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CONDITIONAL POISSON DISTRIBUTIONS

Published online by Cambridge University Press:  12 December 2005

M. Zhou ,
D. Yang ,
Y. Wang  and
S. Nadarajah
M. Zhou
Affiliation:
Department of Statistics, University of Nebraska, Lincoln, NE 68583, E-mail: snadaraj@unlserve.unl.edu
D. Yang
Affiliation:
Department of Statistics, University of Nebraska, Lincoln, NE 68583, E-mail: snadaraj@unlserve.unl.edu
Y. Wang
Affiliation:
Department of Statistics, University of Nebraska, Lincoln, NE 68583, E-mail: snadaraj@unlserve.unl.edu
S. Nadarajah
Affiliation:
Department of Statistics, University of Nebraska, Lincoln, NE 68583, E-mail: snadaraj@unlserve.unl.edu
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Abstract

Compared to the known univariate distributions for continuous data, there are relatively few available for discrete data. In this article, we derive a collection of 16 flexible discrete distributions by means of conditional Poisson processes. The calculations involve the use of several special functions and their properties.

Information

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 20 , Issue 1 , January 2006 , pp. 95 - 102
Copyright
© 2006 Cambridge University Press

1. INTRODUCTION

Compared to the great multitude of continuous univariate distributions, there are relatively few choices available with respect to univariate discrete distributions. This is evident from the length of the compendiums of distributions available in the literature; see Johnson, Kotz, and Balakrishnan [2,3] for continuous distributions and Johnson, Kotz, and Kemp [4] for discrete distributions.

In this article, we present a collection of new discrete distributions. These are generated by means of conditional Poisson processes (Ross [6]); suppose {N(t),t ≥ 0}, where N(t) denotes the number of events during a time period of length t, is a Poisson process with rate parameter Λ. If g(λ) denotes the probability density function (p.d.f.) of Λ, then the unconditional distribution of N(t) can be written as

Now a discrete distribution for N(t) can be generated by substituting a valid form for g(λ). In this article, we generate a collection of discrete distributions for N(t) by taking g(λ) to belong to 16 flexible families. The calculations use several special functions, including the integral cosine defined by

the integral sine defined by

the incomplete gamma function defined by

the error function defined by

the modified Bessel function of the third kind defined by

the parabolic cylinder function defined by

the 1F1 hypergeometric function (also known as the confluent hypergeometric function) defined by

the 1F2 hypergeometric function defined by

and the Kummer function defined by

where (f)k = f (f + 1)···(f + k − 1) denotes the ascending factorial. The properties of these special functions can be found in Prudnikov, Brychkov, and Marichev [5] and Gradshteyn and Ryzhik [1].

The details of the derivation for (1) are not given in this article and can be obtained from the authors. The structural properties of N(t) are also not given since they can be obtained directly from those of Λ. For example, the mean and the variance of N(t) are

respectively. Thus, these follow directly by knowing E(Λ) and E(Λ2); see Johnson et al. [2,3].

2. DISCRETE MODELS

In this section, we provide a collection of formulas for Pr{N(t) = n} by taking g to belong to 16 flexible families.

Beta distribution: If g takes the form

for 0 < λ < 1, then

If g takes the form of the generalized beta distribution given by

for a < λ < b, then

Uniform distribution: If g takes the form

for a < λ < b, then

Inverted beta distribution: If g takes the form

for λ > 0, then

Exponential distribution: If g takes the form

for λ > 0, then

Gamma distribution: If g takes the form

for λ > 0, then

Rayleigh distribution: If g takes the form

for λ > 0, then

Stacy distribution (c = 2): If g takes the form

for λ > 0, then

If 2α is an integer, then the above reduces to the simpler form

Pareto distribution of the first kind: If g takes the form

for λ > k, then

Pareto distribution of the second kind: If g takes the form

for λ > 0, then

Inverse Gaussian distribution: If g takes the form

for λ > 0, then

Half Normal distribution: If g takes the form

for λ > 0, then

Half logistic distribution: If g takes the form

for λ > 0, then

Half Cauchy distribution: If g takes the form

for λ > 0, then

Half t distribution: If g takes the form

for λ > 0, then

If (1 + ν)/2 is an integer, then the above reduces to the simpler form

where

Fréchet distribution: If g takes the form

for λ > 0, then

Pearson type VI distribution: If g takes the form

for λ ≥ b > a > 0, then

3. CONCLUSIONS

We have generated a collection of 16 flexible discrete distributions. The definition of the conditional Poisson process is used as the mathematical tool. We hope that this work will help to address the inadequacy of the number of distributions available to model discrete data.

References

REFERENCES

Gradshteyn, I.S. & Ryzhik, I.M. (2000). Table of integrals, series, and products, 6th ed. San Diego: Academic Press.
Johnson, N.L., Kotz, S., & Balakrishnan, N. (1994). Continuous univariate distributions, Vol. 1, 2nd ed. New York: Wiley.
Johnson, N.L., Kotz, S., & Balakrishnan, N. (1995). Continuous univariate distributions, Vol. 2, 2nd ed. New York: Wiley.
Johnson, N.L., Kotz, S., & Kemp, A.W. (1992). Univariate discrete distributions, 2nd ed. New York: Wiley.
Prudnikov, A.P., Brychkov, Y.A., & Marichev, O.I. (1986). Integrals and series, Vols. 1, 2, and 3. Amsterdam: Gordon and Breach Science.
Ross, S.M. (2000). Introduction to probability models, 7th ed. Burlington, MA: Harcourt/Academic Press.