Discrete Fit Distributions

This section contains statistical details for the options in the Discrete Fit menu.

Poisson

The Poisson distribution has a single scale parameter λ > 0.

pmf:

for

; x = 0,1,2,...

E(x) = λ

Var(x) = λ

Since the Poisson distribution is a discrete distribution, the overlaid curve is a step function, with jumps that occur at every integer.

Gamma Poisson

This distribution is useful when the data is a combination of several Poisson(μ) distributions and each Poisson(μ) distribution has a different μ. One example is the overall number of accidents combined from multiple intersections, when the mean number of accidents (μ) varies between the intersections.

The Gamma Poisson distribution results from assuming that x|μ follows a Poisson distribution and μ follows a Gamma(α,τ). The Gamma Poisson has parameters λ = ατ and σ = τ+1. The parameter σ is a dispersion parameter. If σ > 1, there is over dispersion, meaning there is more variation in x than explained by the Poisson alone. If σ = 1, x reduces to Poisson(λ).

pmf:

for

;

; x = 0,1,2,...

E(x) = λ

Var(x) = λσ

where

is the Gamma function.

Remember that x|μ ~ Poisson(μ), while μ~ Gamma(α,τ). The platform estimates λ = ατ and σ = τ+1. To obtain estimates for α and τ, use the following formulas:

If the estimate of σ is 1, the formulas do not work. In that case, the Gamma Poisson has reduced to the Poisson(λ), and

is the estimate of λ.

If the estimate for α is an integer, the Gamma Poisson is equivalent to a Negative Binomial with the following pmf:

for

with r = α and (1-p)/p = τ.

Run demoGammaPoisson.jsl in the JMP Samples/Scripts folder to compare a Gamma Poisson distribution with parameters λ and σ to a Poisson distribution with parameter λ.

Binomial

The Binomial option accepts data in two formats: a constant sample size, or a column containing sample sizes.

pmf:

for

; x = 0,1,2,...,n

E(x) = np

Var(x) = np(1-p)

where n is the number of independent trials.

Note: The confidence interval for the binomial parameter is a Score interval. See Agresti and Coull (1998).

Beta Binomial

This distribution is useful when the data is a combination of several Binomial(p) distributions and each Binomial(p) distribution has a different p. One example is the overall number of defects combined from multiple manufacturing lines, when the mean number of defects (p) varies between the lines.

The Beta Binomial distribution results from assuming that x|π follows a Binomial(n,π) distribution and π follows a Beta(α,β). The Beta Binomial has parameters p = α/(α+β) and δ = 1/(α+β+1). The parameter δ is a dispersion parameter. When δ > 0, there is over dispersion, meaning there is more variation in x than explained by the Binomial alone. When δ < 0, there is under dispersion. When δ = 0, x is distributed as Binomial(n,p). The Beta Binomial exists only when

pmf:

for

;

; x = 0,1,2,...,n

E(x) = np

Var(x) = np(1-p)[1+(n-1)δ]

where

is the Gamma function.

Remember that x|π ~ Binomial(n,π), while π ~ Beta(α,β). The parameters p = α/(α+β) and δ = 1/(α+β+1) are estimated by the platform. To obtain estimates of α and β, use the following formulas:

If the estimate of δ is 0, the formulas do not work. In that case, the Beta Binomial has reduced to the Binomial(n,p), and

is the estimate of p.

The confidence intervals for the Beta Binomial parameters are profile likelihood intervals.

Run demoBetaBinomial.jsl in the JMP Samples/Scripts folder to compare a Beta Binomial distribution with dispersion parameter δ to a Binomial distribution with parameters p and n = 20.