
critical value with degrees ν of freedom (that is, the inverse chi-squared distribution function evaluated at 1 ‑ 
). The routine CHIIN (see Chapter 17, “Probability Distribution Functions and Inverses”) is used to evaluate the critical values.
For more than one observation, the estimates are obtained as above and then divided by the number of observations, NOBS.
Comments
1. Informational error
|
Type |
Code |
Description |
|
3 |
1 |
CONPER is 0.0 or too small for accurate computations. The confidence limits are both set to THAT. |
2. Since the Poisson is a discrete distribution, it is not possible to construct an exact CONPER% confidence interval for all values of CONPER. Let α = 1 ‑ CONPER/100, and let k be a single observation. Then, the approximate lower and upper confidence limits θ L and θ U (TLOWER and TUPPER) are solutions to the equations
Example
It is assumed that flight arrivals at a major airport during the middle of the day follow a Poisson distribution. It is desired to estimate the mean number of arrivals per minute and to obtain an upper one-sided 95% confidence interval for the mean. During a half-hour period, the number of arrivals each minute was recorded. These data are stored in IX, and POIES is used to obtain the estimates.
USE POIES_INT
USE UMACH_INT
IMPLICIT NONE
INTEGER NOBS
PARAMETER (NOBS=30)
!
INTEGER IX(NOBS), NOUT
REAL CONPER, THAT, TLOWER, TUPPER
!
DATA IX/2, 0, 1, 1, 2, 0, 3, 1, 2, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, &
0, 1, 2, 0, 2, 0, 0, 1, 2, 0, 2/
!
CALL UMACH (2, NOUT)
! For a 95 percent one-sided ! .I.,
! CONPER = 100.0 - 2.0*(100.0-95.0)
CONPER = 90.0
CALL POIES (IX, CONPER, THAT, TLOWER, TUPPER)
WRITE (NOUT,99999) THAT, TUPPER
99999 FORMAT (' Point estimate of the Poisson mean: ', F5.3, /, &
' Upper one-sided 95% confidence limit: ', F5.3)
END
Output
Point estimate of the Poisson mean: 0.800
Upper one-sided 95% confidence limit: 1.125
