Description

Estimate quantiles of an Poisson distribution, and optionally contruct a confidence interval for a quantile.

Usage

Arguments

Details

The function eqpois returns estimated quantiles as well as the estimate of the mean parameter.

Estimation
Let X denote a Poisson random variable with parameter lambda=λ. Let x_{p|λ} denote the p'th quantile of the distribution. That is,

Pr(X < x_{p|λ}) ≤ p ≤ Pr(X ≤ x_{p|λ}) \;\;\;\; (1)

Note that due to the discrete nature of the Poisson distribution, there will be several values of p associated with one value of X. For example, for λ=2, the value 1 is the p'th quantile for any value of p between 0.14 and 0.406.

Let \underline{x} denote a vector of n observations from a Poisson distribution with parameter lambda=λ. The p'th quantile is estimated as the p'th quantile from a Poisson distribution assuming the true value of λ is equal to the estimated value of λ. That is:

\hat{x}_{p|λ} = x_{p|λ=\hat{λ}} \;\;\;\; (2)

where

\hat{λ} = \bar{x} = \frac{1}{n} ∑_{i=1}^n x_i \;\;\;\; (3)

Because the estimator in equation (3) is the maximum likelihood estimator of λ (see the help file for epois), the estimated quantile is the maximum likelihood estimator.

Quantiles are estimated by 1) estimating the mean parameter by calling epois, and then 2) calling the function qpois and using the estimated value for the mean parameter.

Confidence Intervals
It can be shown (e.g., Conover, 1980, pp.119-121) that an upper confidence interval for the p'th quantile with confidence level 100(1-α)\% is equivalent to an upper β-content tolerance interval with coverage 100p\% and confidence level 100(1-α)\%. Also, a lower confidence interval for the p'th quantile with confidence level 100(1-α)\% is equivalent to a lower β-content tolerance interval with coverage 100(1-p)\% and confidence level 100(1-α)\%.

Thus, based on the theory of tolerance intervals for a Poisson distribution (see tolIntPois), if ci.type="upper", a one-sided upper 100(1-α)\% confidence interval for the p'th quantile is constructed as:

[0, x_{p|λ=UCL}] \;\;\;\; (4)

where UCL denotes the upper 100(1-α)\% confidence limit for λ (see the help file for epois for information on how UCL is computed).

Similarly, if ci.type="lower", a one-sided lower 100(1-α)\% confidence interval for the p'th quantile is constructed as:

[x_{p|λ=LCL}, ∞] \;\;\;\; (5)

where LCL denotes the lower 100(1-α)\% confidence limit for λ (see the help file for epois for information on how LCL is computed).

Finally, if ci.type="two-sided", a two-sided 100(1-α)\% confidence interval for the p'th quantile is constructed as:

[x_{p|λ=LCL}, x_{p|λ=UCL}] \;\;\;\; (6)

where LCL and UCL denote the two-sided lower and upper 100(1-α)\% confidence limits for λ (see the help file for epois for information on how LCL and UCL are computed).

Value

If x is a numeric vector, eqpois returns a list of class "estimate" containing the estimated quantile(s) and other information. See estimate.object for details.

If x is the result of calling an estimation function, eqpois returns a list whose class is the same as x. The list contains the same components as x, as well as components called quantiles and quantile.method.

Note

Percentiles are sometimes used in environmental standards and regulations. For example, Berthouex and Brown (2002, p.71) state:

The U.S. EPA has specifications for air quality monitoring that are, in effect, percentile limitations. ... The U.S. EPA has provided guidance for setting aquatic standards on toxic chemicals that require estimating 99th percentiles and using this statistic to make important decisions about monitoring and compliance. They have also used the 99th percentile to establish maximum daily limits for industrial effluents (e.g., pulp and paper).

Given the importance of these quantities, it is essential to characterize the amount of uncertainty associated with the estimates of these quantities. This is done with confidence intervals.

The Poisson distribution is named after Poisson, who derived this distribution as the limiting distribution of the binomial distribution with parameters size=N and prob=p, where N tends to infinity, p tends to 0, and Np stays constant.

In this context, the Poisson distribution was used by Bortkiewicz (1898) to model the number of deaths (per annum) from kicks by horses in Prussian Army Corps. In this case, p, the probability of death from this cause, was small, but the number of soldiers exposed to this risk, N, was large.

The Poisson distribution has been applied in a variety of fields, including quality control (modeling number of defects produced in a process), ecology (number of organisms per unit area), and queueing theory. Gibbons (1987b) used the Poisson distribution to model the number of detected compounds per scan of the 32 volatile organic priority pollutants (VOC), and also to model the distribution of chemical concentration (in ppb).

Author(s)

Steven P. Millard (EnvStats@ProbStatInfo.com)

References

Berthouex, P.M., and L.C. Brown. (2002). Statistics for Environmental Engineers. Second Edition. Lewis Publishers, Boca Raton, FL.

Berthouex, P.M., and I. Hau. (1991). Difficulties Related to Using Extreme Percentiles for Water Quality Regulations. Research Journal of the Water Pollution Control Federation 63(6), 873–879.

Conover, W.J. (1980). Practical Nonparametric Statistics. Second Edition. John Wiley and Sons, New York, Chapter 3.

Forbes, C., M. Evans, N. Hastings, and B. Peacock. (2011). Statistical Distributions. Fourth Edition. John Wiley and Sons, Hoboken, NJ.

Gibbons, R.D. (1987b). Statistical Models for the Analysis of Volatile Organic Compounds in Waste Disposal Sites. Ground Water 25, 572-580.

Gibbons, R.D., D.K. Bhaumik, and S. Aryal. (2009). Statistical Methods for Groundwater Monitoring, Second Edition. John Wiley & Sons, Hoboken.

Johnson, N. L., S. Kotz, and A. Kemp. (1992). Univariate Discrete Distributions. Second Edition. John Wiley and Sons, New York, Chapter 4.

Pearson, E.S., and H.O. Hartley, eds. (1970). Biometrika Tables for Statisticians, Volume 1. Cambridge Universtiy Press, New York, p.81.

Zar, J.H. (2010). Biostatistical Analysis. Fifth Edition. Prentice-Hall, Upper Saddle River, NJ.

See Also

epois, Poisson, codeestimate.object.

Examples