Zero-Modified Distribution

< List of probability distributions > Zero-Modified Distribution

What is a zero-modified distribution?

A zero-modified distribution is a probability distribution that starts at zero. More specifically, it is a distribution modified to put extra probability mass at zero. In other words, the probability mass function (PMF) is weighted so that it produces a heavy tail on the left side, which means that the probability of observing a zero is higher than it would be in a regular distribution. Thus, zero-modified distributions are commonly used to model data with a high occurrence of zero counts.

Any distribution of this type has the “zero-modified” prefix added to the regular distribution’s name. For example, the lognormal distribution becomes the Zero modified lognormal distribution (also called the delta distribution) and the normal distribution becomes the Zero modified normal distribution.

A similar distribution is the zero-truncated distribution. The difference is that the the probability of zero occurring in a zero-truncated distribution is 0:

• Zero-modified distribution: p(0) > 0
• Zero-truncated distribution p(0) = 0.

All zero-truncated distributions are members of the zero-modified family.

Zero-modified counting distribution

Counting distributions — such as the binomial, negative binomial and Poisson distributions — are discrete distributions that have a domain of only non-negative integers {1, 2, 3, …). These types of distribution are often used to model the number of event occurrences such as the number of hurricanes in a calendar year.

Counting distributions cannot be used to model events that include zero. For example, if zero hurricanes happen in a year, the Poisson distribution — without modification — is not suitable. That’s because probabilities are always greater than 0 — never equal to zero. Thus, one option is to modify the distribution to include zero in the domain, giving a mixture model of a standard Poisson distribution and a degenerate distribution at zero.

For example, the pmf of a zero-modified Poisson distribution is p(0) = p₀ and:

Where f(x), the Poisson distribution pmf, is modified by a constant. The constant assigns a large value to p(0) and modifies the other values of p(x).

Zero-Modified Distribution & (a, b, 1) distributions

In actuarial science, zero-modified counting distributions are called class (a, b, 1) or the zero-modified distribution of the base (a,b,0) distribution [3].

A member of the (a, b, 0) class has two parameters, a and b, in the recursive relation (a recursive relation defines a sequence of numbers in terms of previous terms in the sequence):

P_k / P_k-1 = a + b/k            k = 1, 2, 3, …

Where:

• P(k): Probability of k occurrences.
• a and b:Parameters specific to the distribution.
• P_k-1 : Probability of (k-1) occurrences.

The initial probability P₀ is fixed (the sum of all P_k must equal 1).

Similarly, a counting distribution belongs to the (a, b, 1) class of distributions if the following recursive relation, which starts at k = 2 , is satisfied by certain constants a and b:

P_k / P_k-1 = a + b/k            k = 2, 3, 4, …

The initial probability P₀ is an assumed value, which means that is not directly observed from data. Instead, it is assumed based on prior knowledge or beliefs about the data. Once P₀ is assumed, other probabilities can be calculated.

Like most probability distributions the sum of all probabilities must equal 1. For example, if P₀ is assumed to be 0.1, then the sum of the other probabilities in the distribution must be equal to 0.9. Given that we are assuming P₀, then the probability P₁ is the value such that the sum P₁ + P₂+,… = 1 – P₀. This means that P₀ is a third parameter for the (a, b, 1) class of distributions.

The three members of the (a, b, 0) class are the binomial distribution, negative binomial distribution, and Poisson distributions; The (a, b, 1) distribution contains the zero-truncated and zero-modified version of one of these distributions.

The (a, b, 1) distribution shares the same name as the (a, b, 0) distribution, but with the prefixes “zero-truncated” or “zero-modified.” For instance, starting from the (a, b, 0) Poisson distribution, the derived distributions in the (a, b, 1) class are the zero-truncated Poisson and the zero-modified Poisson. In other words, there are two (a, b, 1) distribution subclasses with three modifications of the (a, b, 0) distributions in each subclass:

• P₀ = 0: zero-truncated distribution (includes zero-truncated binomial, zero-truncated negative binomial, zero-truncated Poisson)
• P₀ > 0: zero-modified distribution ((includes zero-modified binomial, zero-modified negative binomial, zero-modified Poisson)

The (a, b, 1) class also contains distributions that are not modifications of the (a, b, 0) distributions: logarithmic distribution, Sibuya distribution and the extended truncated negative binomial (ETNB) distribution.

The ENTB distribution is a generalization of the negative binomial distribution — a discrete probability distribution used to model the number of successes in a sequence of Bernoulli trials before a fixed number of failures has occurred. The ETNB distribution has pmf [4]

 

References

1. R-forge distributions Core Team. Handbook on Probability Distributions.
2. R Documentation. The Zero-Modified Poisson Distribution. Retrieved March 5, 2023 from: https://search.r-project.org/CRAN/refmans/actuar/html/ZeroModifiedPoisson.html
3. Discrete Distributions Chapter 6.
4. Zero-modified distribution. Retrieved August 19, 2024 from: https://actuarialmodelingtopics.wordpress.com/tag/zero-modified-distribution/