Zero-Truncated Distribution

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Counting distributions such as the binomial distribution and Poisson distribution are biased when it comes to modeling data without zero counts, such as modeling the number of people traveling in a plane. A zero-truncated distribution avoids bias by “truncating” — cutting off — the distribution so that zeros are excluded.

Zero-truncated distributions include the zero-truncated binomial distribution and the zero-truncated negative binomial distribution. However, the Zero-truncated Poisson (ZTP) distribution is the most widely used because it is the simplest to understand and work with.

Zero-truncated distribution and (a, b, 1) class

The (a, b, 1) class of distributions is a class of discrete probability distributions that are derived from the zero-modified or (a, b, 0) class of distributions by conditioning on the event that a random variable is nonzero. “Conditioning on the event” refers to focusing only on the outcomes where the random variable is not zero.

Three zero-truncated distributions are included in the (a, b, 1) class – the zero-truncated Poisson, binomial and negative binomial distributions. However, the (a, b, 1) distribution is not synonymous with zero-truncated distribution, as it contains the zero-truncated and zero-modified version of a distribution.

The Zero-Truncated Poisson (ZTP) distribution

The zero-truncated Poisson distribution, developed in 1952 by David and Johnson [1], is a modified Poisson distribution conditioned on being nonzero. This means that the probability of observing a value of zero is zero. In other words, the zero-truncated Poisson only considers Poisson random values that are greater than or equal to one [2].

The distribution has a parameter θ, which belongs to the interval −∞<θ<∞, and the data x falls within the set of positive integers {1, 2, 3, …}.

Let’s say we have

where m is the mean of the untruncated Poisson distribution and μ is the mean of the zero-truncated Poisson distribution.

To compute the log likelihood and its derivatives, we can use the following formulas:

In statistics and in the programming language R, the function log denotes the natural logarithm.

The second derivative formulas have different parameter values, and these can cause issues with “catastrophic cancellation”. Thus, calculating them is a highly challenging problem to calculate accurately [2].

Related distributions that have been developed for non-zero data include

• The zero-truncated Poisson-Amarendra distribution [3],
• The zero-truncated Poisson-Akash distribution [4],
• The zero-truncated two-parameter Poisson-Lindley distribution [5].

References

1. David, F. N, & N. L. Johnson, ”The Truncated Poisson.” Biometrics, 8(4), 275-285 (1952).
2. Geyer, C. (2017). Stat 3701 Lecture Notes: Zero-Truncated Poisson Distribution. Licensed under a Creative Commons Attribution-ShareAlike 4.0 International License (http://creativecommons.org/licenses/by-sa/4.0/).
3. Shanker, R., ”A zero-truncated Poisson-Amarendra distribution and its application,” International Journal of Probability and Statistics, 6(4), 82-92(2017).
4. Shanker, R., & K. K. Shukla, ”THE POISSON-WEIGHTED AKASH DISTRIBUTION AND ITS APPLICATIONS.” Journal of Applied Quantitative Methods, vol. 13, no. 2, 2018, p. 23.
5. Shanker, R., & K. K. Shukla, ”A Zero-Truncated Two-Parameter Poisson Lindley Distribution with an Application to Biological Science.” Turkiye Klinikleri Biyoistatistik, vol. 9, no. 2, 2017, pp. 85-95