Zero-truncated Poisson distribution

In probability theory, the zero-truncated Poisson (ZTP) distribution is a certain discrete probability distribution whose support is the set of positive integers. This distribution is also known as the conditional Poisson distribution^[1] or the positive Poisson distribution.^[2] It is the conditional probability distribution of a Poisson-distributed random variable, given that the value of the random variable is not zero. Thus it is impossible for a ZTP random variable to be zero. Consider for example the random variable of the number of items in a shopper's basket at a supermarket checkout line. Presumably a shopper does not stand in line with nothing to buy (i.e., the minimum purchase is 1 item), so this phenomenon may follow a ZTP distribution.^[3]

Since the ZTP is a truncated distribution with the truncation stipulated as k > 0, one can derive the probability mass function g(k;λ) from a standard Poisson distribution f(k;λ) as follows: ^[4]

${\displaystyle g(k;\lambda )=P(X=k\mid X>0)={\frac {f(k;\lambda )}{1-f(0;\lambda )}}={\frac {\lambda ^{k}e^{-\lambda }}{k!\left(1-e^{-\lambda }\right)}}={\frac {\lambda ^{k}}{(e^{\lambda }-1)k!}}}$

The mean is

${\displaystyle \operatorname {E} [X]={\frac {\lambda }{1-e^{-\lambda }}}={\frac {\lambda e^{\lambda }}{e^{\lambda }-1}}}$

and the variance is

${\displaystyle \operatorname {Var} [X]={\frac {\lambda +\lambda ^{2}}{1-e^{-\lambda }}}-{\frac {\lambda ^{2}}{(1-e^{-\lambda })^{2}}}=\operatorname {E} [X](1+\lambda -\operatorname {E} [X])}$

Parameter estimation

The method of moments estimator ${\displaystyle {\widehat {\lambda }}}$ for the parameter ${\displaystyle \lambda }$ is obtained by solving

${\displaystyle {\frac {\widehat {\lambda }}{1-e^{-{\widehat {\lambda }}}}}={\bar {x}}}$

where ${\displaystyle {\bar {x}}}$ is the sample mean.^[1]

This equation has a solution in terms of the Lambert W function. In practice, a solution may be found using numerical methods.

Examples

Insurance claims:

Imagine navigating the intricate landscape of auto insurance claims, where each claim signifies a unique event – an accident or damage occurrence. The ZTP distribution seamlessly aligns with this scenario, excluding the possibility of policyholders with zero claims.

Let X denote the random variable representing the number of insurance claims. If λ is the average rate of claims, the ZTP probability mass function takes the form:

${\displaystyle P(X=k)={\frac {\lambda ^{k}e^{-\lambda }}{k!\left(1-e^{-\lambda }\right)}}}$ for k= 1,2,3,...

This formula encapsulates the probability of observing k claims given that at least one claim has transpired. The denominator ensures the exclusion of the improbable zero-claim scenario. By utilizing the zero-truncated Poisson distribution, the manufacturing company can analyze and predict the frequency of defects in their products while focusing on instances where defects exist. This distribution helps in understanding and improving the quality control process, especially when it's crucial to account for at least one defect.

Generating zero-truncated Poisson-distributed random variables

Random variables sampled from the zero-truncated Poisson distribution may be achieved using algorithms derived from Poisson distribution sampling algorithms.^[5]

init:
    Let k ← 1, t ← e−λ / (1 - e−λ) * λ, s ← t.
    Generate uniform random number u in [0,1].
while s < u do:
    k ← k + 1.
    t ← t * λ / k.
    s ← s + t.
return k.

The cost of the procedure above is linear in k, which may be large for large values of ${\displaystyle \lambda }$. Given access to an efficient sampler for non-truncated Poisson random variates, a non-iterative approach involves sampling from a truncated exponential distribution representing the time of the first event in a Poisson point process, conditional on such an event existing.^[6] A simple NumPy implementation is:

def sample_zero_truncated_poisson(rate):
    u = np.random.uniform(np.exp(-rate), 1)
    t = -np.log(u)
    return 1 + np.random.poisson(rate - t)

References

1. Cohen, A. Clifford (1960). "Estimating parameters in a conditional Poisson distribution". Biometrics. 16 (2): 203–211. doi:10.2307/2527552. JSTOR 2527552.
2. Singh, Jagbir (1978). "A characterization of positive Poisson distribution and its application". SIAM Journal on Applied Mathematics. 34: 545–548. doi:10.1137/0134043.
3. "Stata Data Analysis Examples: Zero-Truncated Poisson Regression". UCLA Institute for Digital Research and Education. Retrieved 7 August 2013.
4. Johnson, Norman L.; Kemp, Adrianne W.; Kotz, Samuel (2005). Univariate Discrete Distributions (third ed.). Hoboken, NJ: Wiley-Interscience.
5. Borje, Gio (2016-06-01). "Zero-Truncated Poisson Distribution Sampling Algorithm". Archived from the original on 2018-08-26.
6. Hardie, Ted (1 May 2005). "[R] simulate zero-truncated Poisson distribution". r-help (Mailing list). Retrieved 27 May 2022.