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Zero inflated model

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In statistics, a zero-inflated model is a statistical model based on a zero-inflated probability distribution, i.e. a distribution that allows for frequent zero-valued observations.

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Zero-inflated Poisson

The first zero-inflated model is the zero-inflated Poisson model, which concerns a random event containing excess zero-count data in unit time. For example, the number of insurance claims within a population for a certain type of risk would be zero-inflated by those people who have not taken out insurance against the risk and thus are unable to claim. The zero-inflated Poisson (ZIP) model employs two components that correspond to two zero generating processes. The first process is governed by a binary distribution that generates structural zeros. The second process is governed by a Poisson distribution that generates counts, some of which may be zero. The two model components are described as follows:

Pr ( y j = 0 ) = σ + ( 1 − σ ) e − λ Pr ( y j = h i ) = ( 1 − σ ) λ h i e − λ h i ! , h i ≥ 1

where the outcome variable y j has any non-negative integer value, λ i is the expected Poisson count for the i th individual; σ is the probability of extra zeros.

The mean is ( 1 − σ ) λ and the variance is λ ( 1 − σ ) ( 1 + λ σ ) .

Estimators of ZIP

The method of moments estimators are given by

λ ^ m o = s 2 + m 2 − m m , π ^ m o = s 2 − m s 2 + m 2 − m ,

where m is the sample mean and s 2 is the sample variance.

The maximum likelihood estimator can be found by solving the following equation

x ¯ ( 1 − e − λ ^ m l ) = λ ^ m l ( 1 − n 0 n ) .

where x ¯ is the sample mean, and n 0 n is the observed proportion of zeros.

This can be solved by iteration, and the maximum likelihood estimator for π is given by

π ^ m l = 1 − x ¯ λ ^ m l .

1994, Greene considered the zero-inflated negative binomial (ZINB) model. Daniel B. Hall adapted Lambert's methodology to an upper-bounded count situation, thereby obtaining a zero-inflated binomial (ZIB) model.

Discrete pseudo compound Poisson model

If the count data Y with the feature that the probability of zero is larger than the probability of nonzero, namely

Pr ( Y = 0 ) > 0.5

then the discrete data Y obey discrete pseudo compound Poisson distribution.

In fact, let G ( z ) = ∑ n = 0 ∞ P ( Y = n ) z n be the probability generating function of y i . If p 0 = Pr ( Y = 0 ) > 0.5 , then | G ( z ) | ⩾ p 0 − ∑ i = 1 ∞ p i = 2 p 0 − 1 > 0 . Then from Wiener–Lévy theorem, we show that G ( z ) have the probability generating function of discrete pseudo compound Poisson distribution.

We say that the discrete random variable Y satisfying probability generating function characterization

G Y ( z ) = ∑ n = 0 ∞ P ( Y = n ) z n = exp ⁡ ( ∑ k = 1 ∞ α k λ ( z k − 1 ) ) , ( | z | ≤ 1 )

has a discrete pseudo compound Poisson distribution with parameters

( λ 1 , λ 2 , … ) = ( α 1 λ , α 2 λ , … ) ∈ R ∞ ( ∑ k = 1 ∞ α k = 1 , ∑ k = 1 ∞ | α k | < ∞ , α k ∈ R , λ > 0 ) .

When all the α k are non-negative, it is the discrete compound Poisson distribution (non-Poisson case) with overdispersion property.

References

Zero-inflated model Wikipedia
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