Estimates the parameter of the Poisson distribution.
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)
NOBS — Number of
observations. (Input)
Default: NOBS = size (IX,1).
Generic: CALL POIES (IX, CONPER, THAT, TLOWER, TUPPER [,…])
Specific: The specific interface names are S_POIES and D_POIES.
Single: CALL POIES (NOBS, IX, CONPER, THAT, TLOWER, TUPPER)
Double: The double precision name is DPOIES.
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 τ critical value with degrees ν of freedom (that is, the inverse chi-squared distribution function evaluated at 1 − τ ). 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.
1. Informational error
Type Code
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
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
Point estimate of the Poisson mean: 0.800
Upper one-sided
95% confidence limit: 1.125
PHONE: 713.784.3131 FAX:713.781.9260 |