< Probability distributions list >* Zero-truncated distribution*

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

- David, F. N, & N. L. Johnson, ”The Truncated Poisson.” Biometrics, 8(4), 275-285 (1952).
- 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/).
- Shanker, R., ”A zero-truncated Poisson-Amarendra distribution and its application,” International Journal of Probability and Statistics, 6(4), 82-92(2017).
- 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.
- 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